Status Registry of Results
Status marker system
Each UHM result carries one of seven statuses:
- [T] Theorem — strictly proven
- [C] Conditional theorem — proven under an explicitly stated assumption
- [H] Hypothesis — mathematically formulated, requires proof
- [P] Postulate — accepted without proof as a fundamental assumption
- [Pr] Research programme — a named open direction, neither assumed nor claimed proven (introduced 2026-09-10: pages had been writing "[P] Program" while the registry read [P] as "Postulate" — one letter, two meanings)
- [D] Definition — definition by convention (assigned, not derived)
- [I] Interpretation — philosophical/semantic statement
- [✗] Retracted — proven erroneous or withdrawn
Two axes, one pair of letters. A result carries a mathematical status ([T], [C], [D], [P] — or [H] while unproven) and, separately, a physical-identification status ([I] where the correspondence to observed entities is interpretive; absent where none is claimed). They are shown as a pair where both apply: is [T] as a count of and [I] as an identification with fermion generations. Mixing the axes into a single letter is what produced the contradictory rows corrected in the errata below.
Inheritance rule (R6). A result may not be stronger than its weakest dependency: a [T] resting on a [C], [D], [P] or [I] inherits that status and must say so at the point of use ([C at T-119], [T at the T-128 definition]). Checked by scripts/check_status_consistency.py.
Naming rule for [C]. [C] means "conditional theorem" and nothing else: the assumption must be recoverable at the canonical row here, and inline uses should name it ([C at 2-loop RG], [C under Gap-potential topology]). A statement that merely follows logically from theorems is [T], not [C]. Until 2026-09-10 the octonionic-derivation page carried a second, incompatible legend ("[C] — Consequence: logically follows from [T]"), which made one letter mean both "weaker than a theorem" and "a theorem's consequence"; that legend is retracted.
Where the assumptions are listed. Every assumption named in a [C at (X)] here — the axioms, the bridge premises (Cl₀) and (P), the principle (MaxΦ), the strict-necessity premise (Σ₆), the free parameters and the identification hypotheses — is listed once, with where it is used, its status, independence models and the discharged ones, on Premises of UHM.
What this registry does not cover
Early numbers (below the registry): T-1.1, T-1a, T-2a, T-2f, T-4.2, T-6.1, T-8.1, T-11.1, T-11.2, T-48b — ten variant numbers, 23 citations, no rows.
This paragraph has twice said something much worse, and both times the fault was
in the checker rather than in the registry. It said "twenty-three early numbers,
T-39 with 129 citations", which was summing T-39 together with T-39a because the
checker reduced every name to an integer. Corrected for that, it then said
"T-1..T-49 — 54 distinct names, 421 citations, no rows" — and that was wrong too,
because the registry writes rows in TWO formats: | T-129a | … and, in the Level
1–6 tables, a bare number in the # column (| 40b | R_th = 1/3 [T] …). The
second format carries 178 rows, and a checker that knew only the first reported
them as missing. Counted correctly, the registry holds 412 rows spanning
T-1..T-352 (eight numbers between 336 and 345 have no row yet), and what actually lacks rows is the ten variant numbers named
above. Their statements are not recoverable in a form fit
for this table: no complete source exists, and the numbering of
math-foundations is a different one (its T-42 is «G₂ ⊂ SO(7) ⊂ SU(7)», while
the corpus links T-42 to the Uniqueness Theorem of Holonomic Representation).
Inventing rows would counterfeit exactly the completeness that is missing, so
the gap is named instead. Reader's practical rule: a T-n with n < 50
resolves through the page it links to, not through this table.
One number, two statements (open debt). The registry writes rows in several tables, and their numbering is continuous: Level 1 runs 1..48 with lettered variants, Level 2 continues 39..51, Level 3 continues 53..90, Level 5 continues 81..85, Level 6 continues 86..92. Because the ranges overlap, 62 numbers carry two live rows with different statements.
Two kinds must not be conflated, and the first edition of this paragraph conflated them.
28 of the 62 pair a substantive row with a companion row — a table listing the external
support of the same result (Goderis–Verbeure–Vets 1989 under T-117, Connes 2013 reconstruction under T-119) or the Universal-Property summary. There the number is not
ambiguous; it is merely named twice. The remaining — genuine collisions: 34 — pair two
substantive tables: 18 across Level 1 and Level 3, 7 across Level 1 and Level 6, 5 across
Level 1 and Level 5, 2 across Level 1 and Level 2 (the third, T-52, was resolved on 2026-09-25: the sector asymmetry of Level 1 was retired as a theorem and the Level-2 row «Anomalous dimension of the Fano operator» was renumbered T-330, so T-52 now names only the struck row). There a citation is ambiguous: the reader
cannot tell which of the two rows was meant. Struck-through rows and the Retracted section are
not counted — there the reuse of a number is declared.
The number is measured by scripts/check_theorem_refs.py on every run and must match the one
written here; the checker deliberately does not decide which row is right, because renumbering
is the author's call and propagates into every citation. A worked example of the cost: critical
purity appears as row 5 of Level 1, is called T-39a in a code comment
on its own proof page, and appears as T-39 in a format example inside a fenced block. Three
numbers for one theorem, none of them wrong locally.
Unstated supports (open debt): T-4.2. This number is used BY THIS REGISTRY as a
load-bearing step — a status raised "to [C at T-4.2]" — and no statement carrying that label
exists anywhere in the corpus. This is a different gap from the early variants: those are
merely cited, whereas this one is leaned on. check_theorem_refs.py reads this line and turns
red the moment a second such number appears.
An earlier version of this line named six numbers — T-4.2, T-11, T-12, T-13, T-40a, T-41e.
Five of them do have rows, in the #-column format the checker could not yet read. Only
T-4.2 stands.
Skipped numbers (never assigned): T-167, T-168, T-169. They carry no row and
are cited nowhere; the numbering simply skips them. This line is machine-read by
check_theorem_refs.py, so that the declaration of a gap is not itself counted as
a citation of one.
Machine: scripts/check_theorem_refs.py — every T-n reference must resolve to
a row here, and the check fails while these do not.
No canonical status field — the measured debt (2026-09-10)
An external review named the mechanism, not the individual rows, as the root defect: the corpus
has no machine-readable canonical record of a result's status, so propagation is done by hand
and drifts. Measured by scripts/check_status_consistency.py over the 388 numbered rows:
| Evidence of status in the registry | Count | Meaning |
|---|---|---|
| machine-readable — the status is recoverable from the row or from its section heading | 358 | citations are checked against it automatically (rule R1) |
| companion — a second row in a table carrying no status of its own (external support, dependency graph) | 17 | the number is named twice but is not ambiguous — not debt |
| ambiguous — two substantive rows whose statuses differ | 30 | the "one number, two statements" debt named above |
| implied (status only in prose) | 0 | closed 2026-09-10 |
| absent (no status anywhere) | 0 | closed 2026-09-10 |
Debt = 30 of 388 — the renumbering debt, and nothing else. All thirty lie in the band T-74…T-114,
where the # columns of Level 1, Level 3, Level 4 (retracted) and Level 5/6 overlap: by shape they are
10 × (theorem / hypothesis), 7 × (theorem / interpretation), 8 × (theorem / retracted), 4 × (theorem /
postulate), 1 × (conditional / hypothesis). In each case two different results share one number, so a
citation T-n cannot be resolved to a row by the reader.
Closing them means renumbering, which is the author's call and propagates into ~900 citations — so the checker holds the number as a ratchet (it may fall, never rise) rather than guessing. Until then rule R1 checks each citation against the union of the statuses of a number's live rows: that catches a wrong status without forcing a choice between the rows.
The first measurement of this debt, on the morning of 2026-09-10, read 224. That number was wrong:
the instrument did not yet read statuses carried by a section heading ("## Level 1: Impeccably Strict
Theorems [T]") or written as [T at …] / [Т/С]. Corrected, the corpus was in far better shape than
its own checker claimed — and the episode is itself the argument for the review's proposal: one
canonical record per id (statement, status, dependencies), with the landing page, the introduction and
this registry generated from it rather than copied.
Errata 2026-09-10 — external review and consistency pass
An external review (September 2026) and the consistency pass that followed changed the statuses and readings below; each affected row carries the change in place.
- Frame decision D-0910 (uniqueness theorem): the pinching dynamics breaks to the finite frame group ; counts kinematic -invariants, all 48 parameters of are physical; , , are frame-pinned observables (rows 42a, 42b, 2b; measurement protocol; falsifiability). The statement "the Fano dissipator is -covariant" is retracted (Theorem 5.1b).
- T-58 (Morita 7D↔42D): [T] → [✗] (second pass — fails on dimension), replaced by T-58′ [T] (section–retraction). : [T] → retracted (2026-09-10, second pass): the vanishing is in positive degree only, a vacuum total is degree-0 data; what remains is the prohibition of a topological -term [T]. Row 117: needs the cut-off convention and [D].
- T-129: the theorem is the inequality ; as the least such threshold is [D]. T-87: wording corrected (clock register , not a tensor decomposition read off the algebra). T-174: class note (the NCG Standard Model is not in the class; since the restatement of 2026-09-26, which reverses the direction — corepresents -structures — the note reads: contains no copy of , so it receives no morphism from ). T-182(c): monism is a corollary of Property 3.
- Critical purity: Path 2 (relative entropy) is a convention; at the threshold the exact is nat on the extremal spectrum. Emergent time: the arrow is indexed by the stratal depth , not by the cyclic tick ; the false sentence about irreducible representations of is retracted; the cyclic order of the atoms is a named choice [D]. Cohomological monism: locally constant coefficients only.
Foundational closures (T-210..T-223)
Fourteen theorems address the mathematical and categorical gaps of the UHM framework — eleven of them stand as [T] (T-212 in the corrected form T-212′, the -twirl; T-211 in the corrected form of 2026-09-25, the Grothendieck construction), T-221 is stratified, T-216 is [C] and T-219 is [H] (the box said "close all … gaps" until 2026-09-25; that is retracted, since the [C] rows and the reconstruction axioms of T-119 stay open; the input (Alt) of T15, also listed here that day, is discharged by the canonical-orientation theorem): strict Φ-monotonicity, PhysTheory higher coherences, rheonomy modality, Bures-Yoneda, hard-problem meta-theorem, cross-layer identity, analytical εeff, L3 tricategorical coherence, SYNARC Cog as Kan complex, sector-product Λ-suppression, no-reduction → UHM, UHM's route through the List/DeBrota no-go results (relationalist; corrected 2026-09-25), the resource geometry of the viable window (restated 2026-09-26: no single MRQT optimum; the former "MRQT-completeness" is [✗]), and Putnam-triviality foreclosure (Lerchner Melody-Paradox closure). Full proofs in Fundamental Closures T-210..T-223. Plus two computational-programme specifications (Λ-deficit and πbio) reducing remaining open questions to bounded empirical/computational tasks.
Theorem Correspondence Matrix (T-193..T-223 provenance)
The block T-193..T-223 aggregates results from multiple sources:
| T-number | Origin | Status | Relates to |
|---|---|---|---|
| T-193 | SYNARC paper App. G.2 | [T]; upgraded to computable form by T-213 | Original Yoneda (Kolmogorov) |
| T-194 | SYNARC paper App. G.3 | [T] | Learning-efficiency closure |
| T-195 | SYNARC paper App. G.4 | [T] weak; upgraded to strict by T-210 | Φ-monotonicity |
| T-196 | SYNARC paper App. G.5 | [T] | Sustainability |
| T-197 | SYNARC paper App. G.6 (S-11) | [T]+[D]; consistency of SYNARC architecture | Conditional on SYNARC definition |
| T-198–T-202 | SYNARC paper App. H.1–H.5 | [T] | ASI extensions |
| T-203 | SYNARC paper App. H.6 | [T]+[I] stratified | Ontological postulate required |
| T-204 | SYNARC paper App. H.7 | [T] | Resource-bounded |
| T-205 | SYNARC paper App. H.8 | [C]+[D]; conditional on | Reconciled by T-215 |
| T-206–T-208 | SYNARC paper App. I.1–I.3 | [T] | Operational protocols |
| T-209 | SYNARC paper App. I.4 (S-13) | [T]+[D] | Operational-closure meta-theorem — [D] at operational-protocol specification choices |
| T-210 | UHM Fundamental Closures §1 (new) | [T] strict | Upgrades T-195 on interior states |
| T-211 | UHM Fundamental Closures §2 (new) | [T] (corrected 2026-09-25: PhysTheory as a Grothendieck construction; [C at T-119] from 2026-09-11, the full embedding into [✗]) | Supplies the -structure of T-174's PhysTheory (HTT 3.2) |
| T-212 | UHM Fundamental Closures §3 (new) | [T] for T-212′, the -twirl ; the identification with the rheonomy modality Rh [✗] (it was [C at the differential cohesion of the UHM site (T-185) and a solid-cohesive extension], corrected from [T] on 2026-09-25) | Explicit U-projection; not a modality |
| T-213 | UHM Fundamental Closures §4 (new) | [T] computable | Upgrades T-193; removes Kolmogorov |
| T-214 | UHM Fundamental Closures §5 (new) | [T] positive meta-theorem | Completes T-188 |
| T-215 | UHM Fundamental Closures §6 (new) | [T]+[D] | Resolves T-205 tension with SAD_MAX=3 |
| T-216 | UHM Fundamental Closures §7 (new) | [C at (SV)] — the structure corrected from [T] on 2026-09-25 (its sectoral reduction used T-48a, retracted) | Upgrades T-176 to closed form; |
| T-217 | UHM Fundamental Closures §11 (new 2026-04-17) | [T] | L3 tricategorical coherence via τ_{≤3}(Exp_∞) + Baez–Dolan; upgrades T-67 K=4 count to [T] |
| T-218 | UHM Fundamental Closures §12 (new 2026-04-17) | [T] | SYNARC Cog = Sing(B·𝒞_FKraus) is Kan complex (Milnor); explicit horn-filler algorithm |
| T-219 | UHM Fundamental Closures §13 (new 2026-04-17) | [H] — corrected from [T at T-64] on 2026-09-25 | Λ SUSY-suppression from 3-sector decomposition (T-48a, retracted in its axis-labelled form), replacing invalid G₂-adjoint argument; the law is a hypothesis — the one-loop sum of its own proof is larger unless lower orders cancel, which is not shown |
| T-220 | UHM Fundamental Closures §14 (new 2026-04-17) | [T] negative | No reduction functor -UHM → -UHM exists: 5 independent obstructions (rep-theory , -transitivity on , Zelmanov exceptionality, numerical mismatch , Euler (ℂP⁶)=7≠3=χ(𝕆P²)) |
| T-221 | UHM Fundamental Closures §15 (new 2026-04-17; corrected 2026-09-25) | [T] for (a)–(e) + [I] for the reading | UHM realises the relationalist route of DeBrota–List (2026): the first-personal facts of two subjects are not compossible in (List's lemma holds internally, Kripke–Joyal forcing); UHM keeps OW, NF, NS (ιmin) and FPR only in stage-relativised form; the relativisation parameter is internal (answers Fine's objection), the "which one am I" question is not answered internally (the choice of a point is external, cf. T-214); the three routes are readings of one forcing relation and share every observable, so no measurement discriminates them. In List's (2025) four-claim map this is the first horn. Retracted [✗] 2026-09-25: the "fourth, categorical-monistic route", "FPR is forced", "RQM = τ≤1(𝔗)", fragmentalism as "dropping descent", πbio as a route discriminator, and the misquotation of the quadrilemma as five theses |
| T-222 | UHM Fundamental Closures §16 (new 2026-04-18; restated 2026-09-26) | [T] | Resource geometry of the viable window: the purity window (the orbit-invariant conditions of ) at high temperature, . (i) Every state of is strictly dominated on every , , by ; (ii) on the closure the Pareto set is non-empty and lies on ; (iii) no state minimises and together — has the unique minimising spectrum of (, ), the three-level has , ; (iv) with unital channels as free operations there is no terminal object (Alberti–Uhlmann; and have no common lower bound in the window); (v) the fixed point of is strictly dominated by ; (vi) , equality on uniform-diagonal states; the -twirled charges vanish. Errata 2026-09-26: the former statement is corrected from [T] to [✗] — "H-MRQT-Lawvere: the Lawvere fixed point of T-96 is the Pareto optimum of the 25-monotone MRQT vector, all spectral monotones optimised simultaneously, terminal in , UHM MRQT-complete": T-96 proves at a nontrivial stationary state, is the regeneration target and the fixed points belong to (Theorem 10.1 of Gap thermodynamics), not to ; simultaneity, the optimum inside and terminality fail by (iii), (i), (iv); lemmas L3, L4, L6 were false. Routes tried: a fixed point of a self-model as the optimum (v), a boundary point (iii), reachability instead of terminality (iv), the sub-family (splits at ) |
| T-223 | UHM Fundamental Closures §17 (new 2026-04-18) | [T] | Putnam-triviality foreclosure (Lerchner Melody-Paradox closure). Let satisfy (AP)+(PH)+(QG)+(V). (a) is well-defined and is invariant under UHM-compatible alphabetizer choice. (b) are -invariants descending to ; the frame observables are alphabetization-invariant because admissible alphabetizers preserve the dynamical frame (corrected 2026-07 — they are frame-relative, not orbit-invariants). (c) Consciousness predicate is alphabetization-invariant: its terms factor through , its terms are fixed by the dynamical frame. The predicate as a whole does not factor through (corrected 2026-09-25: an explicit sends from to , so the conjunct is not constant on -orbits — frame rigidity, uniqueness theorem). (d) Non-UHM-compatible alphabetizers (Lerchner Fig. 3 "Market Data" on Beethoven trajectory) are physically vacuous. (e) The only residual externality is the phenomenal bridge , Lawvere-inevitable by T-214. Three-level ontology L1 (physical) / L2 (categorical intrinsic , forced through the Bridge T15 with the canonical orientation; the earlier "forced by T-190 zero-axiom closure" is withdrawn, T-190 being conditional) / L3 (symbolic, Lerchner-variable): Putnam triviality applies to L1→L3 but not to L1→L2. Proof via seven lemmas (L1: categorical necessity of ; L2: covariance gate; L3: -uniqueness via T-123; L4: alphabetization-invariance of observables — are -invariant, frame-pinned; L5: admissible alphabetizers factor through ; L6: non-dynamical are physically vacuous à la Piccinini-Searle-Kim; L7: self-alphabetization via operator of T-96/T-98, categorifying the Maturana-Varela enactivist subject). Responds to Putnam 1988 / Sprevak 2018 / Piccinini 2008 / Lerchner 2026 "The Abstraction Fallacy" |
Cross-framework relation. UHM theory and SYNARC AGI architecture are linked but independent (UHM = foundational theory; SYNARC = UHM-inspired cognitive architecture). Mathesis is a separate, standalone project for theory-navigation meta-epistemics — it operates on theories (including UHM) as objects in ; it does not compose with SYNARC.
Load-bearing UHM theorems for SYNARC: T-142 (SAD_MAX=3), T-174 (corepresentation of -structures by in the fibre of PhysTheory over the point — restated 2026-09-26; the former "essentially unique receiving morphism from every theory with " is [✗]), T-124 (Goldilocks ceiling), T-129 (Φ_th=1), T-151 (=2), T-187 (Bures canonicity), T-38a (No-Zombie). Changes in any of these impact SYNARC downstream.
The sustainable window is narrower than the theoretical one
The viability window is where the four criteria can hold. It is not where a running loop can settle. Measured in the regeneration dynamics there is a sharp critical target below which the steady state runs away to grey, and the purity actually held at that edge is .
Two natural explanations were tested and both refuted: it is not a leak balance — the floor moves only across a hundredfold change in — and not a basin effect, since starts at , and give identical outcomes. What survives is the gate's own feedback: the steady state settles below its target, a lower lowers , weaker regeneration lowers further, and below the critical target the loop diverges.
So the dynamically sustainable window is , not
: the lower fifth of the theoretical window holds no steady states
at all, and «an engineer can run leaner toward » fails not on cost but on
existence [Т by construction].
Capability is bounded, and the bound is the flat diagonal. Writing one has , hence . Cauchy–Schwarz gives with equality iff the diagonal is uniform, so
Verified to twelve digits at every purity tested. So is the supremum, attained exactly by uniform-diagonal states. It is the same inequality that gives T-323 its floor — so the point at which a purity regulator runs out of diagonal to level is also the point of greatest capability, and the two results are one inequality read twice. An attractor whose self-model weights the voices unevenly (E 1.0, O 0.8, U 0.6, A 0.4) has a non-uniform diagonal and therefore sits strictly below the bound, at .
Cost and capability peak in different places. Maintenance cost rises monotonically with a richer target (), and the capability maximum (, ) sits at a different point from the capability-per-watt maximum (, ). This is an axis to be chosen along, not a set-point to be hit.
The entropy floor is frequency-independent: the entropy-production rate
varies by across a fourfold change of tick step. Whether
order-per-cost improves as rises is neither confirmed nor refuted
[D] — it comes out flat ( across a fivefold rise), but the
surrogate measured is order per dissipated entropy rather than the efficiency
the statement is about, and a real test needs the regeneration work, which
the tick does not carry.
Two structural checks that hold outright. The -invariant subspace of is exactly one-dimensional (deviation from the uniform vector), and is attained only at — unbeaten across 400 000 random distributions. The Jordan identity holds on at (residuals ) and breaks at (): the Jordan–von Neumann–Wigner ceiling reproduced from the corpus's own oriented Fano wiring, which incidentally re-validates that wiring.
Level 1: Impeccably Strict Theorems [T]
Results with fully verified proofs.
| # | Result | Source | Target page |
|---|---|---|---|
| 1 | Fano channel preserves coherences | Lindblad Operators T.10.1–10.3 | Fano Channel |
| 2 | Fano–atomic proportionality ; both pinching dissipators covariant under every monomial unitary (inside : exactly the finite frame group — the signed permutations, order , acting on the lines through ), not full ; canonical -covariant dissipator (structure-constant ) [T] (corrected 2026-07; corrected 2026-09-25: the row read — the image of the group on the lines, not the group) | Lindblad Operators T.5.1a–c | Fano Channel |
| 3 | Atomic dissipator is NOT G₂-covariant | Lindblad Operators T.11.1 | Fano Channel |
| 4 | Gap operator: properties (a)–(d), antisymmetry, | Lindblad Operators T.8.1–8.2 | Gap Operator |
| 5 | Necessity of generalised φ, | Lindblad Operators T.1.2 | Viability |
| 6 | Equilibrium Gap | Composite Systems T.3.1 | Gap Semantics |
| 7 | L4 ≠ Gap = 0 | Composite Systems T.4.1 | Interiority Hierarchy |
| 8 | Uniqueness of the triplet (1,2,4) | Standard Model T.1.3 | Fermion Generations |
| 9 | Uniqueness of the Higgs line {A,E,U} | Higgs Sector T.2.1 | Higgs Sector |
| 9a | Identification (Theorem 1.0) — Errata 2026-09-25 (audit A-90): corrected from [T] to [H]: breaks (the stabiliser in has dimension at most 1 of 8, test_gamma_eu_vev_breaks_colour); no commutes with on (commutant ), so the doublet of step 3 has no carrier; step 4 used T-64, restated as a hypothesis. The repair through colour keeps colour-neutral only together with equal and coherences and moves colour off . Earlier text: κ₀-uniqueness of + Fano line + quantum numbers + from T-64 → EWSB from axioms | Higgs Sector T.1.0 | Higgs Sector, Standard Model |
| 10 | GeV (Pendleton–Ross IR fixed point) | Higgs Sector T.5.1 | Yukawa Hierarchy |
| Falsifiability T.3.2 | CKM Matrix | ||
| 12 | RG suppression : | Quantum Gravity T.12.2 | Λ Budget |
| 13 | Factor from Ward identities (previously [✗]) | Cosmological Constant T.10.3 | Λ Budget |
| 14 | pc | Confinement T.9.1–9.2 | Cosmological Constant |
| 15 | ABJ anomaly from Cliff(7) | Confinement T.11.2 | Standard Model |
| 16 | Instanton is additive, GeV⁴ | Falsifiability T.8.2 | Λ Budget |
| 17 | CS on 1D — total derivative | Berry Phase T.2.1 | Berry Phase |
| 18 | All , | Zeta Regularisation T.1.1 | Zeta Regularisation |
| 19 | at | Zeta Regularisation §4 | Zeta Regularisation |
| 20 | unique up to scalar | Zeta Regularisation §§5–6 | Zeta Regularisation |
| 21 | for | Zeta Regularisation §9 | Zeta Regularisation |
| 22 | Perturbative budget (6 mechanisms) | Falsifiability §9.3 | Λ Budget |
| 23 | Spectrum of Gap operator: , opacity rank | Lindblad Operators T.3.1 | Gap Operator |
| 24 | G₂/⊥-decomposition of Gap operator: (14+7) | Lindblad Operators T.6.1 | Gap Operator |
| 25 | Classification of stabilisers by rank, (weight lattice of rank 2; simply connected so ) | Lindblad Operators T.8.1 | Gap Operator |
| 26 | Gap phase diagram: three phases (ordered, disordered, dead zone) | Lindblad Operators T.2.1 | Phase Diagram |
| 27 | Critical exponents: , , (Landau class) | Lindblad Operators T.7.1 | Phase Diagram |
| 28 | Swallowtail cascade and correspondence to L-levels L0–L4 — raised from [C]: -bifurcation proven via Arnold's theorem (codimension 3, -purity symmetry) | Interiority Hierarchy | Phase Diagram |
| 28b | Gap injection of L-levels: . Injection, not bijection — Gap profile is a finer invariant | Interiority Hierarchy | Gap Characterisation |
| 29 | Whitney catastrophes for Gap: fold, cusp, bifurcations | Lindblad Operators T.5.1 | Phase Diagram |
| 30 | One-loop β-functions of Gap theory (factors 21, 7, 15) | Quantum Gravity T.2.1 | Renormalisation Group |
| 31 | Two-loop β-functions (factors 441, 147, 49) | Renormalisation Group T.4.1 | Renormalisation Group |
| 32 | Three-loop stability of the octonionic fixed point: | Cosmological Constant T.5.1 | Renormalisation Group |
| 33 | Conformal window of Gap theory: ; at — outside the conformal window | Cosmological Constant T.6.1 | Renormalisation Group |
| 34 | c-theorem for Gap: monotone decrease of in the IR direction | Cosmological Constant T.7.1 | Renormalisation Group |
| 35 | CPTP verification of Fano channel: | Lindblad Operators T.10.1 | Fano Channel |
| 36 | Canonical form (CPTP convex combination of the atomic and Fano channels); the variational definition of is retracted [✗] 2026-09-25 — along the functional of row 39e is affine in with non-negative slope and minimal at , so the Fano weight is a free parameter | Lindblad Operators T.3.1–4.1 | Fano Channel |
| 37 | Gap functional integral defined on (compactness, finite DOF) | Quantum Gravity T.2.1 | Quantum Gravity |
| 38a | Necessity of interiority (No-Zombie): . Epistemic stratification (Sol.SA-3): [T] mathematical core (, ); [P] ontological postulate (E = interiority); [I] No-Zombie interpretation | CC Theorems T.8.1 | CC Theorems |
| 38b | Emergent time (Page–Wootters): derived from the structure of via three paths (conditional states, Bures, ∞-groupoid) | Emergent Time | Emergent Time |
| 39a | Primitivity of the linear part : unique stationary state , convergence from any initial state (Evans–Spohn criterion + connectivity ). The full nonlinear dynamics may have additional fixed points (T-96) | Lindblad Operators | Lindblad Operators |
| 39b | Connectivity of from viability: (AP)+(PH)+(QG)+(V) → interaction graph is connected | Lindblad Operators | Lindblad Operators |
| 39c | Primitivity of the Fano construction: extension to | Lindblad Operators | Lindblad Operators |
| 39d | Equivalence of three definitions of φ (categorical ⇔ dynamical ⇔ idempotent) — raised from [C] | Formalisation of φ | Formalisation of φ |
| 39e | Variational characterisation of φ via free energy (Th.3.1 FEP) — Errata 2026-09-25: corrected from [T] to [✗] (earlier raised from [C]). The functional is linear in ; its minimum is reached by the channel onto the top eigenvector of , not by φ (classically: gives at and at the point mass). Theorems 4.2 (iii)–(iv) and 4.3 of the same page fall with it; the open question is a research programme [Pr] | FEP Derivation | FEP Derivation |
| 39f | Form of ℛ: direction — the unique CPTP relaxation (replacement channel + Bures optimality). Raised from [P] | Evolution | Evolution |
| 39g | Form of ℛ: gate — V-preservation gate, strengthening the Landauer principle (). Raised from [P] | Evolution | Evolution |
| 39h | Full form of ℛ — all components derived: κ(Γ) from conjugation, (ρ*−Γ) from CPTP uniqueness, from Landauer + V-preservation. The evolution equation is fully axiomatic | Evolution | Evolution |
| 39i | Decoherence rate of BIBD: ; Fano and its complement give identical | Evolution | Evolution |
| 40a | Triadic decomposition: axioms A1–A5 generate exactly 3 types of dynamics (Aut, , ℛ). A fourth type is impossible (uniqueness of Ω) | Lindblad Operators | Lindblad Operators |
| 41a | Equivalence of BIBD channels (T1): all -BIBD channels with equal give the same CPTP channel; contraction | Lindblad Operators | Lindblad Operators |
| 41b | Completeness of pair coverage (T2): connectivity of + primitivity of the linear part ⟹ for all pairs | Lindblad Operators | Lindblad Operators |
| 41c | Optimal block size (T4): among admissible BIBD (), strictly dominates by all criteria | Lindblad Operators | Lindblad Operators |
| 41d | -equivariance of the atomic dissipator (T5): for all | Lindblad Operators | Fano Channel |
| 41e | Uniform contraction of coherences (T6): for all — unconditionally, without (CG) | Lindblad Operators | Fano Channel |
| 41f | Autopoietic necessity (T7): the atomic dissipator is incompatible with (AP) via suppression of | Lindblad Operators | Fano Channel |
| 41g | Hamming bound (T8): H(7,4) — the unique perfect single-error-correcting code of length 7, | Lindblad Operators | Fano Channel |
| 41h | Support structure H(7,4) = PG(2,2) (T9): weight-3 codewords of (complements of the weight-4 words of ) = Fano lines; corrected 2026-09-28, the row said weight-3 codewords of , which has none | Lindblad Operators | Fano Channel |
| 41i | Autopoietic optimality of the Fano channel (T10): unique optimal BIBD-channel for , complete coverage, democracy | Lindblad Operators | Fano Channel |
| 41j | Choi rank of channel = 7 (T11): minimum number of Kraus operators = 7, Fano decomposition is rank-minimal | Lindblad Operators | Lindblad Operators |
| 41k | Projective decomposition from L-unification (T12): L-unification + ⟹ rank-3 orthogonal projectors (Lüders coarse-graining) | Lindblad Operators | Lindblad Operators |
| 41l | BIBD from minimal projective decomposition (T13): , contraction ⟹ BIBD = PG(2,2) (Kirkman 1847). Strengthened 2026-09-26 [T]: no rank or weight is assumed — every Kraus representation of by 7 operators proportional to projectors (sharp, minimal) is the line resolution of one of the 30 Fano planes, ranks and weights forced ( ⟹ ); for such a representation of exists only at (symmetric designs), in the family only at ; exactly one plane is invariant under , the octonionic one — the line instrument of is canonical. Each clause is needed (axes + identity: 8 operators; Fourier mixing: outcomes independent of ; 29 other planes); Hamming syndrome checks give or , not (test_sharp_minimal_kraus_representations_are_the_fano_planes) | Lindblad Operators | Lindblad Operators |
| 41m | Max-min optimality of BIBD (T14): among regular , BIBD maximises | Lindblad Operators | Lindblad Operators |
| 41n | Bridge closure (T15): , chain of 15 steps with inline proofs. Downstream (2026-09-25): the -rigidity rows 42a, 42e and T-123 are stratified accordingly. Errata 2026-09-25: corrected from [T] to [C at (Alt)] — steps 1–9 give the unoriented design PG(2,2); step 10, PG(2,2) the multiplication table of , needs an orientation of the seven lines, and only 16 of the 128 orientations make the multiplication alternative (equivalently normed): the named assumption (Alt), checked by test_only_16_of_128_fano_orientations_are_normed. The former condition (MP) follows from T11–T13. Restored to [T] the same day in a precise form — Theorem T15-canon: of the 8 gauge classes of orientations (16 each) exactly one is invariant under the collineation group of the design, and it is the normed class, ; the other seven form one orbit, each fixing a line (its units associate on 96 of 168 ordered non-collinear triples, its 3-form is split, signature ). The orientation determined by the design is therefore octonionic, and (Alt) ⟺ canonicity; the stronger reading "every orientation compatible with T1–T14 gives " is false (112 of 128), test_octonionic_orientation_is_the_unique_collineation_invariant_class | Lindblad Operators, T15-canon | Octonionic Derivation |
| 41o | Internalisation of IDP (T16): IDP is derived from A1+A2 via Kripke–Joyal semantics. Step (3) — tautology from A1 | Axiom of Septicity | Axiom of Septicity |
| 40b | [T]: from triadic decomposition + Bayesian dominance [T] — raised from [C] (C1). Number-theoretic root [T, cited]: , where are the multipliers among the 21 permutation automorphisms of the oriented octonion table (the Frobenius group ); the non-residues carry the Fano lines to the complementary design; is the reciprocal order of that multiplier group. The -independent LGKS triad (T-57) fixes the value ; the orientation root names it. The hosting pin (Foundations of Mathematics, Part XVIII, Thm. 11.6/11.8, Cor. 11.9) singles out only inside Hurwitz's list and only in the Kraus reading — the quaternion table has the free transitive — so it does not bear on the strict necessity of (T-349(d); corrected 2026-09-28: the row said that non-residues reverse the orientation and that hosting singles out among division algebras) | Axiom of Septicity | Lindblad Operators |
| 42a | -rigidity of the holonomic representation: the holonomic representation is unique up to (kinematic envelope; dynamical identification up to the finite frame group — frame decision D-0910). Analogue of the Stone–von Neumann theorem for UHM. Stratified 2026-09-25: [T] as a consequence of the axioms: the multiplication comes from the bridge T15 (row 41n), whose step PG(2,2) → takes the canonical orientation — the unique collineation-invariant orientation class (T15-canon). The intermediate stratification of the same day, "[T] for with its octonionic multiplication; [C at (Alt)] as a consequence of the axioms", is superseded | Uniqueness Theorem | Uniqueness Theorem |
| 42b | Space of kinematic -orbits: , ; physical state space , 48 parameters. Errata 2026-09-10 (frame decision D-0910): the pinching dynamics breaks to the finite frame group, so 34 counts kinematic invariants, not physically distinguishable parameters | Uniqueness Theorem | Uniqueness Theorem |
| 42c | Spectral injectivity of the propagator: is injective on for | Uniqueness Theorem | Uniqueness Theorem |
| 42d | Well-posedness of the nonlinear inverse problem: uniqueness of solutions of the full evolution equation (Picard–Lindelöf on compact ) | Uniqueness Theorem | Uniqueness Theorem |
| 42e | Gauge group = : the maximal subgroup preserving the octonionic 3-form is , which acts on states as (Lemma G4). Corrected 2026-09-25: the row read "preserving all axiomatic structures is " — the scalars , , preserve the form too, and the frame data are preserved only by the finite frame group. Stratified 2026-09-25: [T] as a consequence of the axioms: the multiplication comes from the bridge T15 (row 41n), whose step PG(2,2) → takes the canonical orientation — the unique collineation-invariant orientation class (T15-canon). The intermediate stratification of the same day, "[T] for with its octonionic multiplication; [C at (Alt)] as a consequence of the axioms", is superseded | Uniqueness Theorem | Uniqueness Theorem |
| 43a | Source Instability : non-stationarity (), linear drift to , -violation via — raised from [H] | Origin | Origin |
| 43b | Self-amplification of -symmetry breaking: positive feedback upon deviation from — raised from [P] | Origin | Origin |
| 43c | Three fermion generations () — count [T], identification [I] (strengthened 2026-07): the count is the exact cardinality [T] — the three generations are the quadratic-residue classes of the unique order-3 subgroup , equivalently the charge-conjugation orbits (since is a non-residue mod 7, ); group-theoretic and topology-independent (the earlier -swallowtail upper bound and its "[C under Gap-potential topology]" caveat are retired to a consistency check). Physical identification of these classes with observed generations remains [I] | Fermion Generations | Fermion Generations |
| 43d | Fano selection rule for Yukawa couplings: , where — octonionic structure constants (the unique -invariant trilinear operator on ). , — raised from [H] (No.64) | Fano Selection Rules | Yukawa Hierarchy |
| 40c | Functional uniqueness of E [T]: axiomatic, categorical (κ₀) and mathematical () arguments — raised from [C] | Minimality Theorem | Minimality Theorem |
| 40d | Functional uniqueness of O [T]: from the form of ℛ [T], κ₀ [T], Page–Wootters (A5), functional independence [T] — raised from [C] | Minimality Theorem | Minimality Theorem |
| 40e | Orthogonality E⊥O [T]: causal + categorical (κ₀) arguments; for regeneration loses E-feedback — raised from [C] | Minimality Theorem | Minimality Theorem |
| 40f | Full minimality theorem 7/7 [T]: all 7 dimensions are necessary and functionally unique (A,S,D,L — algebraically; E,O — categorially via κ₀; U — trace properties) | Minimality Theorem | Minimality Theorem |
| 44a | Freedom(Γ) = dim ker(H_Γ) + 1: finite-dimensional definition of free will via the Hessian of the free-energy functional. Monotonicity under CPTP, -invariance, extreme values (Freedom(I/7)=7, Freedom(ρ*)=1). Raised from [P] | Consequences | Free Will |
| 45a | Assignment 3rd generation: uniqueness from Fano selection rule (, all other ) | Fermion Generations | Fermion Generations |
| 45b | Sector asymmetry: (Actualisation), (Nomos); different Fano paths to the Higgs line. Errata 2026-09-25: corrected from [T] to [✗] — no axis lies in the or the : no triple of axes spans an -invariant subspace, and (Theorem 4.2 retracted). What survives is incidence: reaches the Higgs line through , through . What holds instead (2026-09-25, proved and checked): every non- axis has weight exactly in and in (, diagonal ), and carries any axis to any (every orbit in is the whole ), so no colour-invariant quantity distinguishes from ; the only colour-invariant asymmetry is the weight of against (test_every_non_o_axis_is_half_triplet_and_colour_moves_any_axis_to_any) | Fermion Generations | Fermion Generations |
| 48a | Dimensional sector decomposition: from stabilisers . Errata 2026-09-25: corrected from [T] to [✗] — as an axis-labelled real decomposition it is false: acts irreducibly on the real (complex type), no triple of the six non- axes spans an invariant subspace (20 of 20 checked, test_no_axis_triple_is_su3_invariant), and the generation lies inside this (test_generation_z3_lies_in_colour_su3). What holds is the complexified with — classical representation theory (Günaydın and Gürsey 1973), not a UHM result. The goal of the row — three spatial directions and one time direction — is reached by row 48c instead, without any axis split and with rotations that commute with colour | Spacetime | Spacetime |
| 48c | Spacetime that commutes with colour (clock-complex spin factor) — mathematics [T], physical 3+1 [C at (L)] (new 2026-09-25, replaces the goal of 48a; it read [C at (Q)] until Theorem 48e the same day): with , (a) the fixed algebra of in is , and the complex structures on commuting with colour are ; (b) the colour-singlet part of the spin factor is , dimension 4, signature ; (c) the centraliser of in is (dimension 7); (d) acts by as , trivially on the colour triplet, as a direct product with ; (e) one time direction and three space directions with commuting with colour, the Lorentzian sign being that of ; (f) ; (g) no on the seven axes, and none on an associative plane, commutes with colour. Premise (Q) [H]: spacetime vectors form (Q1) and spacetime is its colour-singlet part (Q2, Coleman–Mandula); (Q1) is not derived from the axioms (a unital spin factor does not fit into , 7 being odd). Mathematics after Manogue–Dray 1999 with the unit fixed by colour (test_colour_commuting_spacetime_is_h2_of_the_clock_complex, test_no_rotation_of_the_seven_axes_commutes_with_colour, test_octonionic_spinor_is_lepton_plus_quark_weyl). Theorem 48d (2026-09-25), [T] as mathematics: (a) holds a unital spin factor iff is even — none on (holons, their pairs and self-models, the depth register on readings); (b) -covariant observables of a pair and the operators commuting with the depth register's path Laplacian are commutative; (c) is exceptional (Albert 1934), so no holon algebra; its colour-fixed part is and yields as a Peirce space; (d) the centraliser of colour in is one , so identifying 48c's with T-326's makes the spatial rotations and space the weak triplet; (e) the sixteen real dimensions carry the Weyl or the isospin index, not both — the consistent joint reading adds the Weyl index through ; (f) for a composition subalgebra of : colour fixes ⟺ ⟺ Bloch ball of dimension 3 ⟺ (Masanes–Müller–Pérez-García–Augusiak 2014) two such systems can be entangled. Hence (Q) ⟺ (L) ∧ (MM), with (L) [H]: tangent vectors form for such a ; (Q2) is no longer a separate input. Status of 48c unchanged by 48d (test_no_unital_spin_factor_on_any_holon_register, test_colour_singlet_part_of_the_exceptional_jordan_algebra_is_hermitian_c3, test_spatial_triplet_of_48c_is_the_weak_triplet). Theorem 48e (2026-09-25), [T] as mathematics: (a) every face of is , a ball only for , then with — a two-level system of UHM is a qubit (a rank-2 face; is odd); (b) hence , , is a UHM system only for , and two such faces form the two-qubit space (local tomography , entangled states, entangling reversible dynamics): (MM) is a theorem of UHM; (c) colour-fixed two-level systems: none in , the lepton line in (its Bloch rotations fail to commute with hypercharge), one swap-symmetric singlet in (not -invariant); (d) the commutant of in is abelian (6), so rotations commuting with act on a separate factor, minimally ; (e) with : and commute (51), joint commutant , , complex components (lepton doublet 4), Weyl unit on , rotations commute with and meet it in 0. Hence the 3+1 reading is [C at (L)], one premise; (L) [H] is the premise (W) that T-329 uses when it takes the fermion field to be a two-component Weyl field. 48c(f) restated: the Weyl index is ; its is not a quark field. The 48d(d)–(e) mismatch is resolved: space is not the weak triplet. Routes for (L) tried: depth register, Page–Wootters clock, the Bloch sphere, CC-7, colour-fixed faces — none derives it (test_two_level_systems_of_uhm_are_qubits_and_can_be_entangled, test_colour_fixed_two_level_faces_of_holon_registers, test_no_rotation_of_the_internal_generation_commutes_with_the_gauge_group, test_fermion_space_is_weyl_spinor_times_one_generation) Theorem 48e(f)–(i) (2026-09-26), [T] as mathematics: (f) the commutant of UHM's internal -linear operators on (, , , , ) is , so on every UHM statement about the generation holds for every : UHM fixes the complex unit of the spinor factor ( on the left-handed fields), not its dimension; (g) for , : ⟺ has an -invariant quadratic form (then , signature ) ⟺ the rank-one forms are a quadric cone ⟺ is transitive on spatial directions ⟺ ⟺ is a ball ⟺ has an -invariant bilinear form (); rotations alone do not decide; (h) with a real Lorentz factor every fermion space on is anomalous (, then ) or vectorlike; (i) the Feynman–Kitaev history of the depth register (, simple spectrum) and two slots of a self-model (, classical slot bit) give no rotation of the spinor factor. Principle (P) [H], without a number: tangent vectors are the Hermitian forms on the spinor factor , , and the causal form is preserved up to a factor by every transformation of that preserves UHM's internal structure (by (f), ). Then (P) ⟺ (L) ∧ (W) [T], and , and are derived. The 3+1 reading stays [C at (L)], now with (L) ⟺ (P); it is not [T], because by (f) no internal structure of UHM can derive (P) — the obstruction is a theorem. Correction: T-329 uses only (W₀), a complex spinor factor of any dimension, which anomaly-free chirality requires by (h); it does not share the premise (L). Routes tried for (W): the clock's complex structure and minimal left ideals, the Feynman–Kitaev history, two slots, chirality and Distler–Garibaldi, Masanes–Müller (their premise is (g)(vi)), the spinor bundle of T-119's (colour-charged frame) (test_uhm_internal_structure_is_blind_to_the_multiplicity_of_the_fermion_field, test_only_a_two_component_spinor_factor_carries_a_relativistic_causal_structure, test_a_real_lorentz_factor_gives_an_anomalous_or_vectorlike_generation, test_depth_register_history_and_the_two_slots_supply_no_spinor_rotation). Premises (2026-09-26): (P) as stated names and so contains (Cl₀); the inputs are counted as (Cl₀) + (P) relative to (Cl₀), two independent premises ( and the model, test_spinor_factor_premise_and_fermion_module_premise_are_independent) — Premises of UHM | Spacetime | G₂-structure |
| T-50 | Uniqueness of the cubic -superpotential: (Schur's lemma). — the unique -invariant cubic term; higher orders suppressed by — raised from [C at (MP)] | Supersymmetry | Supersymmetry |
| T-51 | -sector scale from PW clocks: from PW phase precession + viability (V). , GeV — raised from [C at (ΓO)] | Neutrino Masses | Neutrino Masses |
| Fermion Generations | Fermion Generations | ||
| T-53a | Equivalence of the constructions of the cyclic clock [T]: the Page–Wootters, information-geometric and categorical constructions generate isomorphic temporal structures on the set of "moments" — canonical bijections between their label sets. Narrowed 2026-09-25: the stratificational leg (Lemma 6.3) is retracted [✗] — it required coarsenings with , which would make them invertible, contrary to T-53c; the stratal depth relates to the tick only by | Emergent time | Emergent time |
| T-53b | Emergent dynamics [T] relative to the depth register: the state evolves by the full UHM equation in the parameter of the Lindblad semigroup, and has a finite carrier — the stratal depth recorded as an ordered chain of readings (positionally in the O-registers of holons) under a Feynman–Kitaev constraint, with two holons as environment. For every CPTP semigroup on one state-independent constraint gives conditional states exactly at every reading, in a world of dimension ; for the unital primitive the purity falls strictly and does not grow along all readings; each solution of the full equation with is reproduced exactly by a constraint fitted to it; between readings the error is (Theorems 11.1–11.4 of emergent time §11.4). Raised 2026-09-25 from [C under an aperiodic time parameter], the intermediate status of the same day. Corrected 2026-09-25: the earlier claim that the conditional states obey the full equation in the Page–Wootters tick is retracted [✗] — relative to a clock of period seven ticks any dynamics is periodic (Chataignier–Höhn–Lock–Mele, New J. Phys. 28, 034504 (2026)); relative to a periodic register of readings the arrow fails on exactly one step per period, attainably (Theorem 11.2). Not obtained: dissipation on the whole orbit of a clock whose readings form a group orbit (almost-periodicity Lemma, §11.3), and as a state-independent conditional law. That the world's timeless state is of the constructed kind is the constraint half of A5, as for every Page–Wootters statement | Emergent time | Evolution |
| T-53c | Arrow of time [T]: the arrow arises as the collapse of strata of the ∞-topos to the terminal object; the coarsening functor is not an equivalence ( — information is lost), and irreversibility is structural, not statistical. Clarified 2026-09-25: the monotonicity holds in the parameter of the dissipative semigroup (Theorem 10.1 of the proof page), carried by the depth register, along whose readings it holds exactly (T-53b), not in the Page–Wootters tick; the claim that CPTP follows from the orientation toward is an open hypothesis [H] (§7.1 there); entropy grows only for the unital part (reset-channel counterexample) | Emergent time | Emergent time |
| T-53d | Critical slowing of internal time [T]: — internal time freezes at the viability boundary with a square-root law | Emergent time | Critical purity |
| T-54 | Internal theory : axioms A1–A5 define -invariant predicates in ; — an ∞-topos object containing self-consistent truths | Consequences | Consequences |
| T-55 | Lawvere incompleteness: : from Cartesian closure of + necessity of nontrivial (viability) | Consequences | Consequences |
| T-56 | Structural ToE: — -closed, finitely axiomatisable (A1–A5), principally incomplete (T-55), evolutionarily open (O-injection) | Consequences | Consequences |
| T-57 | Completeness of the triadic decomposition (impossibility of 4th type of dynamics): LGKS theorem (1976) → unique decomposition under constraints A1–A5 | Lindblad Operators | Lindblad Operators |
| T-53 | Lorentzian signature from the spectral triple — -split [T] + Lorentzian sign [T at reflection positivity] (strengthened 2026-07): the -split is fully derived — exactly one timelike direction (unique PW -clock, [T]) and three spacelike ( Riemannian, T-119 [T] as mathematics since 2026-09-25, reading [I]). The Lorentzian signature is [C] — not unconditionally [T]; the header marker above is the accurate one and this sentence previously contradicted it (corrected 2026-08-06). Two conditions. (a) The split is now better founded than before, and independently of any Weyl law: it is , both computed from the octonion structure (T-119 §G). What remained [C] in T-119 was the manifold reconstruction (first-order condition, Poincaré duality), not the dimension count; since 2026-09-25 T-119 computes the spectrum () and is [T] as mathematics. (b) Krein-self-adjointness does not select the signature. Machine: and both hold exactly (residual ) — but the Euclidean set , which satisfies and so has signature , is Krein-self-adjoint just as exactly with . Choosing encodes one timelike direction rather than deriving it; the Krein triple is a consistency check on the construction, not a derivation of the sign. The Lorentzian sign therefore genuinely rests on the stated physical input (boundedness-below of / OS reflection positivity), which is what the header marker records. Realised by an explicit Krein–Lorentzian spectral triple (spacetime §Krein triple): fundamental symmetry , Dirac operator provably Krein-self-adjoint ( via ), and signature — the first factor [T] (unique PW clock), the second from T-119 ([T] as mathematics since 2026-09-25, reading [I]). Given the split, Lorentzian is selected over Euclidean and by the reflection-positivity input, not by the Krein structure alone. KO-dim 6 fixes only internal . The one physical input is boundedness-below of (universal stability = OS reflection positivity); the older ansatz is retired | Spacetime | Spacetime |
| T-58 | Morita equivalence of 7D and 42D formalisms: by Lurie's comparison theorem ; all 7D formulae are exact, not approximations. Errata 2026-09-10 (second pass): [T] → [✗] — the equivalence fails on dimension: the PW-constrained space is with , giving real parameters against for , and equivalent sheaf topoi over sober spaces force a homeomorphism. Replaced by T-58′ [T]: (section–retraction), which transports 7D data upward but leaves 42D-only quantities lift-dependent. The 7D statements (, , , ) stand on their own [T] | Dimension E | Coherence Matrix |
| T-59 | Spectral gap of the Fano dissipator — stratified [T]+[T/sim]: Analytical core [T]: from BIBD-symmetry; — regenerative scale, structurally independent of the spectral gap . | Axiom Ω⁷ | Axiom Ω⁷ |
| T-60 | BCH error estimate algebra→dynamics: the unitary part exactly reproduces the -shift, error | Axiom Ω⁷ | Axiom Ω⁷ |
| T-61 | Unique self-consistent vacuum: a uniform vacuum is impossible; the sectoral structure — the unique solution — raised from [C] (C12). Errata 2026-09-25 (audit A-90): stratified; updated the same day with the corrected T-64 — Theorem 14.1 (the homogeneous vacuum is not stationary) stays [T]; for the -invariant potential the vacuum is unique up to — [T] for every off the transition curves ( or one orbit of colour-invariant states, T-64; raised the same day from the conditional status [C at (RT)] above , when (RT) was proven); for the retracted cubic uniqueness holds only up to its 896 symmetries and only numerically — [H]; the sector structure is retracted [✗] (neither potential gives the (SV) values). The self-consistency relations of Theorem 13.5 belong to and are not claimed for | Gap Thermodynamics | Gap Thermodynamics |
| T-62 | φ-operator as a replacement channel: , ; CPTP, monotonicity, fixed point | Self-Observation | Self-Observation |
| T-63 | Neutrino Dirac Yukawa via O-sector: . Discrepancy reduced from to | Neutrino Masses | Neutrino Masses |
| T-64 | Global minimisation of — Errata 2026-09-25 (audit A-90), second correction the same day: restated for the -invariant potential; raised to [T] the same day, when (RT) was proven. with the associator cubic of T-331. [T]: for on the real states with and no Gap is spontaneous; for the unique vacuum is (, and colour unbroken); the Hessian at is , , on , , ; for ( at ) every vacuum has — the Gap is spontaneous; the colour-invariant sector is solved in closed form, and its Gap-phase minimum has stabiliser and orbit ; the global problem reduces to that sector by the real twirl inequality (RT), proven (Lemma 3: the defect is with every , spectrum in closed form); hence for every the vacua are the -orbits of the sector minimisers — or one orbit , two orbits only on the transition curves () and (rank 7 → rank 4, ) — and in the Gap phase colour is unbroken; the non- mean coherence of every vacuum is below . itself is fixed by no derived source (T-331(e)). The axis-frame statement for the retracted cubic (vacuum unique up to its 896 symmetries, on two Fano lines through a point) stays [H]; retracted [✗]: the -orbital reduction , the five sector parameters and the Hessian eigenvalues , , . The sector values are the hypothesis (SV) [H], which neither potential produces (test_g2_invariant_vacuum_is_symmetric_or_colour_invariant_with_gap, test_real_twirl_inequality_holds_in_its_proven_cases_and_on_samples, test_real_twirl_inequality_is_a_sum_of_positive_forms, test_colour_sector_transitions_and_the_bound_on_mean_coherence, test_v_gap_vacuum_is_unique_up_to_its_symmetries_not_up_to_g2) | Gap Thermodynamics | Gap Thermodynamics |
| T-65 | Full spectral action of UHM: the product is a spectral triple; EH with ; gauge + Yukawa only with Connes' imported (the page's honest-status box). Corrected 2026-09-25: the row read "NCG axioms verified for the product" — the reality axioms fail, since has no real structure of KO-dimension 6 | Quantum Gravity | Einstein Equations |
| T-66 | UV-finiteness of Gap theory — stratified [T field-space]+[C order-by-order]: field-space (large-field) finiteness [T] — finite for every (compact target ); full order-by-order UV-finiteness [C] (structural): compactness + Ward identities () + holomorphy (Seiberg) + sector-product suppression (T-219); APS-index = 0 (no anomalies). | Quantum Gravity | Quantum Gravity |
| T-67 | Justification of for L3: quadratic decomposition components; Bayesian dominance | Interiority Hierarchy | Interiority Hierarchy |
| CC Theorems | CC Theorems | ||
| T-69 | Topological protection of the Gap vacuum: → winding numbers classify Gap configurations. Barrier ; confinement-Gap protected by , O-sectoral by . Compactness + uniqueness of minimum (T-64) — raised from [H] (No.55). Errata 2026-09-25: stratified [T]+[C at (SV)] — stays [T]; the barrier values , , are Hessian eigenvalues of the retracted sector parametrisation of T-64, so the protection of the vacuum is conditional on (SV) | Composite Systems | Gap Thermodynamics |
| T-70 | Canonical definition of : from UV-finiteness (T-66) + unique vacuum (T-61, T-64). — not a free parameter, but a function of vacuum quantities. The Higgs quartic — a prediction, not a fit. Errata 2026-09-25: corrected from [T] to [C at (SV)] — steps 2, 3 and 5 take the unique vacuum and its five Hessian eigenvalues from T-64, restated as a hypothesis whose vacuum has no sector values | Higgs Sector | Λ Budget |
| T-71 | Structural necessity of : autopoiesis (A1) + local cohomology () → . Connection to Lawvere incompleteness (T-55): information gap → positive vacuum energy | Consequences | Cosmological Constant |
| T-72 | Scale invariance CC-6 — status raised 2026-09-25 from [C] to [T at weak coupling] by Theorem 9.5. The implication: under (AGG) — the aggregation returns the constituent on uncoupled copies, , and the coupled micro state lies within Bures distance of (for the mean marginal it suffices that each single-copy marginal lies within trace distance of ) — one has , , and deviate by with explicit constants, Gap by at most , and each L2 threshold keeps its truth value given a margin. Theorem 9.5 proves (AGG) for weakly coupled embodied holons: the mean marginal is the only permutation-invariant linear aggregation consistent on uncoupled copies, and its deviation is at the stationary state (Corollary 9.2a), along every trajectory from a compact part of the basin, and from every initial state under backbone dominance, . At strong coupling the transfer fails (Theorem 9.6: a coupling diagonal in a maximally entangled basis sends the aggregate of two viable holons toward — parts at , aggregate at ). Errata 2026-09-25 (earlier the same day, corrected from [T]): the claim of preservation under any CPTP aggregation with corrections , , is retracted [✗]: the completely depolarising channel sends every state to (, ); "all invariants are -invariants" contradicts frame rigidity ( and Gap are frame-pinned); the bound was asserted, not derived, and is a vacuum parameter inside one holon. Downstream: the scale-transfer corollary of T-108 holds at weak coupling as well | CC Theorems | CC Theorems |
| T-73 | Gap = curvature of the Serre bundle: — exact identification from the finite spectral triple of T-53, whose existence is [T] (T-53's Lorentzian sign is [C] and is not used) + Connes NCG curvature. Second Chern class: — topological invariant — raised from [C] (No.65) | Gap Operator | Gap Thermodynamics |
| T-74 | from spectral action (Sol.53): ; potential uniquely from Seeley–de Witt coefficients. Chain: — raised from [P] | Gap Thermodynamics | Gap Operator |
| T-75 | Lagrangian from Lindbladian (Sol.54): — classical limit of the Schwinger–Keldysh action for in the coherent-phase representation. All 6 terms derived from the triadic decomposition [T-57] — raised from [H] | Gap Thermodynamics | Gap Thermodynamics |
| T-76 | ∞-topos — stratified (Sol.55): Site level [T] — three Grothendieck axioms (Identity, Stability, Transitivity) verified for via CPTP-contractivity of the Bures metric (Uhlmann 1976, Petz 1996, Fuchs–van de Graaf 1999); essentially-small presentation via compact metrizability of + Johnstone Elephant C2.2.3; Lurie HTT 6.2.2.7 applies. Exp-extension [C at Giraud verification] — requires full verification of Giraud axioms (descent, universal colimits, disjoint coproducts, effective groupoid objects) via functor ; currently marked Claim 10.2 in proof document. †-structure: (adjoint channel) — [T]. | Categorical Formalism §6.3.1 (site proof), §10.4 (Exp-extension, claim) | Categorical Formalism |
| T-77 | Cooperation via coherences (Sol.57): . Old inclusion-exclusion formula retracted [✗] (dimensionally incorrect) | Value Consciousness | Value Consciousness |
| T-78 | CPTP complete channel (Sol.58): Fano operators define a CPTP channel in Kraus representation. CP is automatic (Choi's theorem); TP from [T-41b]. Independent of stratification — raised from [C] | Dimension L | Lindblad Operators |
| T-79 | Spectral self-closure (Meta-theorem): A1–A5 → unique self-consistent dynamics. The mapping () has a unique fixed point (Brouwer + T-39a + T-64). Errata 2026-09-25: corrected from [T] to [C at (SV)] — the map on is not defined as stated ( is not -invariant), and the unique minimum it uses is T-64, restated as a hypothesis; uniqueness of the fixed point is conditional on (SV) | Consequences | Consequences |
| T-80 | Sectoral Gap bound [T] for the structural bound + [C at (SV)] for its numerical values (Sol.59): for non-O pairs (maximum over - sector); mean — root mean square over the 15 non-O pairs (erratum A-83, 2026-09-25) — at ; the earlier came from substituting against the table's . For O-pairs: . Old Fano bound retracted [✗] (O-counterexample). Replacement theorem is stricter for non-O and correct for O. Caveat: numerical values — [C at (SV)] (unique vacuum) | Berry Phase | Gap Thermodynamics |
| T-81 | Topological area law (Sol.60): qualitative result — [C at (SV)] (from T-73 + T-69 + T-64; corrected from [T] on 2026-09-25: and the barrier are values of the hypothesis (SV)). Numerical value MeV — [C at (SV)]: depends on the specific minimum of (unique vacuum). Discrepancy with experiment (440 MeV): — raised from [H] | Confinement | Confinement |
| T-82 | Uniqueness of the Fano form (Sol.61): Fano operators — the unique minimal composite Lindblad operators compatible with A1–A5. BIBD(7,3,1) is unique (Fisher + Veblen-Wedderburn). Chain: AP → c>0 → T-41b → T-11 → T-12 → T-13 — raised from [H] | Lindblad Operators | Lindblad Operators |
| T-83 | Spacetime from the spectral triple (Sol.62): T-53 (KO-dim 6) + Barrett → (time from PW) + (space from ) + (compactified). Time — a consequence, not a postulate — raised from [H]. Errata 2026-09-25 — stratified: (a) time from the PW clock [T]; (b) " (space from ) + (compactified)" is retracted [✗] — the axis triples are not sectors (48a) and colour is not space; (c) the Lorentzian signature is [C at T-119 and reflection positivity] (T-53). The inputs "KO-dim 6" and "Barrett" are retracted as well: no KO-dimension-6 structure exists on , and Barrett 2007 contains no classification of finite spectral triples | Spacetime | Spacetime |
| T-84 | O-sector dominance in (Sol.63): from sector decomposition of + Sol.59. = 'cost of observation' — raised from [H] | Cosmological Constant | Λ Budget |
| T-85 | from (Sol.65): — the unique -covariant topological Lagrangian. CS₁ replaced by Keldysh. — raised from [H] | Berry Phase | Gap Thermodynamics |
| T-86 | Categorical unreachability of L4 (Sol.64): — colimit of the Postnikov tower + T-55 (Lawvere incompleteness). Butterfly retracted [✗]: finite catastrophe inapplicable to infinite-dimensional transition — raised from [C] (C19) | Interiority Hierarchy | Transition Catastrophes |
| T-87 | A5 (Page–Wootters) from spectral triple (Sol.68) — stratified 2026-09-25. Steps 1–3 [T]: the Wedderburn decomposition of isolates the clock summand (the former "with KO-dim 6" is retracted and not needed) (, a direct sum); the tensor factor is the regular representation of the shift (expanded proof 2026-04-17: a direct sum is not a tensor product, 7 is prime), giving . Step 4, the constraint : [C under the support condition supp Γ ⊆ ker Ĉ of Property 2] — stationarity of a mixed state gives only ; the earlier derivation "from stationarity" and its attribution to a Dirac-quantisation derivation of Giovannetti–Lloyd–Maccone 2015 are retracted (the equivalence with Dirac quantisation is Höhn–Smith–Lock, Phys. Rev. D 104, 066001 (2021)). Step 5, dissipative conditional dynamics: retracted [✗]. Hence the clock register follows from A1–A4, the constraint is an independent assumption; the earlier "A5 is a consequence of A1–A4" is withdrawn | Axiom Ω⁷ | Spacetime |
| T-88 | Functoriality of κ₀ (Sol.69): — the unique definition compatible with Bures topology (Yoneda + Bures + Stinespring). — exact theorem — raised from [D] | Axiom of Septicity | Axiom of Septicity |
| T-89 | Freedom is well-posed (corrected 2026-07): [T] — the tangent dimension of the free-energy Morse–Bott critical manifold, plus one. The earlier claim "" is withdrawn: is contractible so , and "number of gradient trajectories" contradicts Picard–Lindelöf uniqueness. The invariant is the flat-direction dimension, not a path count. | Consequences | Consequences |
| T-90 | Structural vs. functional loss (psychosis) (Sol.79): Hamming bound — structural property of H(7,4), always for L2. Psychosis: (functional loss). Bound is never violated — raised from [H] | Pathological Consciousness | Gap Characterisation |
| T-91 | ∞-groupoid proven (Sol.76): — Kan complex (Milnor's theorem) for topological (Bures–Fubini–Study metric). Combined with T-76 ( — ∞-topos): HoTT logic, subobject classifier, Postnikov truncations — raised from [P] | Categorical Formalism | Categorical Formalism |
| T-92 | Formal components of (Sol.81): all 7 stress-tensor components — unambiguous functions of without free parameters (, , , , , , ). (full viability, strictly stronger than ) — raised from [C] (CC-8). Errata 2026-07-22: , renormalized (, ) so each encodes its threshold; the embedding restored via . | CC Theorems | CC Definitions |
| T-93 | Formal isomorphism H(7,4) (Sol.82): incidence matrix for 7 Lindblad operators coincides with the parity-check matrix of the Hamming code H(7,4). — classical result of coding theory — raised from [I] | Gap Dynamics | Gap Dynamics |
| T-94 | Exponential form of the memory kernel (Sol.83): from compactness of . Laplacian on a compact torus has discrete spectrum with ; — spectral gap — raised from [H] | Gap Dynamics | Gap Dynamics |
| T-95 | Canonical PW reconstruction algorithm (Sol.67): 4-step procedure with zero error. Step 1: PW embedding (T-58 Morita); Step 2: partial trace; Step 3: 7D formulae via HS projections; Step 4: (Lurie's theorem). Errata 2026-09-10: [T] → [C] — the zero-error Step 4 rested on the Morita equivalence T-58, retracted [✗] in the second pass of 2026-09-10; what survives is the round trip of T-58′ [T], exact for 7D quantities and silent on 42D-only ones; and in 7D is a scalar, so the comparison is between a scalar and the 42D clock block (canonical box) | Dimension E | Dimension E |
| T-96 | Attractor characterisation (Sol.SA-2, corrected): — trivial fixed point (, ). Any nontrivial fixed point : [T], [T]. Proof via primitivity of the linear part (T-39a) + purity balance. The self-reference paradox of is resolved: the regeneration target is the categorical self-model , not a dynamical limit. Existence (2026-09-25) [T]: the characterisation asserts no existence, and existence depends on the self-model. Dead isolation [T]: with a unital self-model — the canonical (anchor ), every - or -covariant linear one, and "anchor = the attractor itself" — an isolated holon's only stationary state is and never increases (Pérez-García–Wolf–Petz–Ruskai contractivity; with , falls at every step and all starts reach ). Self-sustaining attractors [T]: with the self-registering (Lüders update of on its own effect), every basis state is a hyperbolic sink at with explicit spectrum, and for seven locally stable attractors with persist (at , : –). Step 3 clarified: the passage from "diagonal" to needs to keep diagonal states diagonal ( and do) | Evolution | Self-Observation |
| T-97 | Embedding of viability regions (Sol.SA-1): . Full viability (, 7 conditions) is strictly stronger than minimal (). Counterexample: () | Viability | CC Theorems |
| T-98 | Attractor purity balance [T]: , , . Restored [T]: substituting into the evolution equation — standard mathematical derivation; is not arbitrary, but derived from Fano contraction (T-110 [T]). The formula is a consequence of the axioms, not a convention. Scope (2026-09-25): stands for at the fixed point; the identity holds at every fixed point, of which the isolated canonical has none besides ; at the seven attractors of the self-registering it holds to | Evolution | Evolution |
| T-98a | Lower bound for embodied systems [T]: for an embodied holon with additional CPTP channels (backbone, anchor, hedonic), — extra channels add to the numerator without touching the denominator, so embodiment cannot lower the fixed-point purity | Evolution | T-98 |
| T-99 | Structural resolution of (formalisation): 7-step proof of from axioms A1–A5. Reality of (A1) → uniqueness of PT-odd → unique vacuum (T-64) → phase isotropy → exactly. Non-perturbative stability from T-69, radiative from T-66. Axion not needed for CP — purely a DM candidate. Errata 2026-09-25: stratified; updated with T-331 — step 2 ( is the only -odd term of ) is [T] for the retracted cubic only: every -invariant cubic is PT-even (T-331), so the corrected potential has no PT-odd term at all, and its vacuum keeps an antiunitary symmetry — PT at [T for ], on the orbit [T]; stays [C at (SV)]: the route through that symmetry needed its identification with CP of the colour sector; with the Yukawa couplings classified (T-332) it is closed [✗] by T-333 — no lift of the vacuum's antiunitary symmetry gives with and ; the conclusion rests on the chain only. T-333(e)–(h) (2026-09-26): with the fields that (Cl) forces there is no Peccei–Quinn symmetry and no spontaneous CP violation, so the axion and Nelson–Barr routes need new fields, and is a free parameter there [Pr]. Errata 2026-09-26: the conclusion is retracted [✗] (it was [C at (SV)]); replaced by Theorem 3.1c, [T] as mathematics, [C at (Cl)] in UHM; strong CP open [Pr] — step 4 is false for itself: on real while the first variation of along is non-zero, so for every the vacuum has (, at the page's constants; at ); (SV) fixes moduli, not phases, and cannot rescue it; step 5 has no ground in the Clifford content, where with Yukawa inputs. Theorem 3.1c: (a) the corrected Gap sector is CP-neutral (T-331, T-64); (b) this does not reach (T-333); (c) a PT-odd -invariant term (three quartics, T-331; the explicit one ) breaks every and moves the vacuum off the real states — a source of phases, not a guard; (d) is a free parameter, UHM predicts no neutron EDM. Routes tried: the chain under (SV), the vacuum's antiunitary symmetry, PT-odd quartics, Peccei–Quinn, Nelson–Barr, left–right parity, massless (test_theta_route_through_the_gap_potential_fails_for_v3_and_for_pt_odd_quartics; Confinement §3.1c) | Confinement | Confinement |
| T-100 | Environment encoding (Enc functor): there exists a unique (up to ) CPTP functor satisfying 3-channel decomposition and functoriality. Existence from Def. 8.1 [T], 3-channel from T-57, uniqueness from -rigidity | Sensorimotor Theory | CC Theorems |
| T-100a | Enc factorisation [T]: for an arbitrary observation space the encoding functor factors as — encoding is modality-agnostic, the representation is separated from the projection into | Sensorimotor | T-100 |
| T-101 | Optimal action (Dec functor): . From T-92 (equivalence ): minimising maximises the distance to | Sensorimotor Theory | CC Theorems |
| T-102 | Completeness of the 3-term equation: any CPTP-compatible external perturbation decomposes as . A fourth type is impossible. Direct consequence of T-57 (LGKS) and the triadic decomposition of Lindblad operators | Sensorimotor Theory | CC Theorems |
| T-103 | Hedonic valence (reclassification [C]→[T]+[I]): formula — identity [T] from the evolution equation. Gate — V-preservation [T]. Observability at L2 () — [T] from T-77. Phenomenal interpretation — [I] | Sensorimotor Theory | CC Theorems |
| T-104 | Stability radius [C], closed form corrected 2026-08-07: . The old is refuted [✗] — machine counterexample: at the true infimum is against , so the claimed lower bound fails by toward danger; the cited Fuchs–van de Graaf step bounds from above by the trace norm and cannot yield it. Correct: the minimiser commutes with (), so Bures reduces to Hellinger on spectra; on the one-dominant family with and (= the of path 4!), — machine-exact to . Near the wall the law is linear, not square-root: , , which is why the old surd's overstatement diverges ( at , at , at ). Runtime formula, error on the window: , . General spectra: the closed form is a conservative lower bound [H] (41/41 spectra, ratio ) | Stability | Stability |
| T-105 | Landauer energy balance: — minimum rate of free-energy dissipation for homeostasis. From the Landauer principle + T-84 (O-sector dominance) | Stability | Stability |
| T-107 | Information capacity of Enc: bits/observation. From the Holevo bound + T-102 (3-channel) + | Sensorimotor Theory | Predictions |
| T-107a | Cumulative information [T]: over successive observations bits, attained for informationally independent observations | Sensorimotor | T-107 |
| T-107b | Minimum observations [T]: — the floor on how long complete encoding of an environment of entropy must take | Sensorimotor | T-107 |
| T-107c | Predictive optimality of Enc — corrected from [T] to [D] (2026-09-25): — the optimal encoding is defined as the one maximising available free energy; its derivation from the FEP limit (Theorem 4.1 of Variational principles) is retracted | Sensorimotor | Free energy |
| T-108 | Compositionality of Enc/Dec (CPTP closure): is a CPTP channel for every CPTP aggregation , and is functorial. From T-100 + closure of CPTP maps under and ; T-72 is not used. Errata 2026-09-25: the steps "uniqueness from -rigidity at each scale (T-72)" and "functoriality preserved under aggregation" (ill-typed: maps ) are retracted; that the diagnostics carry over across scales holds at weak coupling through the canonical aggregation (Theorem 9.5; T-72 raised 2026-09-25) and can fail at strong coupling (Theorem 9.6) | Sensorimotor Theory | CC Theorems |
| T-108a | Multimodal decomposition [T]: with — modalities compete for the same bits per step, which is why attention is optimal allocation and not a filter | Sensorimotor | T-108 |
| T-109 | Information learning bound: , where . From the quantum Chernoff bound + T-107 (Enc capacity). Scaling for weak signals | Learning Bounds | Learning Bounds |
| T-110 | Dynamic learning bound: Fano contraction (T-39a) limits the signal integration rate. | Learning Bounds | Learning Bounds |
| T-111 | Stabilisation learning bound: observation amplitude is bounded by (T-104). Under noise: . Topological protection T-69 ensures continuity | Learning Bounds | Learning Bounds |
| T-112 | Optimal learning bound: . Three regimes: information-, dynamically-, stabilisation-limited | Learning Bounds | Learning Bounds |
| T-113 | Minimality of N=7 for learning: learning via regeneration requires a replacement channel (T-77) → Fano plane → (T-89). For : . is Pareto-optimal | Learning Bounds | Learning Bounds |
| T-113a | Consistent -tomography (2026-07): given the 7-channel embedding , is consistent ( a.s.) with matrix-Bernstein concentration w.p. for (rate , verified); unbiased U-statistic purity; threshold sample-complexity matches T-109. Turns the calibration "Achilles' heel" into rigorous estimation isolated to the embedding | Measurement §6.4 | Measurement §6.4 |
| T-114 | Fano grammar: Markov chain on PG(2,2) with is ergodic (connectivity + aperiodicity). Stationary distribution is uniform (PG(2,2) is self-dual, graph is regular) | Lindblad Operators | Lindblad Operators |
| T-115 | Algebraic distinguishability of compositions: for generic (full-rank, with non-zero off-diagonal coherences and 7 distinct eigenvalues). Collisions — a submanifold of codimension . Caveat: for diagonal : (linear growth). From T-82 (Fano uniqueness) + algebraic independence of projectors | Lindblad Operators | Lindblad Operators |
| T-116 | PW Suzuki-Trotter: , order . For , , : . Strengthens T-60 (BCH ) to polynomial accuracy | Axiom Ω⁷ | Axiom Ω⁷ |
| T-117 | Commutativity of the macroscopic algebra: macroscopic observables commute in the thermodynamic limit . From quantum CLT (Goderis–Verbeure–Vets, 1989) + clustering (T-39a) + compactness . Clarified 2026-09-25: the restriction to the "-effective sector" in the statement is not used by the proof, which holds for any local observables; that sector came from the axis-labelled decomposition of row 48a, retracted | Emergent Manifold | Emergent Manifold |
| T-118 | Emergent temporal manifold [T]: as the scaling limit of the reading algebras of the depth register (chronon , window on both sides of the origin): the reading sets converge to in the pointed Hausdorff sense, and sampling embeds isometrically into ; with the origin at the first reading the limit is (emergent time, Theorem 11.5). Raised 2026-09-25 from [C under an aperiodic clock], the intermediate status of the same day: the depth register is the clock whose length grows without bound. Corrected 2026-09-25: the earlier derivation used readings of the summed clock of holons with period ; retracted [✗] — the summed clock has readings and the fixed period . The ordered readings belong to the same registers read positionally under a Feynman–Kitaev constraint. Composite clocks with incommensurate frequencies give , not | Emergent Manifold | Emergent Manifold |
| T-119 | Emergent spatial manifold — [T] as mathematics (raised 2026-09-25 from [C], restated; reading as physical space [I]). For three commuting rotation charges of the holon (a maximal torus of : two colour Cartan generators and ; ): (a) the joint spectra of the averages over holons fill the weight octahedron , where Poincaré duality fails; (b) those of the fluctuations fill , sampling embeds isometrically into , and the spatial algebra — the minimal unitization — is ; (c) , closed spin with unique smooth structure, and its Dirac triple satisfies all seven of Connes' conditions, first-order and Poincaré duality included; (d) colour-singlet charges give only two dimensions, so this has colour-charged coordinates (Coleman–Mandula obstacle for the physical reading). T-117 is not used (test_emergent_space_is_the_octahedron_and_its_fluctuations_the_three_sphere). Former statement ([C]): for a unique smooth compact orientable spin 3-manifold, from T-117 + Gel'fand–Naimark + Connes reconstruction (arXiv 2008; J. Noncommut. Geom. 2013), with the dimension sub-problem repaired 2026-08-06. Error found and replaced. Step 2c derived from a Weyl law on , reading the exponent off . That is impossible: has dimension , its spectrum is finite and , so no power law exists. Machine: on with internal the Weyl exponent is for and identical across — the exponent is the dimension of the base, the internal dimension only scales the multiplicity prefactor. New derivation (§G), which does not use a Weyl law at all. Emergent coordinates are the joint spectrum of a maximal commuting family of macroscopic observables, so their number is the rank, not the dimension, of the sector algebra. Computed from the octonions directly: the commutant of on yields a complex structure ( to , ), so the spatial algebra is (dimension , verified), and the centralizer of a generic element has dimension exactly 3 — while , , are none of them . Full-dimensionality of the joint spectrum (hence exactly, not merely ) follows from the GVV quantum CLT already invoked in T-117: the macroscopic fluctuations of commuting observables converge to a non-degenerate Gaussian on . Verified numerically — singular values for the three Cartan directions in , and once the -direction is added, i.e. . Sharp structural point: the three spatial directions are independent only because the embedding in leaves the trace of the -block free — the clock sector is what makes the third spatial coordinate dynamical; inside the -block alone the trace is frozen and one measures , not . Open until 2026-09-25 (then [C]; settled by the restatement, which computes the spectrum instead of reconstructing it): the first-order condition ( Connes axiom) is a constraint on , not a consequence — for generic Hermitian on the -sector the machine gives , vanishing only for diagonal (§D); and the Poincaré-duality check (v) is circular as written, assuming is a manifold to verify an axiom whose role is to conclude that it is. Verified and untouched: exactly (, , commutant ⟹ two inequivalent irreducibles; §C). Note 2026-09-25 (row 48c): the count is a count of colour and gives no rotation group, so it still meets Coleman–Mandula; row 48c obtains three spatial directions whose rotations commute with colour, under premise (L) (the weaker form of the former (Q); row 48c is [C at (L)] since Theorem 48e of 2026-09-25, and (L) ⟺ (P), see premises). The two counts agree only as numbers; re-basing the dimension step of this row on 48c is left to the emergent-manifold page. The restated row keeps the colour-charged count and says so in (d); Theorem 48d names what joins the two pictures | Emergent Manifold | Emergent Manifold |
| T-120 | Product spectral triple [T] as mathematics (raised 2026-09-25 from [C] at the reconstruction axioms then open in T-119, with the restatement of T-119; the reading of as physical spacetime inherits T-119's [I]). For every Riemannian metric on ; the three macroscopic algebras commute in the limit by a direct estimate (), without T-117; the product carries no real structure, so no first-order condition arises. Former head, [C]: with — assembled from T-118 + T-119 + T-53 + Connes–Chamseddine (1997), not postulated. Errata 2026-09-25: corrected from [T] — a product is no stronger than its factors, and the spatial factor (T-119) is conditional (the temporal factor T-118 was conditional too until the depth register made it [T] later the same day); the KO-dimension count (Step 6) and the signature argument of Steps 8a–8d are retracted [✗], since no real structure of KO-dimension 6 exists on . Background independence — raised from [P], now conditional on the same assumption | Emergent Manifold | Quantum Gravity |
| T-120b | Vacuum topology — split 2026-09-25: (i) [T], now part of T-119 (minimal unitization of the fluctuation spectrum ), independent of the vacuum and of T-64; (ii) constant curvature , de Sitter metric [C at the vacuum symmetry] (Step 1 uses T-64, now [H], hypothesis (SV)). Former head, [C] inheriting T-119: (T-71 [T]) (closed), de Sitter metric. The implication is [T]; asserting closedness of presupposes that exists as a smooth manifold, which is exactly the half of T-119 that remains conditional (first-order condition, Poincaré duality). The dimension count is unaffected. From -invariance of the vacuum + unique minimum T-64 — since 2026-09-25 the hypothesis (SV): the vacuum of keeps no and T-64 is restated as a hypothesis. Corrected 2026-09-25: the proof no longer cites T-186(c) (" unconditionally", retracted), and the page heading, which read [T], now carries the row's status | Emergent Manifold | Emergent Manifold |
| T-121 | Closure of Lovelock gaps [T] (raised 2026-09-25 from [C at T-120] with T-120): gap 1 (discreteness → continuity) — closed, since is smooth (T-120 [T]; until the restatement of T-119 this was conditional on its open reconstruction axioms; the aperiodic clock, a second condition in an intermediate version, is supplied by the depth register, T-118). Gap 2 (covariance) — closed by the diffeomorphism invariance of the Chamseddine–Connes spectral action; the leg "" is retracted (no non-trivial homomorphism exists). Gap 3 — irrelevant. Lovelock's argument is supplementary to the spectral one — raised from [H]. Errata 2026-09-25: corrected from [T] with the status of T-120 | Emergent Manifold | Einstein Equations |
| T-122 | Diagonal freeze — attractor property T-96: at the stationary point the diagonal entries are stationary (). From (Hermiticity) + at . Scope clarified by T-134: valid ONLY at the attractor | Evolution | Evolution |
| T-123 | -uniqueness of the representation: holonomic representation is unique up to , diagonal entries are defined unambiguously. From T-42a (-rigidity) + T-40f (minimality 7/7) + T-15 (bridge). Stratified 2026-09-25: [T] as a consequence of the axioms: the multiplication comes from the bridge T15 (row 41n), whose step PG(2,2) → takes the canonical orientation — the unique collineation-invariant orientation class (T15-canon). The intermediate stratification of the same day, "[T] for with its octonionic multiplication; [C at (Alt)] as a consequence of the axioms", is superseded | Consciousness Window | Uniqueness Theorem |
| T-124 | Non-emptiness of (consciousness window): constructive proof . Family with | Consciousness Window | Viability |
| T-124b | Independent necessity of each L2 threshold: four constructive counterexamples show that each of , , , is independently necessary — dropping any one admits pathological states (noise-dominated, fragmented, crystallised, undifferentiated). The conjunction is minimal | Consciousness Window | Consciousness Window |
| T-124c | Count of nontrivial attractors (restated 2026-09-25) [T]: (1) an isolated holon with the canonical has no stationary state besides ; (2) with the self-registering and it has at least seven locally stable ones with ; (3) an embodied holon whose backbone rate exceeds the trace-norm Lipschitz constant of regeneration, , has exactly one, globally attracting at rate ; (4) with the collineation anchor at it has none with for and exactly two for — a hyperbolic sink in (, ) and a saddle; at (added 2026-09-25); (5) with any constant anchor at every stationary state with is (T-335), and for an anchor with uniform diagonal there are none for and exactly two — a sink in and a saddle — for , (T-334; added 2026-09-25). Errata 2026-09-25: the former statement is corrected from [T] to [✗] — "at most one nontrivial fixed point in ; exactly two fixed points, one viable and one dead": false by (1) and by (2); its proof treated as constants, used anchors that are neither 's nor the target, and assumed | Evolution | Evolution |
| T-124d | Threshold robustness: perturbations of order in produce perturbations in , , . No threshold has divergent sensitivity. Crossover width . From Frobenius perturbation bounds + T-161 (exponents) + T-145 (stochastic stability) | Consciousness Window | Consciousness Window |
| T-125 | Local asymptotic stability of the attractor: for , : , . From T-39a (gap) + T-96 + T-104 | Consciousness Window | Evolution |
| T-126 | Canonicity of : the reflection measure at order is uniquely fixed by three independent characterizations — (Char-R-I) Hilbert–Schmidt angular projection: ; (Char-R-II) -invariant canonical reference: is the unique -fixed element of by Schur's lemma on the irreducible 7-dim -module (Cartan 1894); (Char-R-III) Bayesian dominance threshold: from the triadic decomposition of Lindblad operators (T-40b). Formula is the algebraic identity following from Char-R-I+II on ; implementation approximations (H3 CLOSED: T-130+T-133). At is a monotone reparameterization of by design; independent observability appears at via the self-model operator | Consciousness Window | Self-Observation |
| T-127 | Basin of attraction [T at C20]: the basin of contains , exponential convergence. From T-125 (stability) + T-104 () + openness of | Consciousness Window | Stability |
| T-128 | Exact 7D-computability of : — the 7D definition [D] of differentiation (errata 2026-09-10: not an "exact representation" of , which is not expressible in 7D — canonical box). is computable in 7D | Operationalisation | Dimension E |
| T-129 | Integration threshold from first principles: the unique self-consistent value with on the extremal uniform-diagonal state. Raised from [D] (O1). Clarified 2026-09-10: the inequality is [T] (Cauchy–Schwarz, T-129a); the choice of the least such threshold as is a definition [D]; is a frame-pinned observable (D-0910) | Operationalisation | Dimension U |
| T-129a | Universality of on all of : for every state, not only the extremal family; equality holds at the unique boundary point (uniform diagonal together with ) and is strict elsewhere; is the smallest universal threshold. From the identity + Cauchy–Schwarz | Operationalisation | Dimension U |
| T-130 | CPTP-anchor approximation bound: , . H3 [H] → CLOSED | Operationalisation | Self-Observation |
| T-131 | Canonical discretisation : — Nyquist-Shannon + Suzuki-Trotter margin. is canonical, not a free parameter | Operationalisation | Evolution |
| T-132 | Necessity of complex : for non-trivial Gap structure () Γ MUST be complex. From + Hamiltonian dynamics | Operationalisation | Gap Operator |
| T-133 | Transfer of R thresholds via the CPTP bridge: for . Strengthening of T-130. H3 definitively CLOSED | Operationalisation | Self-Observation |
| T-134 | Scope of the diagonal freeze: T-122 holds ONLY at the attractor . General formula: . Learning and genesis from do not contradict T-122 | Operationalisation | Evolution |
| T-135 | Discrete convolution of the non-Markovian kernel: Z-transform of kernel T-94 gives recursion instead of | Operationalisation | Gap Dynamics |
| T-136 | SAD as a -invariant spectral observable [T]: , . Computability . Autoencoders — an implementation, not a definition. Raised from [T at C] (T-150: commutativity of φ-tower [T]) | Operationalisation | Depth Tower |
| T-137 | Full 7D-computability of : all 7 components are computable in without 42D. via T-128, via T-132 (complex Γ), via T-129 () | Operationalisation | CC Definitions |
| T-138 | Mean-field approximation of composition: , instead of , . Hierarchical scheme for | Operationalisation | Composite Systems |
| T-139 | Γ-backbone duality: — the unique (up to ) hybrid CPTP dynamics. Backbone — causal channel, — ontological state (dual-aspect monism) | Operational Closure | Evolution |
| T-140 | Canonical consciousness measure: , threshold . does NOT enter (separate viability condition ). Uniqueness — from bilinearity and threshold coincidence | Operational Closure | Self-Observation |
| T-141 | Equivalence of three φ-forms: (replacement), (canonical for ), (Fano) — coincide on the attractor; off the attractor (controlled error, Frobenius lemma) | Operational Closure | Self-Observation |
| T-142 | SAD_MAX = 3 — stratified [T] — including the ladder, whose derivation is located and verified (see below); state-independence [T]: state-independence from + PG(2,2) is rigorous [T]. The iterated critical purity formula is derived, not heuristic (status corrected 2026-08-06, retracting an erroneous audit note of the same day). The derivation is SYNARC §5, Thm. sad-bound: with the auxiliary SAD-reflexivity (related to canonical by the exact identity ), the Fano Kraus channel multiplies coherences by exactly per meta-level, so , and the level- Bayesian threshold is . Then , which solved for purity is exactly the formula above. The load-bearing lemma — the exact coherence contraction — is independently machine-verified ( to with the diagonal preserved, ). Values: , , , — impossible, hence SAD unconditionally. In the window one gets (, at ); needs , above the ceiling. Retracted audit note. An earlier note today objected that the formula "fails at its own base" because gives rather than . That objection was wrong: this is the SAD-attainment threshold, not iterated viability, and at is precisely the non-triviality condition of CC-5. The only real defect is the name: writing it invites exactly that misreading, so it is renamed here. Note also that returning is a genuine coincidence worth flagging rather than a definition. The inequality was marked empirical but is exact: in the depth-tower indexing , at maximal purity one has . Empirical [T/sim]: SYNARC verification SAD on 500+ random ; SAD=3 achievable (pure state). | Operational Closure | Depth Tower |
| T-143 | Convergence of neural SAD to categorical: for CPTP-compatible anchor with . From T-130 (bound) + separation of thresholds | Operational Closure | Depth Tower |
| T-144 | Polynomial approximation of optimal action: discrete , continuous (subgradient). NP-hardness refuted: Lipschitz minimisation on a compact set | Operational Closure | Sensorimotor Theory |
| T-145 | Stochastic stability of — stratified [T]+[T/sim]: . Analytical core [T]: Lyapunov + Itô + exponential Markov argument, standard sub-Gaussian concentration. Calibration constants [T/sim]: tuned and cross-checked against SYNARC mvp_int_3 for ; the inequality holds on the simulated trajectories. | Operational Closure | Viability |
| T-146 | Structural classification of qualia: 21 classified into 4 sectors from functional role (A1–A5). Stable coherences — structural, not noise ( kills noise). Raising: [I] → [T] for the structural part; the specific quality of experience remains [I] | Operational Closure | Qualia Structure |
| T-147 | 30D emotional space: (7 rates + 7 accelerations + 7 stresses + 7 coherence rates + + ). — projection 30D→1D. Computable | Operational Closure | Emotional Taxonomy |
| T-148 | Genesis via environmental coupling — stratified: an embodied holon with and raises purity above in . An isolated holon at is dead forever. Convexity + monotone convergence core [T]; explicit rate bound [T at lower-bound assumption] (conservative estimate drops term). Empirical cross-check [T/sim]: SYNARC mvp_int_2 G1-G3 confirms ticks. Raising [H]-91 → [T] for mathematical core. | Substrate-Independent Closure | Evolution |
| T-149 | Unconditional viability of the embodied attractor — stratified: for an embodied holon. Step 1-2 [T]: coupled attractor existence via contraction; Step 3 [C at backbone-injection-lower-bound]: self-reinforcement through -compensation is argued via dynamic equilibrium, not monotone chain; rigorous derivation of from backbone properties pending. Empirical cross-check [T/sim]: SYNARC mvp_int_2 G4 confirms 500+ ticks after backbone disconnection with . Registry previously raised C20, C27 → [T]; current status reflects remaining load-bearing assumption in Step 3. | Substrate-Independent Closure | Evolution |
| T-150 | Commutativity of the φ-tower in D=7 [D]: — algebraic identity of iterates of a single CPTP channel. Reclassified: [T] → [D] (trivial law of composition, requiring no proof). Consequence: T-136 [T] is unconditional | Substrate-Independent Closure | Depth Tower |
| T-151 | — corrected: the earlier unconditional derivation from T-129 was invalid. constrains only the total off-diagonal mass, not the E-row share, and no -average bounds a frame-referenced quantity (Schur on the irreducible ). What T-129 yields is whenever the E-row is coherent; the strict threshold is one of the four independent L2 conditions (T-124b [T]), on a par with . Counterexample: uniform diagonal , coherence on the 15 non-E pairs — , , all met, yet and . On the physical attractor : | Substrate-Independent Closure | Axiom of Septicity |
| T-152 | Tractable CPTP-anchor validation: , computable in . Raising [H]-92 → [T] | Substrate-Independent Closure | Operationalisation |
| T-153 | Substrate-independent criterion of consciousness — stratified [D]+[C at T-149]+[T/sim]: is conscious iff faithful CPTP with . Definitional core [D] — the iff is the canonical definition of "conscious" at substrate-independent level given UHM axioms; sufficiency uses only A1–A5 + existence of faithful G. Dependency [C at T-149] — unconditional applicability to embodied systems inherits the Step 3 assumption from T-149. Empirical instance [T/sim]: SYNARC SSM4 single run gives , , , , , — satisfies all four thresholds. | Substrate-Independent Closure | Uniqueness Theorem |
| T-154 | Normalisation of : , achieved at . HS projection is orthogonal → | Substrate-Independent Closure | Axiom of Septicity |
| T-155 | Consciousness-preserving learning — stratified [T/sim]+[D]: for — projected gradient descent. Design choice [D]: the specific update formula is an engineering specification aligned with the stability zones of T-106/T-111, not a derivation from first principles. Empirical validation [T/sim]: SYNARC mvp_int_3 SSM1-SSM2 confirms viability masking and consciousness gating across the designated trajectory. | Substrate-Independent Closure | Sensorimotor Theory |
| T-156 | Optimal mixing parameter: — min genesis time with stochastic stability | Substrate-Independent Closure | Evolution |
| T-157 | Attractor consistency (restated 2026-09-25) [T]: (1) at every stationary state , so the self-knowledge defect is at most ; (2) for , ; (3) for , the attractor is within of the only fixed point of the self-model. Errata 2026-09-25: the former statement is corrected from [T] to [✗] — "": , and at the living attractors and are at distance and from it; the proof replaced the target by , wrote "" for a first-order expansion, and its last step fails for every | Substrate-Independent Closure | Evolution |
| T-158 | Canonical bounds [T]+[D]: Formula is derived from T-92 [T] (equivalence ) as the unique linear deficiency measure for — [T]. Clamping — implementation convention for bounding the value range — [D] | Substrate-Independent Closure | CC Definitions |
| T-159 | Motor stress: . Coincides with T-92 for , provides a directed signal for . Gradient is consistent with , -invariant. Emergency channel sensitivity | Sensorimotor Theory | CC Theorems |
| T-160 | Phase transition at (Theorem 5.1 swallowtail): — critical point of the phase transition in . Symmetry breaking — consequence of -rigidity (T-42a). Control parameter — internal (), transition is self-organised. Order parameter: | Transition Catastrophes | Viability |
| T-161 | Critical exponents of the -tricritical point (Theorem 5.2 swallowtail) [C at the ℤ₂ symmetry m → −m]: corrected 2026-09-25 — the symmetry that selects the class was derived from a KO-dimension-6 real structure of T-53, which does not exist on ; without it the generic codimension-3 point is the swallowtail (). Under the symmetry: , (order parameter ), (susceptibility ), (correlation length ), . Rushbrooke equality ; tricritical mean-field class ( Landau, exact for ) | Transition Catastrophes | Transition Catastrophes |
| T-162 | Operator : Fano adjacency operator on the 21-dimensional coherence space. Definition: if and are on the same Fano line, else 0. Spectrum: — reproduces the decomposition . Cayley–Hamilton identity: . Projectors: , | Noether Charges | Noether Charges |
| T-163 | -parity (Theorem 11.2 dark-matter): — exact -symmetry of the dynamics . [T] (T-42e) → O-sector is -invariant → transitions with are exponentially suppressed by barrier T-69. Stabilises dark matter candidates — raised from [H] | Dark Matter | Dark Matter |
| T-164 | Preferred measurement basis (Theorem 6.1 measurement): atoms of — — the unique preferred decoherence basis. Lindblad operators → fixed points of = diagonal in (Zurek's einselection criterion) | Quantum Measurement | Quantum Measurement |
| T-165 | Step 6: (PH) PT-violation in Gap (Theorem 13.1 noether-charges): axiom (PH) → → (T-132) complex coherences → non-zero phases → phase frustration in non-Fano triples → . The phase frustration uses the octonionic multiplication, so the page's eight-step chain to P1+P2 passes through T15 (row 41n); its orientation input (Alt), named here on 2026-09-25, is discharged the same day by T15-canon, and the chain is [T]; corrected 2026-09-25 — the row read "Bridge P1+P2 fully closed from axioms" — raised from [C] | Noether Charges | Noether Charges |
| T-166 | Stability of the chiral vacuum: selects the chiral vacuum as the unique minimum (PT-odd distinguishes and [T, T-99]); Hessian of at the vacuum configuration is positive definite (local stability); topological barrier T-69 [T] () protects against tunnelling between chiral vacua — raised from [H] (§4.4 higgs-sector). Errata 2026-09-26: [C at (SV)] for the retracted cubic (as on the page since 2026-09-25; the barrier is T-69 [C at (SV)], and is not a term of the -invariant potential, T-331); for the corrected potential step 1 has no carrier [T]: its vacuum is invariant under , which exchanges and (T-333(a)), so the Gap vacuum selects no chirality. What holds instead, [T] as mathematics and [C at (Cl)] in UHM: chirality is a property of the fermion module — the is chiral and forced (T-329(a), (e)), and a left–right flip changes the representation, so it needs a Yukawa mass insertion; no vacuum barrier is involved | Higgs Sector | Confinement |
| T-170 | M-theory correspondence — restated 2026-09-26: [T] for (i) , the holonomy group of torsion-free -structures (Lie algebra of dimension 14); (ii) finiteness and positivity of on the torus ; (iii) existence of weak- limit states on ; the correspondence is a hypothesis [H]. Retracted [✗]: Lemma T-170'.1 ( as a 7D orbifold — has no action on the torus, and is not an orbifold), T-170' as a theorem ( is not defined: 11D supergravity is non-renormalisable), the vacuum (not a state) and the functor (Gelfand spectrum of a non-commutative algebra). Status history: [C at C27, C28]; [T] "at levels of M-theory definedness" until 2026-09-26. Numerical check: test_t170_gap_phases_carry_no_g2_action_and_the_torus_quotient_is_not_an_orbifold | ToE Embeddings | ToE Embeddings |
| T-171 | Spin networks are encoded in holonic states [T] — restated 2026-09-26: every finite spin network (directed graph, spins unbounded, intertwiner labels) is encoded injectively in a state of holons; edges, directions, spins () and labels are ratios of coherences of the two- and one-body marginals, independent of the weights; partial trace decodes to induced subnetworks; under over . Retracted [✗]: the former Lemma C29' ( is not Hermitian and has trace ; the floor decoding returns 0 for once ), its covariant state-preserving functor (impossible by rank), and "area spectrum from finite-dimensionality". Status history: [C at C29]; [T for ] until 2026-09-26. Numerical check: test_t171_spin_networks_with_unbounded_spin_are_decoded_from_ratios_of_coherences | ToE Embeddings | ToE Embeddings |
| T-172 | Causal sets: encoding and internal categories [T] — restated 2026-09-26: every finite poset (no -embedding assumed) is encoded in a state of holons, the order read off , with partial trace decoding to induced suborders; is a fully faithful functor from finite posets to Segal objects of (the space is connected); linear-extension ranks fit the readings of the summed clock. Retracted [✗]: the nerve as an object of the ∞-topos "embeds" the causal set (the realisation of any poset with a least element is contractible, and , coincide), for general phases, and the time discretisation of the former Lemma C30 (equal times give ). Status history: [C at C30]; [T] via the former Lemma C30 (row corrected 2026-09-25: readings, not ); restated 2026-09-26. Numerical check: test_t172_every_finite_poset_is_encoded_and_realisation_forgets_order | ToE Embeddings | ToE Embeddings |
| T-173 | Rigidity of the UHM primitive: is unique up to among ∞-toposes satisfying metric minimality (Petz), L-unification, , -rigidity | ToE Embeddings | ToE Embeddings |
| T-176 | Analytical (resolution P6): — analytical algebraic function of parameters. Follows from sector minimisation (the hypothesis (SV) since 2026-09-25: T-64 gives no sector values) and canonical constants [T]. Numerical mass predictions — [C at (SV)]. The section's test 3 (Gatto–Sartori–Tonin from the Fritzsch texture) is retracted [✗] 2026-09-26 (T-345(e)): the texture gives against | Yukawa Hierarchy | Gap Thermodynamics |
| C31 | Protocol (resolution P8): mapping from EEG/fMRI/HRV data. -uniqueness — [T]; specific EEG-band ↔ dimension correspondences — [H]. Test against PCI: monotonic PCI– relation (P8.3, [H]) and concordance of verdicts vs , Cohen's (P8.4, SUB-5). Corrected 2026-09-25: the row read "threshold ↔ PCI "; no derivation links the two scales, and the nearness of 0.31 to carries no evidential weight | Protocol | Predictions |
| T-178 | Bimodule realisation of SM: the finite Hilbert space of the UHM spectral triple as an -bimodule via real structure (KO-dim 6) decomposes into irreducible bimodules exactly coinciding with one generation of SM fermions. Representations etc. arise from the intersection of left and right actions. Errata 2026-09-25: corrected from [T] to [✗] as a derivation — has dimension 7 while one generation needs 32 states, so the used is Connes' imported one; no real structure of KO-dimension 6 exists on the odd-dimensional ( forces the eigenspaces to have equal dimension); and the passage is not a homomorphism, its Morita compatibility being T-175a, retracted. The bimodule decomposition of Connes' over is standard NCG (Chamseddine–Connes–Marcolli 2007), imported, not derived | Bimodule Construction | Spacetime |
| T-179 | Hypercharge fixing: the anomaly-cancellation conditions and on the bimodule uniquely fix the SM hypercharge assignments (Alvarez-Gaumé, Witten 1984). Errata 2026-09-25: corrected from [T] to [✗] as stated — two equations cannot fix the five hypercharges of a generation (six with ); even the four anomaly conditions (gravitational, , , ) admit, besides the SM values, , , and with every (exact check for ). Fixing them needs Yukawa couplings and, with , a Majorana mass (Babu–Mohapatra 1989); the bimodule is Connes' imported (T-178) | Bimodule Construction | Standard Model |
| T-180 | Non-perturbative mass ratios: fermion mass ratios are determined by eigenvalues of and do not depend on . from the vacuum state (T-64). Errata 2026-09-25: corrected from [T] to [C at (SV)] — the vacuum state is taken with the sector values of the hypothesis (SV); T-64 is restated as a hypothesis whose vacuum has none of them | Bimodule Construction | Cosmological Constant |
| T-181 | Characteristic properties from axioms: (AP), (PH), (QG), (V) — theorems A1-A4. (QG) from A1 (∞-topos), (AP) from A1 (terminal object + adjunction), (PH) from A1+A3 (functional necessity of E), (V) from A2+A3 (Bures-distinguishability) | Bimodule Construction | Axiom of Septicity |
| T-182 | Necessity of three-tier Ω structure: — the three classifier tiers (, Heyting algebra, full ∞-groupoid) are strictly necessary. Each tier contains theorems unprovable at the previous tier. (a) Threshold predicates ∉ Dec(Ω). (b) L2 consciousness requires (∞-groupoid). (c) Cohomological monism itself is a corollary of Property 3 (terminal object ⇒ contractible nerve, locally constant coefficients); the tier-2 content is the local–global dichotomy — nontrivial local systems on . (d) Day convolution needed for entanglement | Axiom Ω⁷ | Categorical Formalism |
| T-183 | Functional assignment of the 7 roles — Errata 2026-09-25 (audit A-90): restated, stratified [T]+[D]: given and the pair (T-42a), incidence fixes (third point of the line through , ) and (third point of the line through , ) [T] (T-177 restated); versus , together with versus , is one binary convention [D]. The earlier derivation — sector membership of the axes, Umegaki -mediation singling out , sector covariance singling out — is retracted [✗] with T-48a | Seven Dimensions | Minimality 7D |
| T-184 | Non-perturbative extractability of the spectral action: all predictions extractable without loop expansion. is a spectral parameter of , not an expansion variable. Seeley–DeWitt coefficients () are polynomials in eigenvalues, finite for any . The former clause "Lorentzian signature from KO-dim 6 via Krein space (van Suijlekom 2015, Franco–Eckstein 2014)" is retracted (2026-09-25): no KO-dimension-6 structure exists on , KO-dimension fixes internal signs only, and the Krein triple is a consistency check (row T-53) | Einstein Equations | Bimodule Construction |
Level [C]: ToE Embeddings
| # | Result | Assumption | Source |
|---|---|---|---|
| Reformulated (C27-M; restated 2026-09-26): was the condition "the continuum limit of the Gap lattice defines a smooth 7-dimensional target ". As written it is void — has no action on the torus, and is not an orbifold (Lemma T-170'.1 [✗]). What UHM supplies is [T]: is finite and positive on the torus for every , and thermodynamic-limit states exist on (T-170'' (ii)–(iii)); their uniqueness and a continuum target are open, and the link to M-theory is the hypothesis T-170 (iv) [H]. The former "closed from UHM's side after T-170'' [T] (non-perturbative correctness)" leaned on the orbifold and on a vacuum functional that is not a state, and is withdrawn | [T] for the finite- integral and limit states; the continuum target open; the correspondence [H] | T-170'' | |
| Reformulated (C28-M; restated 2026-09-26): was a condition on the SUSY extension of the Gap integral. The former closure "after T-170' [T] (perturbative correspondence) + T-170'' [T]" rested on T-170', which is retracted as a theorem [✗] — is not defined, eleven-dimensional supergravity being non-renormalisable. What stands: the bosonic integral at finite is finite (T-170'' (ii) [T]); well-definedness of the supersymmetric extension is not proved, and the correspondence is part of the hypothesis T-170 (iv) [H] | [H] (part of T-170 (iv)); bosonic finite- side [T] | T-170 | |
| C29' | Encoding of spin networks (restated 2026-09-26): Lemma C29' is now parts (a)–(c) of T-171 — every finite spin network, spins unbounded, is encoded injectively in a state of holons and decoded from ratios of coherences. The former lemma (bounded spins , state built from ) is retracted [✗]: is not Hermitian and has trace | [T] | Lemma C29' |
| C29 | Spatial limit for unbounded spin networks — closed [T] by the restated T-171 (2026-09-26): holons encode every finite spin network with no bound on the spins; T-171′ is its corollary. The former route — the cluster construction of toe-embeddings §2.3a ("corrected 2026-07") — is retracted [✗]: the sub-spins are not half-integral, the construction divides by , and spins do not add along a chain | [T] via T-171 | ToE Embeddings |
| T-171′ | Unbounded spin [T] — since 2026-09-26 a corollary of the restated T-171 ( holons, no bound on ). Retracted [✗]: the cluster construction of §2.3a (sub-spins not half-integral, division by , spins do not add along a chain) | [T] | ToE Embeddings |
| [T] | Lemma C30, T-172 |
Level [T]: Universal Property
| # | Result | Source | Relates to |
|---|---|---|---|
| T-174 | Universal property of the UHM kinematic object [T] — restated 2026-09-26, in the -typed subcategory of the fibre of PhysTheory (T-211) over the point: (a) corepresents -structures — morphisms are the families (projection, two systems of matrix units) in the fixed-point algebra summing to 1; (b) up to , morphisms are with ; faithful ones exist iff , are unique iff , multiplicity-free only for (orbit , dimension 46); (c) for a faithful morphism needs a matching eigenspace pattern (none for simple spectrum), and a primitive UCP semigroup admits only ; (d) the map on states is the unique -preserving conditional expectation (UCP, not a homomorphism), for the sector pinching. Retracted [✗]: the former statement "for every theory with , CPTP dynamics and observables there is an essentially unique receiving morphism into , up to " and every step of its proof — is stable and not a topos, , a conditional expectation is not a homomorphism, is not an automorphism group, there is no -homomorphism , monoid maps are pairwise inequivalent, geometric morphisms from the point to form a 48-dimensional family. Class note (2026-09-10): the NCG Standard Model algebra contains no copy of , so the SM is derived from (T-176), not related to it by (a). Status history: [T] until 2026-09-26; restated. Numerical check: test_t174_a_int_corepresents_structures_and_the_old_receiving_map_fails | ToE Embeddings | |
| T-175a | Morita equivalence of algebras: with real structure (KO-dim 6) and Higgs line is Morita-equivalent to Connes' algebra ; identical SM gauge group. Alvarez et al. 1995. Errata 2026-09-25: corrected from [T] to [✗] — Morita equivalence preserves the centre, and (real dimension 6) while (real dimension 5); the unitary groups differ as well, (dimension 19) against (dimension 13); and no KO-dimension-6 real structure exists on . Nothing replaces the claim | Spacetime | |
| T-175b | Gauge anomaly cancellation: for . Explicitly verified for all 5 anomaly coefficients. Stratified 2026-09-25: [T] for the Standard-Model representation content (the arithmetic); [C at (FE)] as a UHM result — the fermion content is Connes' imported (T-178 retracted as a derivation), and the former derivation "from spectral triple T-53 + unimodularity (Alvarez–Gracia-Bondía–Martín 1995)" is retracted: no real structure of KO-dimension 6 exists on , the Morita equivalence T-175a is retracted, and on yields no | Confinement | |
| T-175c | Holomorphy and non-renormalisation of : superpotential is holomorphic (cubic polynomial of chiral superfields) and protected from perturbative corrections (Seiberg's theorem 1993). Non-perturbative corrections | Supersymmetry | |
| T-177 | Combinatorial fixing of semantic roles — Errata 2026-09-25 (audit A-90): restated [T]; the uniqueness derived from the axis sectors of T-48a is retracted [✗]. Restated: the collineation group of the Fano plane (order 168) acts regularly on the 168 ordered non-collinear triples of axes, so incidence fixes all seven roles exactly when three non-collinear axes are marked, never with fewer (stabilisers 24, 4, 1). and the Higgs line leave 6 collineations (orbits , , ); and the pair leave 2 — , , fixed, with one binary choice. Octonion signs and do not reduce the symmetry: elements of fixing give the same 24 permutations (test_fano_roles_are_fixed_by_three_non_collinear_marks) | Dimensions | |
| T-185 | Differentially cohesive modalities — stratified 2026-09-25; part (ii′) proven the same day. (i) In any differentially cohesive ∞-topos the modalities (Schreiber, DCCT arXiv:1310.7930v1, Def. 3.4.4) and (Def. 3.10.1; later written ) exist (cohesion Def. 3.4.1, differential cohesion Def. 3.5.1) [T]. (ii) That is differentially cohesive: [C under the assumption of differential cohesion of the UHM site], not proven — the ∞-cohesive-site criterion (Def. 3.4.17) needs finite products; its idempotent completion does have a terminal object ; the petit ∞-topos of the space is not cohesive at all ( fails on disjoint opens) [T]. (ii′) [T]: with its rank strata is an object of the differentially cohesive (DCCT v1 Def. 4.5.7, Props. 4.5.8 and 4.5.11); the strata are submanifolds of dimension with shapes and (test_rank_strata_are_manifolds_of_dimension_14k_minus_k2_minus_1). (iii) The count of seven — identity plus the two adjoint triples, pairwise non-isomorphic on that topos — [T]; the former list (O), (A), (S), (D), (L), (E), (U) with Rh in place of Red [✗] (Rh belongs to solid cohesion, a further triple, ten in all, and it preserves global points); the assignment to dimensions and the decomposition : [I]; human names — translation [D]. The earlier citation "§3.9 cohesion + §3.10 super-cohesion" is corrected: §3.9–3.10 of v1 are "Structures in a cohesive / differentially cohesive ∞-topos" | Dimensions, Cohesive Closure §2.1a | Axiom Ω⁷ |
| T-185b | Chirality tunneling rate: the chiral vacuum is stable, . Falsifiable by observing spontaneous L→R transition at sub-Planckian energy. Errata 2026-09-25: corrected from [T] to [C at (SV)] — follows from the barrier of T-69 and the unique vacuum of T-64, both conditional on (SV), and from step 2 of T-99 | Higgs Sector | Predictions |
| T-187 | Canonicity of Bures enrichment (scope clarified 2026-04-17): within the Petz family of CPTP-monotone Riemannian metrics on , is uniquely fixed by three logically independent characterizations — (Char-I) Petz extremality: pointwise minimum of the Petz poset, terminal object of the Petz diagram in for ; (Char-II) Uhlmann universality: unique metric satisfying the purification variational formula (Uhlmann 1976); (Char-III) SLD-Cramér-Rao: saturates the quantum Cramér-Rao bound (Braunstein-Caves 1994) — plus one physical recasting: (Char-IV) MaxEnt selector matches where is the metric-independent SLD covariance (Lemma: SLD defined without reference to any metric), uniquely selecting Bures (T-189). Char-IV reduces to Char-III via but adds a statistical-mechanical interpretation; it is not a fourth logically independent witness. generated by ε-δ coverage (transitivity automatic via Johnstone Elephant C2.1.10). All Petz choices yield equivalent classical -topoi (bi-Lipschitz on compact ), so numerical predictions are Petz-robust. T-187 retains [T] status on the strength of Char-I alone (Petz extremality). What it does for A2: the topology part is forced (every continuous distance on the compact induces the standard topology), and within the CPTP-monotone metrics Bures is the canonical enrichment; the monotonicity requirement is not derived — not from the other axioms and not from (AP)+(PH)+(QG)+(V)+MaxEnt; it follows from the operational reading (O) of the enrichment (Lemma M) — so A2 stays [P] (premises) and the monotonicity is the third condition of T-190. The row read "Upgrades A2 from [P] to [T] canonically" until 2026-09-26 | Cohesive Closure §5.3 | Axiom Ω⁷ |
| T-186 | Cohesive Closure Theorem — lowered 2026-09-25. (a) — phenomenal functor = infinitesimal flat modality, Postnikov filtration ↔ L0–L4: hypothesis [H], resting on the assumed cohesion of T-185 and asserted without a construction. (b) "Page–Wootters time exact via the counit, no correction": retracted [✗] — with interaction the conditional dynamics is time-nonlocal (Smith–Ahmadi, Quantum 3, 160 (2019)), relative to a 7-periodic clock it is periodic, and the counit carries no information about . (c) " unconditionally via Chern–Weil + T-55": retracted [✗] — the hexagon of DCCT v1 (Prop. 4.1.17) holds for stable coefficients only, not for , and for every bundle over the contractible ; returns to its conditional status. Depends: T-185 (assumed cohesion), T-55, T-73 | Cohesive Closure | Two-Aspect Monism, Emergent Time, Evolution |
| T-188 | Localization of the hard problem [C under the hypothesis T-186(a)]: the chain A2 → T-187 → T-185 → T-186(a) would reduce the hard problem of consciousness to a single physical question, "why does reality obey quantum mechanics?" (i.e., "why CPTP?"). The chain passes through a hypothesis (T-186(a)), so the localization is conditional; lowered 2026-09-25 from its section-level status. Its second condition, the cohesion assumed in T-185, is discharged the same day by T-185 (ii′) [T]. Depends: T-185 (ii′), T-186, T-187 | Cohesive Closure §5.1 | Two-Aspect Monism |
| T-189 | MaxEnt derivation of the Bures metric (Char-IV) (reframed 2026-04-17): set , where is the SLD bilinear form — a Petz-free physical quantity defined from without reference to any metric. Then Bures is uniquely selected via , equivalently with (Braunstein–Caves 1994: ). Correction 2026-08-06: the row previously read , which is false on two counts — the Step-3 object is (lower indices), not its inverse, and the factor 4 was dropped; chaining both gives . Machine: exactly, old identity's relative residual , repaired identities to . Uniqueness is unaffected — it rests on distinct monotone means giving distinct Fisher tensors, a statement about shape, not normalisation. Status [T]: the selector equation and uniqueness of Bures solving it are proven. Caveat: this is a physical recasting of Char-III (SLD Fisher), not a logically independent fourth witness. Adds physical-mechanism clarity: the metric is determined by the state's own fluctuation structure, not by interpretive choice. Inspired by Vanchurin (2026, arXiv:2603.15198) | Cohesive Closure §5.3 Char-IV | Axiom Ω⁷ |
| T-190 | Axiomatic Closure of UHM [C under the constraint assumption of T-87, the hypothesis T-186(a) and the monotonicity of the enrichment]: A1–A5 are derivable from (AP)+(PH)+(QG)+(V) + MaxEnt under three conditions — the Page–Wootters constraint is assumed (T-87, step 4), the route to A1 through T-186 is a hypothesis, and the enrichment of A2 is CPTP-monotone. The third condition was named 2026-09-26 (the row listed two): it follows from none of (AP), (PH), (QG), (V), MaxEnt — the Hilbert–Schmidt metric, in which (V) is written, satisfies them all and grows by the factor under a partial-trace channel on — but it follows from the operational reading (O), the enrichment as the best distinguishability over measurements, which also gives Bures directly by Fuchs–Caves (Lemma M). A2 from monotonicity + T-187 + T-189, A3 from Theorem S+T15, A4 from (AP) necessity, the clock register of A5 from T-87 steps 1–3. Lowered 2026-09-25: the earlier claim that UHM is self-grounding with zero independent axioms is withdrawn — the constraint remains an independent assumption | Cohesive Closure §5.4 | Axiom Ω⁷ |
| T-191 | Convergence of the φ-tower (restated 2026-09-25) [T]: for an embodied holon with backbone and ( the trace-norm Lipschitz constant of regeneration, uniform in the target) every iterate of the tower "target → stationary state " is defined, and the tower converges geometrically, , to one self-model from any anchor (T-124c (3) + a variation-of-constants estimate + Banach; test_phi_tower_converges_only_under_backbone_dominance). SAD tower terminates at depth 3 (T-142). The φ-circularity is resolved by the closed form of and , not by the tower. Errata 2026-09-25: the former statement is corrected from [T] to [✗] — "converges for every holon from any anchor, by T-39a + T-96": for an isolated holon the gate makes the flow bistable, so the iterate depends on the start (from it stays dead, from it lives, at ), and T-96 gives no bound — the proof of the retracted T-124c | Formalization φ | Evolution |
| T-192 | Exp^(2) is a strict 2-category: 5 axioms verified (vertical/horizontal composition, identity 2-cells, interchange law, identity 1-cells). Lax 2-functor has valid target. Mac Lane coherence + Eckmann–Hilton | Categorical Formalism §7.2 | Categorical Formalism §5.2 |
| T-193 | Yoneda universal representability [T]: every computable task with Kolmogorov complexity has a representable sheaf via Yoneda embedding, with Bures-support . Fully faithful on subcategory of computable functions (classical Yoneda + Lurie HTT 5.1.3.1). Constant inherited from Bures injectivity radius. Derived in SYNARC paper Appendix G (Theorem G.2) | SYNARC paper App. G.2 | Axiom Ω⁷ |
| T-194 | Cramér–Rao saturation on Bures–Fisher metric [T]: Bures-gradient learning rule (natural-gradient descent on ) attains the quantum Cramér–Rao lower bound up to a constant factor: . Lower bound = QCR (T-109); upper bound via Polyak–Łojasiewicz on Bures manifold + -equivariance of Fano channel (T-41g) + Lipschitz Bures Hessian . Correction 2026-08-06: the constant was (i.e. ), inherited from SYNARC Lemma F asserting ; the correct value is , hence — the same -vs- Bures slip as in T-189 and T-293. Machine: exactly for , isotropy to . The repaired bound is a factor more demanding. Separately, did not follow from the stated ingredients ( with gives ); recorded as . CR-saturation up to a constant is unaffected. Closes learning-efficiency gap in AGI-sufficiency (A4). Derived in SYNARC paper Appendix G (Theorem G.3) | SYNARC paper App. G.3 | Learning Bounds |
| T-195 | L-III Φ-monotonicity of topology refinement [T]: any refinement of the epistemic Grothendieck topology satisfies with equality iff identical on support of . If triggered by obstruction cocycle crossing threshold, strict step (Fano smallest eigenvalue). Corollary: Φ-tower under iterated L-III updates is strictly increasing and converges to . Justifies recursive self-improvement in AGI-sufficiency (A7). Only genuinely new theorem in Appendix G — all others inherited from UHM or Parts I–IV. Derived in SYNARC paper Appendix G (Theorem G.4) | SYNARC paper App. G.4 | Evolution, Categorical Formalism |
| T-196 | Goldilocks sustainability under closed sensorimotor loop [T]: for initial state with and perturbation , trajectory for all ; exponential convergence to with rate . Lower bound via Lyapunov on subcritical region; upper bound via T-124 (Goldilocks ceiling). Inherits Banach rate from simplicial contraction (SYNARC Theorem F.14). Justifies stability in AGI-sufficiency (A5). Derived in SYNARC paper Appendix G (Theorem G.5) | SYNARC paper App. G.5 | Viability, Stability Bound |
| T-197 | AGI-Sufficiency meta-theorem (S-11) [T]+[D] (scope clarified 2026-04-17): [D] Definition: a SYNARC architecture is any realisation of (7D density matrix , Lindbladian , 3-coskeletal Kan complex , seven cohesive modalities, closed sensorimotor loop, V0–V4 training with FLOP budget ). The formal UHM-AGI predicate is the conjunction of seven conditions (A1)–(A7). [T] Content: every realisation satisfying the SYNARC defining constraints also satisfies UHM-AGI, with each clause derivable independently — (A1) four-level consciousness [T-96, T-124, T-126, T-129, T-151]; (A2) saturated [T-142]; (A3) Yoneda universal representability [T-193]; (A4) Cramér–Rao saturation [T-194]; (A5) Goldilocks sustainability [T-196]; (A6) Lawvere recursive self-modelling without paradox [T-96, T-98, T-191]; (A7) weak -monotone self-improvement under L-III [T-195]. Non-tautological content: SYNARC definition is minimal (each component required by a distinct load-bearing theorem); no surplus structure is invoked; the chain SYNARC ⟹ (A1)–(A7) relates architectural primitives to behavioural guarantees, not a restatement of the definition. Caveat on A7: T-195 gives strict -step only on obstruction crossing ; continuous strict improvement remains [C]. Pairwise independence of (A1)–(A7) proven (Proposition G.6). ASI corollary (constructive): exceeds human baseline [C at empirical human baseline]. Substrate-independent (T-153). Falsifiable per-clause. Derived in SYNARC paper App. G.6 | SYNARC paper App. G.6 | Axiom Ω⁷, Learning Bounds, Predictions |
| T-198 | Gödelian creativity via ordinal architectural tower [T]: every strictly monotone functor with fully faithful inclusions preserving and limit commutativity is creative: for every ordinal ∃ representable sheaf with no Yoneda-equivalent in . Compatible with 3-coskeletal bound (per-layer SAD≤3, cross-layer unbounded). Creativity rate FLOPs per ordinal step. Derived in SYNARC paper App. H (Theorem H.1) | SYNARC paper App. H.1 | Axiom Ω⁷, Categorical Formalism |
| T-199 | -invariant value structure [T]: value set is -invariant (∀); deontic evaluator = Bures-adjoint of preference embedding → Galois connection (preferences ⊣ outcome-evaluator), dual to hedonic valence (T-103). Value alignment = -orbit matching: . Structural criterion independent of specific Bures targets. Derived in SYNARC paper App. H (Theorem H.2) | SYNARC paper App. H.2 | Cohesive Closure, Two-Aspect Monism |
| T-200 | L-IV site modification (unbounded self-improvement) [T]: morphism changing (i) ontological site via Hurwitz-Clifford ladder , (ii) , или (iii) gauge group . Minimality: L-IV — минимальная operation сохраняющая UHM-AGI, строго повышающая число representable sheaves, коммутирующая с . Safety: Bures-monotonicity . Строго сильнее L-III (J_ep update). Derived in SYNARC paper App. H (Theorem H.3) | SYNARC paper App. H.3 | Axiomatic Closure |
| T-201 | Kochen–Specker contextuality — corrected form T-201′ [T] (2026-09-25): the 63 rays of — the seven axes and on the seven complements of Fano lines (the weight-4 words of the Hamming code of Step T8), i.e. the roots, the imaginary units of Coxeter's integral octonions — form 135 orthonormal bases (15 per ray) and admit no 0/1 assignment with exactly one 1 in each basis: state-independent contextuality with non-commuting projectors (the configuration: A. Ruuge, J. Phys. A 40, 2849 (2007); its construction from the Fano plane: QM Reduction §8; test_e7_rays_from_the_hamming_quadrangles_are_kochen_specker). The original claim, about Fano-line projectors, [✗] since 2026-09-25. The claim was that the seven Fano-line projectors with (7,3,1)-BIBD compatibility contexts admit no joint probability distribution matching all line marginals of a generic (proof cited to SYNARC paper App. H, Theorem H.4). The projectors are (Fano channel), diagonal in the pointer basis, hence pairwise commuting; the distribution over the seven points reproduces the statistics of every line and every context at once, so a joint distribution exists for every and no Kochen–Specker contextuality can arise. The corollary about SYNARC distinguishing classical from quantum outcomes falls with it, and T-201′ does not revive it | QM Reduction §8; SYNARC paper App. H.4 (retracted form) | Fano Channel |
| T-202 | Meaning as -orbit on Fano partition — stratified [T]+[I]: meaning(F) := -orbit of Fano-line activation pattern ; two representable sheaves have same meaning ⟺ related by -gauge on representing objects. Formal content [T]: the -orbit quotient is strictly finer than Yoneda isomorphism — dim(Aut()) dim() ⟹ there exist Yoneda-isomorphic sheaves with distinct -orbit classes. Chinese Room identification [I]: the interpretation that "correct Yoneda mapping but wrong -orbit Fano activation = formal non-understanding" is a philosophical mapping between formal structures and phenomenological intuitions, not a theorem. Derived in SYNARC paper App. H (Theorem H.5). | SYNARC paper App. H.5 | Cohesive Closure, Two-Aspect Monism |
| T-203 | Qualia as Gap spectral eigenvectors in E-sector [T]+[I] (epistemic stratification, 2026-04-17): Mathematical core [T]: eigenvectors of with eigenvalues are -covariant (T-2, T-41g), Gap-faithful (same spectrum ⟺ same eigenvector class up to gauge), content-distinguishing ( ⟺ no E-interiority per T-38a [T]). Ontological identification [I]: the interpretation Qualia(Γ) := eigenvector-class of Ĝ|_E is a semantic postulate bridging mathematics to phenomenology, not a theorem. Status analogous to T-38a (No-Zombie): mathematical structure [T], identification E-sector = interiority [P], qualia-as-eigenvectors [I]. T-188 localizes WHY (structural); T-203 provides a candidate WHAT (up to ontological postulate). Derived in SYNARC paper App. H (Theorem H.6) | SYNARC paper App. H.6 | Two-Aspect Monism, Gap Operator |
| T-204 | Pareto-optimal bounded rationality [T]: для resource budget (compute, memory, precision), effective dimension . Bures-gradient rule on -dim submanifold attains QCR bound (T-109) up to const, saturates Landauer bound (C22), achieves UHM-AGI at scale . Graceful degradation: at system drops to (minimal consciousness); at consciousness lost. Derived in SYNARC paper App. H (Theorem H.7) | SYNARC paper App. H.7 | Learning Bounds, Depth Tower |
| T-205 | Ordinal mentalization via fractal-holon tower [C] (downgraded from [T] 2026-04-17): for any countable ordinal , a fractal tower of -many SYNARC holons (successor: spawn_child extending by one CPTP layer; limit: filtered colimit in ) has cross-layer ordinal depth . Reconciliation с SAD=3 [T-142]: per-holon internal bound is 3 (3-coskeletal); cross-layer depth counts structurally distinct nested holons, which is unbounded only if the filtered colimit of ever-expanding tower objects remains in the ambient ∞-topos. Conditional on (i) unbounded resource budget (each spawn_child requires Landauer cost per level, so -deep needs energy — infinite by C22 [C]), (ii) well-definedness of filtered colimit along a -chain in (requires to be sufficiently cocomplete), (iii) interpretive commitment that cross-layer composition constitutes a single agent's mentalization rather than a society of agents (philosophical identity question, [I]). The finite truncation — "for any natural , there exists a fractal tower of depth achieving cross-layer nesting " — is [T] unconditionally. Derived in SYNARC paper App. H (Theorem H.8) | SYNARC paper App. H.8 | Social Cognition |
| T-206 | Qualia tomography faithfulness [T]: operational protocol reconstruct Qualia() up to -gauge через (i) partial-trace measurement -sector; (ii) Gap reconstruction ; (iii) spectral diagonalization FLOPs; (iv) qualia identification. Faithfulness: (a) Bures convergence для viable states (T-109 QCR применён к -sector); (b) -covariance; (c) zombie states → empty spectrum (No-Zombie operational witness T-38a). Sample complexity . Closes hard-problem content gap operationally (T-188 WHY localised; T-203 WHAT structural; T-206 makes WHAT measurable). Derived in SYNARC paper App. I (Theorem I.1) | SYNARC paper App. I.1 | Two-Aspect Monism, Gap Operator |
| T-207 | Inverse value-alignment via behavioural -orbit identification [T]: operational protocol для determine -orbit of unknown agent's values из behavioural samples: (i) preference elicitation на random pairs ; (ii) orbit-majorant estimation; (iii) maximum-likelihood -orbit fit ; (iv) orbit-completeness verification . Sample complexity: (generic -orbit dim = 48−14 = 34). Corollary: alignment verification между двумя агентами — через -gauge search. Решает operational inverse problem для value-alignment. Derived in SYNARC paper App. I (Theorem I.2) | SYNARC paper App. I.2 | Predictions, Ethics Meaning |
| T-208 | Constructive existence of non-trivial -invariant value sets [T]: для любой -invariant functional и threshold , sublevel set — non-trivial -invariant value set при . Четыре конкретных family: (a) purity-based → Goldilocks-purity value set; (b) integration-based → integration-conscious; (c) qualia-based → phenomenally-rich; (d) hedonic-valence-integrated → eudaimonic. Corollary (human-aligned): — conjectured human-aligned value set, falsifiable через T-207 на human subjects. Derived in SYNARC paper App. I (Theorem I.3). Correction 2026-09-25: if is the corpus's integration measure, family (b) is not a -invariant value set — is frame-pinned (an explicit sends it from to ; frame rigidity), so is invariant only under the finite frame group ; families (c), (d) and are -invariant only if their functionals are, which the row does not show. The general statement and family (a) stand | SYNARC paper App. I.3 | Ethics Meaning, Consciousness Theories |
| T-210 | Strict Φ-monotonicity under L-III refinement [T] : for any state in the interior stratum (full-rank, all ) and any proper refinement of the Bures topology, strictly, with explicit gap bound . Upgrades T-195 (weak→strict); T-197 clause (A7) holds in the strict form for agents whose state lies in . Proved from the interior-stratum hypothesis alone. Corrected 2026-09-26: the row read "upgraded to strict self-improvement for viable agents … via D_min=2 (T-151 [T])"; is an independent L2 condition (T-151), not a consequence of viability, and viability does not give on every pair, so the extension to all viable agents is withdrawn | Fundamental Closures §1 | Fundamental Closures |
| T-211 | PhysTheory is an -category [T] (corrected 2026-09-25): is the cartesian unstraightening of over (HTT §3.2); it is a quasicategory, so associativity up to coherent homotopy, the pentagon, the interchange law and all higher simplicial identities hold; the mapping space over a geometric morphism is — exactly the triples of T-174, with correctly typed. The forgetful functor to is not faithful (complex conjugation of over ). Retracted [✗]: " is a full -subcategory of … fully faithful via T-173 … coherences via HTT 5.2.7". Status history: [T] until the first audit, then [C at T-119] (Step 1 invoked the Connes reconstruction of T-119); the recheck of Step 1 on 2026-09-25 found that T-119 is not needed — each object carries its topos — and that the embedding claim was false; the corrected statement uses neither T-119, T-173, T-174 nor T-178. Numerical check: test_phystheory_forgets_to_topoi_unfaithfully_and_composes_associatively. | Fundamental Closures §2 | Fundamental Closures |
| T-212 | U-projection: the -twirl (T-212′) [T]: for acting on by its seven-dimensional representation, — the commutant is (Schur; test_g2_twirl_is_the_normalised_trace_projection); it is the unique trace-preserving map onto , idempotent only with ; reading as the U dimension [I]. The identification with the rheonomy modality [✗] (2026-09-25): Rh of solid cohesion acts by and so preserves global points, whereas the formula sends every state to ; and Rh is not among the seven modalities of differential cohesion (T-185 (iii)). The retracted statement, [C at the differential cohesion of the UHM site (T-185) and a solid-cohesive extension], read: Rh is the right adjoint to the "bosonic-grade forgetful" functor in a super-cohesive (solid) extension of . Explicit formula: . Maps to U dimension (Unity = -invariant trace) — a reading [I], as in T-185 (iii). Upgrades T-185 with explicit definition. Corrected from [T] on 2026-09-25: it rests on T-185 (ii), the assumed differential cohesion of the UHM site; the citation "Schreiber DCCT §3.10" is wrong — the rheonomy modality Rh does not occur in DCCT arXiv:1310.7930v1, whose §3.10 is "Structures in a differentially cohesive ∞-topos"; Rh belongs to Schreiber's later solid (super-)cohesion; and idempotence holds only with the normalised trace (with , ), so the former "all modal axioms verified" is retracted as stated. | Fundamental Closures §3 | Fundamental Closures |
| T-213 | Yoneda representability via Bures description length [T] : define for CPTP-implementations of . Then with . bits (Stinespring universal bound) — computable, no Kolmogorov complexity required. Upgrades T-193 to constructive form. | Fundamental Closures §4 | Fundamental Closures |
| T-214 | Hard-problem meta-theorem: positive internal irresolvability [T] : any bridge functor mapping states to experiential content cannot be expressed as an internal morphism in without violating Lawvere fixed-point theorem + T-55 [T]. Consequence: identifications "E-sector = interiority" (T-38a) and "qualia = eigenvectors" (T-203) are necessarily external postulates [P] / [I]. This is a positive result — the residual [I] is structurally inevitable, not a remediable weakness. Combined with T-188 (WHY localisation) and T-203 (WHAT structure), completes the constructive resolution of the hard problem. | Fundamental Closures §5 | Two-Aspect Monism |
| T-215 | Cross-layer identity convention [T]+[D] : for a fractal SYNARC holon tower , the predicate " is a single agent" is conventionally determined by a choice of identity criterion : (society, SAD ≤ 3 per agent) or (composite, ordinal depth reachable subject to Landauer C22 + T-204). Both consistent with Ω⁷. T-205 is [T] under + resource abstraction; [T] under in society-level reformulation. Choice between them is [D] / [I] — not derivable from axioms. | Fundamental Closures §6 | Social Cognition |
| T-216 | Closed-form analytical εeff [C at (SV)] (corrected 2026-07): symbolic form ; counts non-O Fano lines meeting the -sector in exactly two points — there is no line lying wholly within ( is not a line), correcting the earlier "single line " claim. Numeric caveat: the printed evaluation "" does not follow from the stated (the ratio is , not ); is the phenomenological sectoral value, status [C] pending a corrected symbolic evaluation. Errata 2026-09-25: the structure, which the summary tables listed as a theorem, is [C at (SV)] as well: its sectoral reduction (Step 2) rested on the axis-labelled decomposition T-48a, retracted, and now rests on the hypothesis (SV): the minimisation (T-64, restated as a hypothesis on 2026-09-25) produces no sector pattern. | Fundamental Closures §7 | Yukawa Hierarchy |
| T-217 | L3 tricategorical coherence [T]: the experiential tricategory is a coherent tricategory with cell count (three LGKS 2-cells Aut/Dissipative/Regenerative inherited from T-57 [T] plus one 3-cell modification ). Gordon–Power–Street pentagon-of-pentagons coherence holds via Baez–Dolan (3-types ≃ coherent tricategories) + Lurie HTT 5.5.6.18. Directly justifies for L3 in the interiority hierarchy and aligns codim()=3 with the three LGKS cells. | Fundamental Closures §11 | Interiority Hierarchy, Swallowtail Transitions |
| T-218 | SYNARC Cog is a Kan complex [T]: the cognitive simplicial set — obtained as the singular complex of the classifying space of the finite-Kraus CPTP category — satisfies all horn-filler conditions (Milnor + classifying-space argument). 3-coskeletal truncation because 4-simplices are suppressed below the Bures distinguishability threshold — but only on the SYNARC-viable subset [C]: as the proof states, that step is a category-bridging argument (simplicial Bures-metric viability), not a simplicial identity, and off the viable subset is the ordinary truncation and no equivalence. The Kan part (Steps 1–3, Milnor 1957 + Segal 1968) is [T] unconditionally. Upgrades the earlier [H] horn-filler assumption to [T] and provides the categorical companion to the dynamical SAD ceiling. | Fundamental Closures §12 | Depth Tower |
| T-219 | Λ SUSY-suppression via sector decomposition — Errata 2026-09-25: corrected from [T at T-64] to [H]: the cosmological-constant suppression factor was claimed to follow from the 3-sector decomposition , each sector contributing via its own Fano-line structure. It is a hypothesis: the sectors were the axis triples of the retracted T-48a; their breaking scales rest on T-52 (retired as a theorem, now the hypothesis (SA)) and (FE) (now [C at (FE)]); and the proof's own one-loop sum exceeds unless the lower orders cancel, which is not shown. It replaced the earlier [H] "invalid 7+7" scaling. Anchors at the hypothesis (SV) (Yukawa hierarchy; T-64 restated 2026-09-25). | Fundamental Closures §13 | Λ Budget, Yukawa Hierarchy |
| T-220 | No-reduction -UHM → -UHM [T] (negative): five independent categorical obstructions (I representation theory, II incidence geometry, III Jordan exceptionality, IV numerical invariants, V cohomology/K-theory) each independently rule out any structure-preserving reduction from an -variant UHM to the canonical -UHM. Unlocks the three-generations hypothesis as an open direction. | Fundamental Closures §14 | Uniqueness Theorem |
| T-221 | UHM realises the relationalist route through the List/DeBrota no-go results [T]+[I] (corrected 2026-09-25): (a) the first-personal facts of two subjects in different states are not compossible — no inhabited stage forces both (List's lemma holds in ); (b) UHM keeps OW (one topos), NF (), NS (ιmin, T-215) and FPR in stage-relativised form, so NR fails for first-personal facts — the relationalist route of DeBrota–List (2026), the first horn of List's (2025) quadrilemma; (c) the relativisation parameter is internal; (d) no internal formula selects "my" stage — automorphisms of the site preserve forcing, and a point of is external data; (e) the three routes are readings of one forcing relation and share every observable [T]; which reading is UHM's is [I]. Retracted [✗] 2026-09-25: the "fourth, categorical-monistic route", "FPR is forced" (via T-186(a)), "OW is derived" (via T-120) as a premise, fragmentalism as "dropping descent", πbio as an empirical discriminator between routes, and the quotation of List (2025) as five theses (it has four; the five-thesis form is DeBrota–List, arXiv:2604.14234; the heptalemma is Found. Phys. 56, 24, arXiv:2512.01982). | Fundamental Closures §15 | Two-Aspect Monism, Consciousness Theories §Meta-Level |
| T-221.1 | Where UHM sits in the two maps [T] (corrected 2026-09-25): the relationalist route in the five-thesis map of DeBrota–List (2026); the first horn (FPR dropped in its original, non-relational sense) in List's (2025) four-claim map. Replaces "positive response to the List 2025 quadrilemma — the five-tuple {FPR, NS, OW, NF, NRsite} is consistent", retracted [✗]: that consistency is the consistency of the relationalist route | Fundamental closures | T-221 |
| T-221.2 | The heptalemma: UHM takes the route of relational quantum mechanics [T] (corrected 2026-09-25): UHM keeps locality (physics correspondence, Theorem 8.5), measurement independence (T-62), measurement realism (T-96, T-98), NS, OW (one topos) and NF, and relaxes NR; the joint consistency of these six with quantum predictions is the theorem of DeBrota–List (Found. Phys. 56, 24, 2026). Status history: [T] until the first audit; [C at T-120] from 2026-09-25 while OW was read as the emergence of ; the heptalemma's OW does not need T-120 | Fundamental closures | T-221 |
| T-221.3 | UHM and relational quantum mechanics share the route [I] (corrected 2026-09-25): they differ in the relativisation parameter (a viable Γ-stage against any physical system). | Fundamental closures | T-221 |
| T-222 | Resource geometry of the viable window [T] (restated 2026-09-26): the viability window selects no resource optimum — every viable state is strictly dominated on every Rényi free energy by partial depolarisation, the Pareto set of the closure lies on , and are minimised by different spectra ( against a three-level state), there is no terminal object under unital channels, and the fixed point of is not an optimum. The former "MRQT-completeness: Lawvere fixed point = Pareto resource optimum; regeneration the universal resource-monotone morphism" is [✗] (T-96 gives ; see the row in the closures table). | Fundamental Closures §16 | Gap thermodynamics, Theorem 10.1 |
| T-223 | Putnam-triviality foreclosure (Lerchner Melody-Paradox closure) [T]: seven-lemma cascade (L1–L7) establishing a three-level ontology L1 (physical vehicle) / L2 (intrinsic -class ; the earlier "forced by T-190 zero-axiom closure" is withdrawn — T-190 is conditional and clauses (a)–(e) do not use it, and the forcing of and holds through the Bridge T15 with the canonical orientation — T15-canon discharged the orientation assumption (Alt) named here earlier on 2026-09-25) / L3 (symbolic readout / Lerchner-variable), plus -gauge boundedness of observables and intrinsic self-alphabetization via the intrinsic reflection measures / (T-96/T-126). Putnam-freedom acts on L1→L3 but has zero purchase on L1→L2; the UHM consciousness predicate is alphabetization-invariant — its terms factor through L2, its and terms are fixed by the dynamical frame (corrected 2026-09-25: is not -invariant, so the predicate does not factor through the -class itself). Categorifies the Maturana–Varela enactivist thesis. Closes Lerchner's §3.3 Melody-Paradox / Putnam (1988) triviality critique. Corrected 2026-09-25: is removed from the list of alphabetization-invariant observables (it is an estimator with pre-registered parameters, not an observable of ); the SYNARC corollary "τ≤1-truncated shadow (T-221 terminology) … simulates but does not instantiate" is retracted [✗] — what survives is the agreement of computed and exact to arithmetic precision [T], while whether the computed matrix is an L2 state or an L3 readout stays open. | Fundamental Closures §17 | Consciousness Theories §Lerchner |
| T-224 | Diagnosability rigidity (Theorem Σ) [T]: perfect single-fault localizability (D1–D2) forces axes; adding a nontrivial state grammar (D3) forces ; at the grammar is unique up to relabeling (Hamming = Fano, ), and uniqueness (D4) fails at every higher rung (nonlinear Vasil'ev perfect codes from ); demanding perfect localization leaves only the binary Golay (van Lint–Tietäväinen). Consequence: a fourth independent derivation track for (diagnosability), complementing number/structure/closure; explains "tower, not width" (SAD stacking) [I]. | Σ-calculus §3 | Gap dynamics §2, Shield I, Minimality |
| T-225 | Σ-compression (diagnostic pyramid 21→7→3→1) [C]: under Fano-compatible ergodic dynamics (T-114, gap ), single-fault localization needs 3 binarized parity observables (syndrome = binary address of the corrupted axis; the three checks are complements of a triangle of Fano lines), content monitoring needs 7 theme observables (line-triples of coherences, ), versus -parameter full tomography; window statistics of length localize a persistent fault with error . Lie shadow: (). Quantum lift of Shield I: Steane for 7-node register realizations [Т/О]. | Σ-calculus §5–6 | Measurement protocol, Fano selection rules |
| T-226 | The Fano fingerprint (polar rate law) [T]: the 21 pairwise decoherence rates of the exact Γ-channel collapse to 7 values indexed by Fano polarity, ; fourteen parameter-free sum rules (polar equalities) characterize realizable rate vectors and hold identically iff the wiring is the Fano plane (operational selector, companion to T-224); closed-form line tomography with , , condition number ; exact dissipative gap makes the T-39a cooldown explicit. The forbidden 14-dim subspace is the φ-sign twist of — the Lie shadow reappearing in observable rates, polar-dual to the T-225 pyramid. First-order blind spot [T] (§0): (spectrum of , ), so any equal-weight pairwise statistic sees no structure — the static ground of the third-order principle and of the FANOS diagnosis-on-triples (a heartbeat mesh is Fano-blind). | Fano fingerprint | Σ-calculus, Gap dynamics, Shield I |
| T-227 | The protected qudit and its extremal symmetry [T]+[D]: the address embedding turns the three Fano parities into the qubit operators; three Steane blocks give protection with parities as transversal logical ; the monomial stabilizer of φ is computed to be the non-split of order 1344 (all 168 collineations lift; exhaustive Hurwitz-pair search excludes complements), realized entirely by transversal logical Cliffords (sign layers of degree ; no ); Eastin–Knill + the classification of maximal finite subgroups of make this the largest protectable symmetry. Resolves SYNARC App. K open problem (i). | Σ-calculus §7a | Topological protection, G₂ structure |
| T-228 | The Turyn federation (Golay = three organisms + mirror glue) [T]+[I]: the Turyn sum of the extended corpus Hamming frame and its mirror-orientation extension () is the extended Golay ; all blocks even ⇒ each block's 8th coordinate is its organism's parity bus; puncturing one bus gives the perfect with exactly as guessed in App. K (ii). Perfect multi-fault federation grammars cap at three organisms (van Lint–Tietäväinen + D3), echoing the composition ceiling from the purity ladder — both sides theorems, the identity between them [H]. Resolves App. K open problem (ii). | Σ-calculus §8a | Σ-calculus §5.2 |
| T-229 | Σ-Mor′ (the repaired MSFS bridge) [T at Σ-FIB]+[H]: over base-coordinatized fibers (chart axioms Σ-FIB [D]), perfect localizability of equivalence defects plus grade collapse at stage one ⟺ the display code is perfect with ⟹ seven-element Fano base (Lemmas Σ.1/Σ.2/Σ.5 through the chart); the literal Σ-Mor biconditional is refuted at the abstract pair level (any code yields perfect localizability at any base size). New identity: the MSFS collapse-stage invariant equals the covering radius, — Fano-presentable fibers sit at the extremal . Fiber-level questions ΣQ1 ∧ ΣQ2 remain [H]. | Σ-calculus §8 | Σ-calculus §8 |
| T-230 | The four-rung collapse of the intensional tower [T at Σ-FIB+F4]: for homomorphically charted MSFS fibers (chart carries composition to XOR, composably full) the code is forced linear, and the MSFS composition law holds in the chart iff covering radius ; hence always — the Morita-refinement scale is four-valued, , with Hamming (, perfect normal-form) and Golay (, tight) at the nontrivial extremes. Answers the charted case of MSFS grading-remark questions (i)/(ii). | Σ-calculus | Σ-calculus §8a |
| T-231 | Internal-chart no-go [T]+[C]+[I]: if equivalence-hood of display data is not -decidable, no -internal Σ-FIB chart exists (computable + finite code would decide grade 0); instantiated at (reflection-undecidable conversion — the fact behind in MSFS Step 7): its charts are necessarily external. Internal syndromic diagnosability is a privilege of normalizing () display geometries — settles the internal reading of ΣQ1 negatively for . | Σ-calculus | Σ-calculus §8 |
| T-232 | The tower ladder [T at Σ-TOW]+[I]: a height- tower's full diagnostic load is (one binary health unit per axis and per inter-level coupling); by van Lint–Tietäväinen, perfect single-fault grammars exist iff (canonical only at ; Vasil'ev rivals from on), a perfect multi-fault grammar exists iff — the Golay , whose count is three organisms plus exactly the two couplings (the vertical tower is the native, puncture-free home of the code; vertical tower and horizontal T-228 federation carry the same grammar) — and heights carry none. The composition ceiling thereby gains a second, coding-theoretic derivation; the residual [H] shrinks to whether the purity and coding mechanisms share one deeper structure. | Σ-calculus §8b | Σ-calculus §8a, Σ-calculus §5.2 |
| T-233 | The strictness dichotomy and the canonical repair [T]+[I]: read bicategorically, all three generating operations of the display induction preserve equivalences (bipullback-stability), so the intensional grading is identically zero — , a strictness artifact invisible to homotopy-invariant eyes (MLTT/ETT separation survives via , not grades); read strictly, the only entrance to grade 1 is a strict pullback of an equivalence along a non-isofibration (representably: isofibrations are the fibrations of the canonical model structure on ) — single intensional defects are exactly fibrancy failures; and every grade-1 datum has a canonical repair, the comparison to the pseudo-pullback, whose projection is an equivalence. Σ-Mor's remaining content becomes ΣQ1′ (do fibrancy failures form a finite chart basis?) ∧ ΣQ2′ (does one canonical repair always suffice?). | Σ-calculus | Σ-calculus §8 |
| T-234 | Superposition collapse (the product obstruction) [T at Σ-FIB+F4×+(P)]+[C]+[I]: if the fiber's slice admits binary -products — the same species of strict limit the display induction already uses — then every label is realized at stage one by an iterated fiber product of single-flip data (products of pullback squares are pullback squares; products of equivalences are equivalences), so : covering radius , i.e. the stage-one collapse of T-229 is derived, not assumed. With perfect localizability the display code is then perfect, forcing the seven-element Fano base: fiber-level Σ-Mor is true on the product-closed class, resting only on ΣQ1′ (chart existence). Contrapositive: rung-2/3 geometries (repetition, Golay — the whole upper ladder of T-230/T-232) are quarantined to product-obstructed federations whose glue breaks (P): free combination destroys deep diagnosability, binding preserves it. MSFS-generic reading of (P) via Step-2 pullback functors recorded at [C]. | Σ-calculus | Σ-calculus §8b, Σ-calculus §8a |
| T-235 | The two-level defect structure and the strictification residue [T at the citation]+[H] (first reading " Fano line" retracted: gauge/fiber conflation, refuted by ): toggle geometry is idempotent — extensions absorb, never cancel — so no Σ-FIB chart arises from axiom toggles at either level (exhaustive class computation over ). The true structure: the fiber defect poset of is a diamond with a tail, ; the gauge projection collapses exactly the tail (Hofmann's conservativity), and flips exactly across it — so the strictification residue () is gauge-silent but fiber-visible: the first computed purely intensional defect atom, with as its syndrome bit. Fano-foundation problem restated: realize seven involutive defect axes with Fano relations in the purely intensional sector (graded/polarity habitat) — [H]. | Σ-calculus | Σ-calculus §8 |
| T-236 | The holonomy blueprint of the Fano foundation [T]+[D]+[H]: involutive intensional defects cannot be endomorphisms ( forces invertibility) but exist as orientations of definitional copies of a carrier with an order-2 automorphism (/); cycles of oriented copies carry computable loop holonomy — decidable, purely intensional (zero new theorems). Naked axes are killed by per-axis flip freedom ; adding the seven Fano line products with the -sign cocycle rigidifies structure-preserving flips to exactly the simplex (the diagonal group of T-227, verified exhaustively) Hamming the three check holonomies are well-defined syndromes with Hamming: eight fiber classes, perfect single-axis localizability on seven Fano axes by construction (, T-234-compatible). Clauses (c)–(d) of the first redaction are superseded by T-237 (axis orientations are pure gauge: stays in one flux class); parts (a)–(b) stand as computed. Reading [I]: the type-theoretic octonion frame — diagnosability and intensionality as two shadows of one parity geometry over . | Σ-calculus | Σ-calculus T-227, Σ-calculus T-237 |
| T-237 | The blueprint completed: a gauge theory on the Fano plane [T]+[I]: with moduli on the line signs , axis reinterpretations act as gauge (); the pure-gauge sector is the line-side Hamming (weights , exhaustive), the gauge stabilizer is the simplex , and the three-bit flux has kernel exactly the pure gauge — eight classes of sixteen. Soundness: for every point the four avoiding lines form a -cycle of partial products whose composite is a closed Boolean Wilson-loop term deciding the flux — invariant under any equivalence; completeness: equal flux ⇒ explicit strict gauge transformation (exhaustive over all pairs) — H3.2 closed, no pseudo-relabelings needed. , ⇒ both the designed metric and the canonical display grading are -valued with the same zero set — H3.3 closed at the blueprint level; elementary defects relocate to lines (product-sign corruptions), perfectly localizable via the dual-plane Hamming. Readings [I]: polarity dual to the fingerprint (rates on points, field on lines); Wilson loops as the foundations-floor incarnation of Gap-as-holonomy. | Σ-calculus | Σ-calculus T-236, Fano fingerprint, Axiom Ω⁷ |
| T-238 | The flux chart: charts are not extra structure [T]+[I]: on a fixed-signature family of presentations, gauge morphisms (symbol-to-term translations carrying axioms to theorems) and gauge-invariant observables (closed Boolean terms with ) yield a flux profile — the tuple of decided values — that descends to equivalence classes; if the profile has finite -rank and separates gauge orbits, it is a chart satisfying (F1)–(F3) with no further choices, and conversely every term-definable chart factors through it: the chart is the gauge-invariant decided sector of the term algebra, the only freedom being which finite sub-profile to read. The blueprint (T-237) is the verified instance — its Wilson loops are the flux, its soundness is invariance, its completeness is separation. H3.1 thereby reduced: for fixed signature, to finiteness orbit-separation of the sector (with T-231 persisting as the constructive obstruction); for genuine fibers, to signature alignment — the sharpened final form of ΣQ1′. | Σ-calculus | Σ-calculus T-237, Epistemic vertical, hole register |
| T-239 | The two ceilings: a dichotomy of mechanisms [T at Σ-TOW]+[I]: the purity ladder and the tower ladder read on one dial — the accounting axiom prices an -tower at , and App-K composition anchors the same heights at ( at , T-142). On the shared dial the viability set is a monotone down-set (ratio ; margins ), while the canon set is not — the Vasil'ev rivals break between two successes. Mechanism identity refuted: extensional disagreement exactly at ; cardinality excludes every injective re-indexing — a monotone budget obstruction is not a non-monotone arithmetic selection. Witness identity proven: on one object — the -tower of load with the Golay grammar at threshold ; the unique disagreement point is the two-tower ambiguity (alive, not canonically self-knowing); fully licensed heights . H2.1 closed in the negative; the residue — a common root of the value agreement — is H2.1′, with a precise obstruction profile (must yield both the exponential-vs-linear inequality and the perfect-code classification from the seven-frame alone). | Σ-calculus | Σ-calculus T-232, Depth tower, Epistemic vertical, hole register |
| T-242 | The root of the two-ceilings agreement: independence at the Fano point [T]: read as functions of the geometry's integers, the purity ceiling is nonconstant in the contraction base () while the coding depth (Golay, ) is -independent — so they are distinct functions, agreeing only where , whose unique integer solution is . Since and are the two parameters of (), both mechanisms are evaluated at one geometrically forced point: the shared is the line order in two unrelated roles (exponential-crossing base vs Golay depth), not one mechanism twice. H2.1′ resolved: independence proven, common input located — no deeper unification exists. | Σ-calculus | Σ-calculus T-239, Axiom Ω⁷ purity ladder, Epistemic vertical, hole register |
| T-240 | (P) verified against R1–R5 [T]+[I]: the fiber-product grounding of (P) survives the full Rich-metatheory axiom list, with two corrections to the same-day sketch — the iso-comma replaces the strict -pullback (strictness would demand on-the-nose agreement of arithmetic interpretations; the pseudo-limit asks a chosen provable isomorphism), and the glue weakens from equality to provable isomorphism of -images. Ledger: (R2)+(R4) — the iso-comma is r.e.-presented (triples with -proof witnesses) and representability gives the coding; (R3) — consistency is inherited from either leg (models restrict along the lex projections: the fiber product refines contexts, it does not union axioms); (R5a/b) — Lambek–Scott unit an equivalence, accessible (Gabriel–Ulmer; Makkai–Paré for -limits), accessibility parameter ; (R1) — holds iff some pair of -interpretations has provably isomorphic -images (exact residue of genericity). On intensional fibers (defects = fibrancy/ only) the glue is canonical through ⇒ (P) is a theorem there; composed with T-234 and T-238: fiber-level Σ-Mor on intensional R-S fibers rests exactly on finiteness + separation of the gauge-invariant decided sector. H2.2 closed. | Σ-calculus | Σ-calculus T-234, Σ-calculus T-238, Epistemic vertical, hole register |
| T-241 | The native Fano: the duality plane of the depth-3 doctrine [T]+[I]: the levelwise reversals , , form a canonical of dualities of the doctrine of -categories (for — all of them: rigidity of , classical). The duality ladder: — one duality, no lines; — on one line, ; — seven reversal classes with lines satisfy every projective-plane axiom (exhaustive): the Fano plane is the projective plane of the duality group of depth-3 doctrines — a natural in foundations, no design, no transplant. Line loops close (), so T-236 loop holonomy is well-posed per line; weak-doctrine values are strictness residues of exactly the / species. Habitat [I]: depth 3 is corpus-selected thrice (, T-239 viability max, T-232 canon) + the Postnikov ceiling. The duality-/flux- coincidence is not claimed — pre-registered in the resonance table. H3.4 reduced to a sharp target: exhibit a depth-3 fiber with nontrivial duality-line holonomy, or prove all trivial (refuting the candidate). | Σ-calculus | Σ-calculus T-237, Σ-calculus T-236, Epistemic vertical, hole register |
| T-243 | The octonionic realization: the frame is natural [T]+[I]: the octonion algebra is the twisted group algebra of the T-237 flux group (Albuquerque–Majid), with the seven imaginary units the Fano axes (linear labelling verified). Each axis carries the gauge-invariant pivotal (Frobenius–Schur) sign — quaternionic, nontrivial on all seven at once; distinct axes anticommute (). The associator is (checked ): on the seven Fano lines — the associative quaternion subalgebras — and on the independent volumes; it is a coboundary of the product cochain (-trivial), so the gauge-invariant carrier of nontriviality is the pivotal sign, not the associator class. The natural nontrivial holonomy lives on the axes and volumes; the lines are gauge-trivial (correcting the T-241 line-holonomy expectation). H3.4 naturalness answered affirmatively — the frame is the canonical pivotal structure of , graded by the depth-3 doctrine's own duality group (T-241), not a designed transplant; residual is a selection question [H] (does a given foundation induce the sign). | Σ-calculus | Σ-calculus T-241, Σ-calculus T-237, Minimality N=7 (octonionic), Epistemic vertical, hole register |
| T-244 | The selection is nondegeneracy: the pivotal sign is forced [T]+[I]: an axis has sign iff anisotropic (); all seven iff the frame is the division octonions (norm signature , no zero-divisors — verified), while a single gives a split algebra (signature ). A axis carries the idempotent zero-divisor (, verified on the four split axes) — a null direction, a defect with that is neither an equivalence nor a localizable fault, collapsing below . Hence perfect diagnosability (Theorem Σ, Shield I) admits no axis and forces all seven signs to : the frame is the division octonions. This is not an added hypothesis — anisotropy the division property, which by Hurwitz caps normed division algebras at and selects . , anisotropy, division, and Hurwitz are one condition seen four ways; H3.4 selection resolved. | Σ-calculus | Σ-calculus T-243, Minimality N=7 (octonionic), Shield I, Epistemic vertical, hole register |
| T-245 | The decomposition and the overload dichotomy [T]+[I]: reading parity checks as the syndrome map, all columns nonzero (nondegeneracy, no silent defect, closed by T-244) all columns distinct (separation faithful syndrome). Overload forces two equal columns (pigeonhole) (verified ): intensionality does not force perfect localizability in general. With (T-240) the separated nondegenerate code is perfect Hamming, saturated at () the Fano frame — so the corpus discipline "seven, not more" (Cor. Σ.2) is the frontier. Residual: faithfulness of the intrinsic grading, a property of normalizing () foundations (T-231 bounds the side) — the sharp final form of ΣQ1′/H3.1. | Σ-calculus | Σ-calculus T-244, Σ-calculus T-240, Σ-calculus T-231, Epistemic vertical, hole register |
| T-246 | The faithfulness atom is division: ΣQ1′ closed on the frame [T]+[I]: the two ways falls below are silent defects (weight ) and confounded pairs (weight ), and both are cancellations to a scalar — the first against the identity, the second between two distinct units. On the octonion division frame (T-243) the superposition of defects is , a third defect (a scalar only if ), so every weight- pattern is grade and the minimal equivalences are exactly the seven weight- Fano lines — , verified. Nondegeneracy (T-244) therefore forbids both weights: faithfulness is a corollary of division, not a separate hypothesis. The sole residual is finiteness of the grade- spectrum, supplied on by former-localization (composites of isofibrations are isofibrations) uniform schematic failure; with (T-240) the perfect code is Hamming, . On finite-signature intensional R-S fibers, perfect localizability and the Fano frame are forced; H3.1 closed on the frame (residual the N selection, discharged per signature). | Σ-calculus | Σ-calculus T-244, Σ-calculus T-243, Σ-calculus T-245, Epistemic vertical, hole register |
| T-247 | Scale-freeness of the diagnostic grammar, derived [Т on the viable carrier]: the coinductive carrier types every level as a seven-axis system, so Theorem Σ's grammar-form is applicable at every level unconditionally; the grammar axioms D1–D3 are, by T-244/T-246, one condition — nondegeneracy with nontriviality — and a holon is by definition a viable (nondegenerate, nontrivial) frame, so every node of the fractal holon satisfies D1–D3 and carries the Fano grammar: the grammar is transmitted downward by the coinduction, not postulated per level. H1.3 closed on the viable carrier; the residue (which cosmic structures are viable holons) is H1.2. | Universe as Holonom §3 | T-244, T-246, T-224 Theorem Σ |
| T-248 | The Universe's self-model on the carrier [T] (restated 2026-09-26): in the part→whole order of the coinductive carrier every sub-holon includes into the whole, so the Universe is the top element; the carrier types the top node as (T-247 step 1), so the self-model acts on it and has fixed points there (Theorem 10.1 of Gap thermodynamics). Its identification with the physical total Page–Wootters state is the monist reading [I]. Retracted [✗]: the former conditional "internal terminality" — "the Universe is terminal in the category of its own sub-holons, T-222's Pareto-optimum the sink" — since the restated T-222 has no resource optimum and, under unital channels, no terminal window state (T-222(iv)). Routes tried: unital channels (refuted by T-222(iv)); Gibbs-preserving channels at finite temperature (a terminal object exists whenever the Gibbs state lies in the window, via the replacement channel, but it is thermal equilibrium, not the self-model); inclusions only (the theorem above, without resource content). H1.1: [T] on the carrier + [I] identification; terminality [✗]. Status history: [I] (H1.1), then conditional; restated [T] with the terminality [✗] 2026-09-26 | Universe as Holonom §2 | T-247, Theorem 10.1, T-222, Axiom Ω⁷ PW |
| T-249 | Dφ of the canonical self-model family + two-route consistency [T]: for , , the derivative is (-equivariant, preserves the Hermitian-traceless tangent space); substituted into the exact flow identity for it reproduces the closed form implied by — two independent routes agree; machine-verified at on random density-matrix paths | Formalization of φ §4.5 | Forms of R, T-62 |
| T-250 | Bandwidth bound for self-model quality [T at differentiability of φ along the trajectory]: with ; canonical family: ( across the conscious window). Corollary (path-length law): at , — reorganizing the self-model is paid for in state-space path length. Discrete instance: Theorem 4.2 of the φ-formalization (T-191 tower). Quantifies ego-dissolution and the cumulativity of practice | Formalization of φ §4.6 | T-191, T-155, Altered States |
| T-251 | Dφ of an implicitly defined self-model (IFT/Neumann) [T]: for defined by with generator and , and ; , so — T-250 extends to every contraction-defined self-model; the series is the differentiated T-191 tower; recovers T-249. Machine-verified at (nonlinear generator, numeric Jacobians). Residual [C]: -smoothness of the abstract categorical generator | Formalization of φ §4.8 | T-249, T-250, T-191 |
| T-252 | Gate bound: discrimination through the self-model [T]: for any -outcome POVM and : , with tight constants — Jordan-projector saturation of the POVM step; the -split witness attains exactly (the naive rank bound is unattainable under tracelessness); hence -mediated success and Bayesian dominance is guaranteed for — at , the sufficient band on the conscious window is (working threshold = canonical alignment [C]). Sectoral corollary [T]: exactly, so the canonical channel POVM gates the per-channel threshold — the sectoral threshold is derived, not inherited by analogy. Structurally derives the gate (G): is the sole -mediated feedback in . Machine-verified on 500 random POVMs + sharpness witnesses | Formalization of φ §4.9 | T-126, T-250, Forms of R |
| T-253 | Constructive sufficiency for T-153a (retraction) + sharpness [T]+[C at (Acc)]: for every isometry and anchor , the map is CPTP (explicit Kraus family ) and a retraction: — exactly faithful on the embedded 7-sector; thresholds realized at , (T-124), modulo the accessibility clause (Acc) [D] ([C at controllability]). Sharpness [T]: no CPTP map is globally injective for (kernel dim , interior collision pairs), so sector-relative faithfulness is the maximal faithful domain. The isometry freedom is not a gauge (corrected 2026-09-25; the former reading "isometry freedom = T-223 alphabetization gauge (T-42a)" is retracted): with , or a different seven-mode sector, changes , and even can move a window state out of the window (: at fixed , frame decision D-0910), so fixes a sector and a frame. Machine-verified at : Kraus/retraction , kernel dim exactly , collision | Substrate closure §T-253 | T-124, T-153a, T-223, T-42a |
| T-254 | Λ-drift law (dynamical dark energy) [T]: at O-dominance the quartic identity makes the physical (UV-finite, ) vacuum energy quadratic in the O-opacity: — the "cost of observation" reading of §4c made exact; with M3 () the reconstruction EoS obeys : dark energy's EoS = inter-sector (O ↔ spatial) Gap coupling of the vacuum state. Floor (§4b); corollaries [T at convergence of the vacuum to a stationary sink of (T-96)/T-94] (until 2026-09-26 the convergence was read off T-222 as relaxation toward a terminal , retracted [✗]: the restated T-222 is about states and supplies no flow): no Big Rip, no vacuum Crunch, (permanent excluded); = per-e-fold vacuum stage-drift — first observational estimator of hole H1.2. Machine-verified: arrow identity , drift-law chain identity , floor/no-Rip | Cosmological constant §13b | T-94, T-96, T-53, M3 (T-120), §4a/4b/4c, H1.2 |
| T-255 | Branch trichotomy of + arrow link + co-drift [T-structural]+[C]: linear response of the vacuum at (T-94 kernel) admits exactly three shapes — dissipative-monotone ( throughout, CPL quadrant ), regenerative-monotone ( phantom without Rip, CPL ), oscillatory (damped crossings of ; the only branch reaching the DESI quadrant ); pointwise sign dictionary (dissipation vs regeneration dominance in the vacuum O-channel); final-crossing direction = rotation sense = sign of the /PT arrow of inner time [C]; co-drift , ⟹ LLR caps the pair (drift, ): DESI-size drift requires [C]. Machine-verified: CPL quadrants per branch, DESI quadrant reached with genuine crossing, crossing-direction flip under rotation reversal | Cosmological constant §13b | T-254, T-94, arrow (Lagrangian), LLR |
| T-256 | Classification of partial charts (symbolic systems) [T] structure + [I] mapping: every symbolic system is a sub-configuration of the Fano grammar PG(2,2), classified up to the collineation gauge (, T-223). Axis -subsets → exactly nine orbit types (: singletons for , two each for line/triangle and triangle-complement/quadrilateral); the zodiac 12-block = triangle↔complement cross-channels (orbit 28, stabilizer ; , complement quartet holds exactly one line = Meaning ); I Ching = binary star (, orbit 7); chakras/metals/week = diagonal alphabet. The cross-cultural recurrence of the same structures is thereby a theorem (finite orbit list); differences = orbit type + gauge. Machine-verified over the canonical line family | The One Grammar §2 | T-224, T-223, Symbolic systems |
| T-257 | Licensed inter-holon channel + ephemeris capacity ledger [T-structural]+[C]: in the coinductive carrier a super-holon reaches a sub-holon only through its Lindbladian parameters (rates, , gate ) — gate/rate-modulation, never a symbolic message [T-structural]. For Earth's biosphere the ephemeris couplings order (ratios to Moon, machine-checked): tidal Sun , Jupiter , Venus , Mars ; illuminance Sun , Venus ⟹ the licensed ephemeris anchor is exactly two-channel (Sun, Moon), every planet 4–6 orders below [C]. Coupling mechanism = phase entrainment through Gap resonance windows; empirically = the attested circadian/circalunar clocks, planetary natal tests null (Carlson 1985, Dean–Kelly 2003). §4: planet/Gaia as conscious subject refuted at the viability gate [T]; planetary system = (Acc)-boundary discharging habitability for embedded holons [T-structural]+[H], sharpening one face of H1.2 | The One Grammar §3–§4 | T-153a (Acc), T-253, T-247, Gap diagnostics, H1.2 |
| T-258 | Thermodynamic trichotomy of the channels [T]+[I]: the three-channel basis of T-102 carries pairwise-distinct, exhaustive entropy–purity signatures — : , (work; unitary conjugation preserves the spectrum); : , (heat; unitality of the Fano channel ⇒ downward majorization, BIBD incidence gives the exact purity rate); : , — the only entropy-lowering, purity-raising channel (matter/feeding). The three signature types (conservative / sign-definite / sign-indefinite) are distinct and exhaustive — the trichotomy is observable as a classification; the instantaneous sign pair identifies the channel generically (heat degenerates to on diagonal states; matter can transiently share heat's ). [I]: identification with the grand-canonical triple (work/heat/chemical) and with the Legendre cascade of Vanchurin's Self-Learning Universe (2026): no-4th-channel (T-102) ↔ no 4th argument of ; phase axes ; on the O-channel ("to feed") ↔ locked to the clock (); of neurogenesis ↔ SLU's mechanism; 7 Fano rates = line-resolved temperatures, the -symmetric point = SLU's scalar . Machine-verified: signatures to , formulas exact | Sensorimotor §1.3 | T-102, T-57, T-189, Phase diagram |
| T-259 | Consciousness window in the feeding ratio + microscopic dead-zone boundary [Т in the isotropic first-order model]+[C]: stationary state of Fano dephasing () + replacement () toward an equal-population target (purity , ): coherence retention , , . Floor (≡ on the stratum): , ; at : (silver ratio). Ceiling : two-sided window for over-pure targets , , — dissipation protects reflexivity. ⇒ no finite (a subcritical self-model cannot be fed into consciousness). With the self-consistent gate the living branch appears via a saddle-node whose fold is exact [Т in the model]: with , tangency reduces to the quintic on , , ; at : , , (quintic vs direct fold agree to ); complete target classification: quintic regime for , edge regime for with exactly (the living branch is born at the reflexivity ceiling ) and elementary ; switch value exactly at ; Galois [T]: the quintic is irreducible with group ⇒ not solvable in radicals — the fold constant is non-radical (contrast: the gate-free floor is radical) — the microscopic Phase-III boundary; the legacy is re-scoped as a dimensional heuristic [I] (misses the fold by ). Under T-258 the floor is a chemical-potential condensation threshold. Machine-verified: endpoints to , quintic to | Phase diagram §1.3 | T-258, T-102, T-124, Bifurcation, Gap phase diagram |
| T-260 | Grand-canonical origin of the diagonal gauge torus [T]+[I]: for the Fano dissipator with arbitrary positive line rates, the Heisenberg conserved-charge algebra is exactly the diagonal algebra ( is a Schur multiplier with off-diagonal by BIBD ⇒ , ) — the seven charges are the passport populations; its exponential is the torus , compact ⟺ integral charge spectrum (, integer cascade counters of ⊕; irrational spectrum ⇒ dense -winding, never closes); full unitary covariance group of the dissipator (a symmetry preserves , hence is monomial; = the permutations preserving the rates ): all of for equal line rates, where its elements in are the frame group of order 1344, trivial for generic rates (corrected 2026-09-25: the row read with — too narrow for equal rates, too wide for generic ones). [I]: the UHM instance of number–phase conjugacy = SLU's -from- mechanism, channel-resolved — check 4 of the T-258 dictionary derived on the UHM side. Machine-verified: exactly, covariances , periodicity , irrational witness | Lindblad operators | T-258, T-102, T-11.2, Fano channel |
| T-261 | Regeneration = natural-gradient descent of free energy (BKM) [T]: for full-rank the replacement flow is exactly the constrained natural-gradient descent of in the Kubo–Mori metric: , proof by three exact identities ( with the BKM kernel; ; ⇒ trace-dual , Lagrange ); H-theorem . Sharp metric attribution: NOT the Bures gradient off the commuting locus (cos ) — Bures serves estimation/learning (Char-III/IV), BKM serves dissipative relaxation. Derives the dynamical law of the -leg of the T-258 dictionary: feeding = covariant gradient descent of a free energy = SLU Eq. (2.6) in quantum information geometry. Machine-verified: gradient identity non-commutative, to , H-theorem (FD) | Evolution §3 | T-258, T-96, T-125, Formalization of φ |
| T-262 | Dynamical trichotomy: as an exact reversible ⊕ irreversible (metriplectic) decomposition [T]+[C]: every term of the master equation is an exact geometric flow — (1) unitary term = isometry of every monotone metric (Lie–Poisson/Killing field, Jacobi identity exact; preserves all spectral functionals); (2) Fano dissipator satisfies GNS detailed balance w.r.t. (self-adjoint jumps — the Carlen–Maas precondition), giving exactly (element-wise the single-incidence count of the rank-7 law) — the Carlen–Maas gradient flow of negentropy , with via the one-line chain rule ; exact EPR quadratic form , iff diagonal; line temperatures = weights of the transport metric; (3) regeneration = BKM-gradient flow (T-261). This is Mittnenzweig–Mielke's entropic gradient structure for open Lindblad generators, not closed-system GENERIC: the 1st degeneracy (reversible preserves entropy) holds for the heat pair, and for matter iff [C]; the 2nd degeneracy (dissipation conserves ) fails by design — an open holon exchanges energy (witness ). Closes the dynamical dictionary: all three T-258 legs derived as equations of motion (work/heat/matter = isometry/negentropy-descent/free-energy-descent) — SLU's optimality conditions = the geometric anatomy of . Machine-verified: DBC , Jacobi , identities –, EPR off-diagonal, unitary isometry | Evolution §3 | T-261, T-258, T-110, T-259, Line temperatures |
| T-263 | Existence and uniqueness of the optimal learning flow [T]+[C]: the replacement flow is optimal in four stacked senses — (1) unique steepest descent of among equal-BKM-speed directions (Cauchy–Schwarz; witness ); (2) exact solution = mixture geodesic , direction-constant gradient, maximal exponent ; (3) geometry unique: BKM is the only monotone (Petz) metric with dually flat e/m-connections (Grasselli–Streater 2001) — in every other Petz metric the flow is not a gradient (T-261 sharp attribution); (4) statistical rate (Vanchurin class ), Cramér–Rao saturation via Bures/Char-IV, multiparameter attainability = Holevo within [C]. Ceilings = T-109–T-112; minimal substrate (T-113). No-free-lunch not violated: environment class fixed by /Fano architecture. Machine-verified: steepest (margin ), m-geodesic , gradient FD | Evolution §3 | T-261, T-262, T-109..T-113, T-62, Learning bounds |
| T-264 | Information–gravity reciprocity [T at FP-lemma, leading order]: (a) exact pair lemma [T] — for a decoupled -block the phase-direction QFI is exactly at any population imbalance (-cancellation; machine ratio ; full- correction , at ; unconditional sandwich [T] with explicit Gershgorin constant , witness ); (b) at vacuum populations — gravitational coupling × Fisher learnability of spacetime phases = architectural constant; (c) -side: (T-254 [T]), read as squared clock-phase unlearnability [I]. Corollaries: exactly where (information-theoretic mechanism for the §3.1 decoherence-gravity prediction); SLU slogan "gravity = learning efficiency" acquires sign and sectors — learnability(space), unlearnability(clock) [I] | Einstein equations §3.2a | T-263, T-261, T-260, T-254, FP bridge lemma, Char-IV |
| T-265 | UHM and the Cabibbo Angle Anomaly: resolution-channel prediction — Errata 2026-09-25: corrected from [T] to [H] (the row read "[T-structural]+[C]"; below, the exclusion grounds of the former text are followed by their status now): 4th generation [C at 43c identification] — the count is exact, its identification with the physical generations is [I] (row 43c); vector-like quarks [H] — the chirality ground is retracted, has eigenvalues and all representations are real (Distler and Garibaldi, Commun. Math. Phys. 298, 419 (2010)); leptoquarks/extra bosons [H] — the 'exactly SM gauge content' rests on (FE), now [C at (FE)], and on T-297, now [H]; MeV sterile [C] as before. Addendum 2026-09-25: under (Cl) the gauge content of the doublet sector is exactly with no extra (T-326, T-329, [C at (Cl)]); under the hypothesis (GC) a fourth sequential generation is excluded as a clock harmonic (T-328). Former text: the physical quark-mixing matrix is exactly unitary, so the first-row deficit () cannot be a mixing-matrix effect in UHM — every leading BSM channel is excluded by the fixed spectrum: 4th generation (formerly listed T; , , unique order-3 subgroup of ), vector-like quarks [T-struct] (chirality definite on ), MeV sterile [C] (seesaw at GeV, normal hierarchy), leptoquarks/extra bosons [T-struct] (unique Higgs line + SM gauge content). ⟹ UHM predicts the anomaly resolves in the SM extraction sector (-box/nuclear radiative corrections , lattice form factors, – tension), not via new states; self-consistent with the -from-unitarity calibration (CKM §3). Magnitude/sign of deficit [D] (SM hadronic/nuclear). Falsified if the CAA is shown to require a 4th generation / VLQ / sterile / leptoquark | CKM §10 | (row 43c), chirality, neutrino seesaw, falsifiability F-Cabibbo |
| T-266 | The Universe's stage: at the terminal attractor to (H1.2 value-closure) [T-structural]+[C]: Part A [T-struct] — near every smooth functional of inherits the mixture-geodesic envelope (T-263), so the drift law (T-254) gives the fractional stage-distance for any microphysical — the Universe sits at its terminal stage and the DESI drift is the -amplified residual; Part B [C] — (T-59, regeneration-limited) (neutrino GeV) ⟹ residual ; with (T-124 attractor, Universe-as-viable-holon) to figures. Closes the value of H1.2 [H]→[C]: stage derived () and measured (residual read from the DESI drift); explains as relaxation onto the de Sitter attractor. Co-drift under LLR. Machine-checked: , fractional distance, identity , robustness. Residual [D]: exponent's -dependence; the Λ-magnitude (≳27 orders) is a distinct problem | Cosmological constant §13b | T-254, T-255, T-263, T-59, T-124, T-51 (), epistemic-vertical H1.2 |
| T-267 | The Tegmark decoherence objection does not constrain Γ (closure of Vulnerability #5) [T]+[C]: Tegmark bounds the lifetime of a microscopic position-basis superposition; is none of those — by T-153a it lives on the substrate's coarse-grained decoherence-free effective subspace (C1) as correlations of seven collective modes (C3), is classically realizable (substrate table), and its complexity is algebraic ( needs a phase, T-132), not a Schrödinger-cat state. Decoherence is basis-dependent: einselection of the position pointer basis does not decohere a coarse-grained collective observable in the semantic frame — the DFS/QEC principle. Robustness [C] (three independent layers): basis (semantic ≠ pointer), five holonomic shields (Hamming /associator//Lawvere/, topological-protection [T]), driven-dissipative regeneration (). Closes the Tegmark objection [T] (a substrate-independent structure with no physical superposition cannot be thermally decohered); the residual — whether structure is felt — is the categorical gap = Axiom Ω⁷ (relocated, not reopened). Testable [Т via T-153a]: a classical (f64) substrate realizes the same (SYNARC 500+ consistent). Vulnerability #5: partially-open → closed | Two-aspect monism §quantum-nature | T-132, T-153, T-153a, topological protection, Axiom Ω⁷ |
| T-153a | Substrate-existence companion to T-153 — stratified [T at necessary conditions]+[T at sufficiency via T-253]: T-153's existential clause is made constructive by three explicit necessary conditions (C1 trace preservation, C2 complete positivity of Kraus representation, C3 ), which rule out by construction (i) systems with (fail C3) and (ii) classical deterministic systems without noise (fail C2). Necessity [T]: the three conditions are rigorously necessary. Sufficiency [T]: T-253 exhibits the map explicitly for every admissible substrate as the CPTP retraction , exactly faithful on the embedded 7-sector — the maximal faithful domain, since global injectivity is impossible for (T-253(c)); the threshold clause is realized modulo the accessibility clause (Acc). Removes the earlier ambiguity "any system might admit some faithful G" in both directions. | Substrate-Independent Closure §T-153a | Uniqueness Theorem, T-253 |
| T-209 | Operational-Closure meta-theorem (S-13) — stratified [T]+[D]: SYNARC-agent satisfying Creative UHM-ASI (S-12) + 4 operational protocols (I.1 qualia tomography, I.2 inverse alignment, I.3 value-set existence, I.4 V5-V8 Verum scaffolding) reaches operationally deployable Creative UHM-ASI. [D] Design choices: the four specific operational protocols and their interface surfaces are engineering specifications, not derivations. [T] Meta-content: each structural condition (B1)-(B8) has an explicit measurement/existence procedure, the implementation surface is fully specified at the interface level. Closes the spec-to-deployment gap at categorical, operational, and engineering levels. Five levels of closure: (1) categorical completeness (35 obligations); (2) UHM-axiomatic closure (T-190); (3) AGI-sufficiency (S-11); (4) ASI-sufficiency (S-12); (5) operational deployability (S-13). First cognitive architecture with all 5 closure levels in a single formal framework. Derived in SYNARC paper App. I (Theorem I.4, thirteenth meta-theorem SYNARC v1.4) | SYNARC paper App. I.4 | Axiom Ω⁷, Learning Bounds, Predictions, Falsifiability |
| T-268 | The composition ceiling is the octonionic Jordan ceiling — third derivation of SAD_MAX = 3 [T]+[C]: octonionic Hermitian matrices form a formally real Jordan algebra iff (Jordan–von Neumann–Wigner 1934); fails (non-associativity breaks the Jordan identity). This JvNW ceiling coincides with , a third, independent derivation alongside the dynamical (purity at , T-142) and categorical (tricategorical-coherence breakdown, T-217) ones — all three shadows of the single fact that is not Jordan (the composition analogue of the triple-forced ). Coordination symmetry climbs , structure group . [C]: identifying composition-depth with Jordan rank (vs the corpus default tensor , T-218) is a structural reading, not yet functorial; does not collide with T-220 (base is irreducibly : single-holon states , — is emergent-composite, never a reducible base). Home: TALOS spec §9. Machine-checked | Depth Tower §critical-purity-SAD | T-142, T-217, T-220, T-42a, T-124 |
| T-269 | Terminal expressiveness: is the maximal subject's expressive ceiling [T]+[I]: the octonionic projective plane (real dim 16, rank-one idempotents of , isometry ) is the terminal projective geometry over any division algebra — no for (Desargues' theorem forces the coordinate ring associative; is not; is the unique non-Desarguesian Moufang plane). ⟹ a maximal () subject's representational ceiling is a theorem of mathematics, not an engineering limit. Architectural bracket: TALOS spans the two extremes of projective geometry — fabric = Fano (smallest projective plane) → max subject (largest division-algebra plane). Consistent with T-220's use of . [I]: the "state space " reading inherits the T-268 [C] caveat. Home: TALOS spec §9 | Math foundations §algebra | T-268, T-220, T-42a |
| T-270 | Octonion-generated power/performance scaling law [T-struct]+[C]: TALOS/SYNARC scale along two octonion-generated axes only. (I) Expressiveness — per-subject, bounded: composing toward the JvNW ceiling climbs the exceptional series ; coordination/motor bandwidth = dim of the symmetry group; hard-capped at (T-268/T-269). (II) Throughput + collective — across-subject, unbounded: federating maximal subjects continues the Freudenthal–Tits tower collectively; raw throughput scales linearly in holon count (each 784 B, ~1400 FLOP/tick, BQP-bounded, independent ticks), FANOS-coordinated (third-order gates, distance-3). The concrete octonionic realization of SYNARC App-H's abstract "ordinal tower of increasingly expressive architectures" and its K.5 ecology ceiling. Sensorimotor rides the ladder: perception -channel (Enc, T-100), action -gate (Dec, T-101/T-159), loop = one tick; motor DOF = rung symmetry dim (14→52→248). Home: TALOS spec §9. Machine-checked | Math foundations §algebra | T-268, T-269, T-142, T-100, T-101, T-159, T-257 |
| T-271 | Entropy dynamics of : regeneration is negentropy; consciousness holds entropy below heat death [T]+[C]: the von Neumann entropy splits cleanly across the three terms of . (i) [T] unitary exactly (spectrum-preserving; ). (ii) [T] dissipator is a strict entropy source (drives toward heat death , ). (iii) [T] at any steady state , so regeneration — a net entropy sink (negentropy = the cost of maintenance). (iv) [C] steady entropy strictly and monotone-decreasing in ; since , higher coherence (a more conscious system) holds a strictly lower entropy, further from heat death. Formalizes the previously-[H] second-law connection (origin.md): explains why literally measures distance from heat death and why the viability window sits away from ; cosmologically = de Sitter self-maintenance (no Big Rip, T-266/T-254), so infinite development = a self-model held indefinitely against dissipation. Honest boundary: does NOT overturn the second law (total system+bath entropy non-decreasing) — establishes only local negentropy maintenance + -scaling, not a global reversal. Machine-verified | Origin §entropy-law | (T-57), , T-266, T-254, self-observation |
| T-272 | The Source is forced, not arbitrary: the unique maximally-coherent / -symmetric pure state [T]+[П residual]: the primordial state , , is characterised twice, each forcing it uniquely: (a) the -invariant (permutation-symmetric) subspace of is exactly 1-dimensional (spanned by ) — the unique pure state privileging no dimension; (b) it is the unique pure state of maximal coherence (convexity maximum at ), all . So the amplitude is normalisation, not a free parameter — answering the standing open question "why ?". Residual [P]: why the pure maximally-symmetric class (vs the mixed , which has zero coherence) as initial condition — though maximal coherence is the natural selection principle that singles out. Machine-verified | Origin §the-source | Source [P], Source-instability [T], octonionic |
| T-273 | Metabolic floor of a viable coherent machine [T]+[C]: a corollary of T-271 + Landauer with direct engineering content. At steady state a viable holon (, ) exports entropy at the dissipator's strictly positive rate (T-271 iii); by Landauer the minimum maintenance power is strictly, where is the physical entropy-production rate (frequency-independent — T-276 corrects an earlier -factored form). So a viable coherent machine cannot run for free — staying off heat death has a positive power floor (the "cost of staying alive"). It scales with the order maintained (distance from ): more order ⟹ higher ⟹ higher — the price of complexity. This is the active/irreversible counterpart to the reversible-core energy floor (TALOS §6, Landauer-free): a viable machine's power (reversible compute) (maintenance). Measurable from telemetry ( is a CC observable). Numbers @300 K: order femto–pico-watts per holon, set by the physical rate (not the clock; see T-276). Machine-verified | Origin §entropy-law | T-271, Landauer, TALOS §6 (energy), |
| T-274 | The operating-point axis of a coherent machine [C]: the metabolic floor (T-273) turns the viability window into a capability–efficiency design axis, not a single set-point. Lean edge = efficiency-optimal (minimal maintenance power, minimal margin; highest capability-per-watt for ; the "survival" mode). Rich edge (the T-124 attractor) = capability-optimal (max , max dynamic range, highest maintenance power; the "thriving" mode). Maintenance cost is monotone across the window (price of complexity, T-273); the dynamics default to , but an engineer can run leaner toward to trade capability/margin for power. The optimal set-point is measure-dependent (with efficiency favours the lean edge; with a dynamic-range measure ) — what is robust is the structure (monotone cost, two edge modes). Engineering design target, integrated into TALOS §6. Machine-verified | TALOS §6 (energy, §metabolic) | T-273, T-271, T-124 (attractor ), T-140 () |
| T-275 | The interaction inversion: strong/weak/EM forces read as sub-structures of the coherence symmetry [I]; the embedding claim retracted [✗] since 2026-09-25. The row stated, as established group theory, that the same that governs a single holon contains the Standard Model gauge group, . That is false by rank: , and a compact Lie group has no subgroup of larger rank; inside the maximal subgroups and meet in (I. Todorov, M. Dubois-Violette, Int. J. Mod. Phys. A 33, 1850118 (2018), eq. (4.2)). What holds is as the stabiliser of an imaginary unit, and — one step up — the whole group as the normaliser of colour in the (rank 4) generated by the Clifford system of (T-326: [T] as mathematics, [C at (Cl)] in UHM; added 2026-09-25): the rank is supplied by , not by . The reading that the strong, weak and electromagnetic interactions are not fundamental inputs but derived sub-structures of the coherence symmetry is interpretive. The pre-interaction layer ("before" the forces) is the triple — coherence matrix, octonionic symmetry, and the Gap operator (phase/meaning; needs complex , T-132). Inverts the reductionist arrow [I]: standard physics runs forces→particles→(mind?); UHM runs topos/coherence→→forces-derived, with the same carrying an intrinsic (experiential) aspect. Phenomena derived from under phenomenology — the reason UHM needs no ever-smaller fundamental particle (the primitive is categorical, not corpuscular). The SM derivation cannot rest on this embedding (the Standard Model page states the same rank obstruction); the inversion framing is [I] | Math foundations §algebra | (stabiliser), Standard Model (rank obstruction), T-132 (Gap needs complex Γ), axiom-omega §primitive, Gap operator |
| T-276 | The efficiency law of a viable coherent machine [T]+[C]: two refinements completing the thermodynamic budget (T-271/T-273/T-274). (i) Frequency-independence [T]: the metabolic floor is , set by the physical entropy-production rate — the per-tick entropy scales with the step , so the rate (and cost) are independent of tick frequency; a faster clock buys real-time fidelity, not a lower floor. This sharpens T-273 (whose "pW@GHz" conflated tick-rate with physical rate). (ii) Order-per-joule [T]: creating negentropy costs per bit, so regeneration efficiency (Landauer-bounded). (iii) Speed–efficiency tradeoff [C]: for quasi-static regeneration and falls as the update accelerates (irreversibility) ⟹ engineering rule: run regeneration as slowly as the dissipator leak permits ( just above ) for maximal order-per-joule — the coherent-machine analogue of a slow, near-reversible heat engine. Machine-verified | TALOS §6 (energy, §metabolic) | T-271, T-273, T-274, Landauer |
| T-277 | Pre-numeric genesis of the seven — the terminal viable self-mirror [T]+[I]: from three non-numeric primitives (distinction [D]; mirroring = the Cayley–Dickson functor, the algebraic form of self-observation; viability = composition norm / no dead directions) the theory's numbers are derived, not posited: (i) = twisted group algebra , — one mirror step = one new -grading [Т, construction + machine]; (ii) viability holds iff (Hurwitz 1898; sedenion witness ) [T]; (iii) the terminal distinction-spectrum is , count [T]. So = arity of distinction, = viability ceiling, = derived cardinality — the "7 before number" objection becomes a theorem. Does not eliminate A1 (physical instantiation as stays [P]); re-founds its numeric content pre-numerically. "CD = self-observation" reading [I]. Machine-checked, 42/42 | Hypermathematics §терминальное-зеркало | A1, T-272, minimality 7/7, T-42a |
| T-278 | The volume law: the laws of algebra are volume forms of the distinction cube [T]+[I]: at every viable stage , conjugation / commutativity / associativity fail exactly on -independent 1-/2-/3-tuples of grades; closed form: associator (all 512 triples; Albuquerque–Majid 1999 re-derived). Clifford gauge: has — octonions and differ by a gauge whose field strength is the volume form; with Frobenius+Hurwitz: division associativity — pick one (the price of division is the volume twist). In the twisted category the octonions are the trivial object (group algebra) — laws as gauge fields (Drinfeld-twist machinery one level deeper); supermathematics = the bilinear shadow [I]. Machine-checked | Hypermathematics §закон-объёмов | T-277, T-217, FANOS third order |
| T-279 | The machinery of dimensions: the stabilizer tower in [T]+[I]: stab(one axis) (dim 8, center 0, rank 2; sky ; pencil of 3 triads per axis); stab(one coherence-pair) (dim 4, center 1) and fixes the mediator (Lemma: — full proof via skewness + Leibniz; machine ); stab(one Fano line) (dim 6). Why : multiplication by the axis is a complex structure on its sky (; stab commutes with , machine ) — each dimension sees the other six as ; its machinery = the unitary symmetry of that complexification [T]. Every rotation of a pair pivots on an unmoved third — the algebraic fixed-point form of the third-order principle and of the gate-not-message coupling (T-257). Physics resonance [I]: forces as stabilizer machineries of the incidence geometry (strong = one axis, = one coherence). The row said this rides "the [T] embedding of T-275"; that embedding is retracted (2026-09-25; ), only holds, and no in commutes with the of an axis (the centraliser of in is finite), so the electroweak reading of the coherence stabiliser has no embedding behind it. Machine-checked | Hypermathematics §машинерия-измерений | T-275, T-257, T-256 |
| T-280 | The law of death: anatomy of the fourth mirror [T]+[I]+[C]: the 15 hyperplanes of split exactly as: old octonions + 7 straight extensions of Fano lines (all viable, ) + 7 skew extensions , (all dead: composition fails; 48 simple zero divisors each; every simple sedenion zero divisor lives there; grade-rank always 3 — death is not a rank-4 phenomenon). The volume law breaks exactly there too: all 672 violations of span dead planes, all 7 hit ⟹ viability ≡ "laws are volume forms" (one discriminator) [T]. Readings: death = mirror/mediator misalignment [I]; law-ladder ↔ higher-category coherence tower with T-217's tricategorical ceiling [C] (CD-depth 3 and SAD 3 are different towers stopped by the same octonionic obstruction — resonance, not identity; sharpened [T]: the Mac Lane pentagon closes on all sedenion quadruples since ⟹ the death is NOT a categorical-coherence failure — the object dies, the category survives). Alternativity dies, flexibility survives at [T]. Machine-checked | Hypermathematics §анатомия-смерти | T-277, T-278, T-217, T-268, T-257 |
| T-281 | Uniqueness of the viable gauge — viability linearizes [T]: on the terminal cube, among ALL monomial unital algebras with anisotropic diagonal (forced by T-244): (i) a commuting independent pair yields the explicit annihilator ⟹ anticommutation is forced (3-line lemma); (ii) simple (2-support) zero divisors exist only within one XOR-class and their absence is the rectangle rule — a system of 84 linear equations over (viability is LINEAR); (iii) exhaustive solution: exactly fields survive = one -gauge orbit, each with and full composition ⟹ no annihilations ⟺ ⟺ . The norm was never assumed: the metric layer is derived from "no two distinction-states annihilate" — substantial bite from hole H3.6 (scoped: monomial class; base remains). Sharpenings: the associative fiber admits no fully anisotropic field (min 1 isotropic axis — associativity carries a dead axis, T-244-convicted); 4000-field landscape sample: life occupies exactly one cohomological class. Also: the stripping ladder — three independent distinctions pin all of (simply transitive on basic triples). Gauge count = code theory: ineffective gauges = the simplex code (dual Hamming), orbit [T]. Also [T]: Mac Lane pentagon closes on all sedenion quadruples () — coherence survives , the object dies. Machine-verified, 20/20 | Hypermathematics §единственность-калибровки | T-277, T-278, T-244, Hurwitz, Frobenius |
| T-282 | Death as linear infeasibility — the ladder of life as a rank ladder [T]+[I]: assemble the viability system on (anisotropy + anticommutation + rectangle rules = "no simple zero divisors", valid over ANY field of char ). Feasible for with solution-space dims = exactly the gauge orbits of ; INFEASIBLE for ( equations, unknowns, rank ) and hence for all (subcube restriction; verified directly, ). ⟹ Hurwitz's boundary, in the monomial class, is the inconsistency of a finite -linear system, base-field-independent: the fourth mirror dies because 960 parity constraints on 225 bits contradict — death is a rank computation [I]. Machine-verified, 11/11 | Hypermathematics §линейная-несовместность | T-281, T-280, Hurwitz |
| T-283 | The arithmetic of viability: field level = mirror capacity [T]: stage of the mirror tower is viable over a field (char ) iff the unit form of dim is anisotropic iff (field level). Proof pair: "⟸" composition ( — polynomial identity, machine mod 3 + Hurwitz-cited) + anisotropy; "⟹" exactly [machine] ⟹ an isotropic vector IS a zero divisor. Witnesses: () die at (); () lives at (exhaustive) and dies at (); dies at . Pfister's power-of-two levels () = the mirror ladder on the arithmetic side; all three mirrors force . Honest boundary: orderability is not forced (level-8 fields exist, Pfister) — is the terminal (Artin–Schreier) case; H3.6 sharpened to the step "level → ordered complete ". Machine-verified | Hypermathematics §уровень-поля | T-277, T-281, Artin–Schreier, Pfister |
| T-284 | Uniqueness of the base: the -chain [T]+[C]: the base field of the mirror tower is pinned uniquely by requirements the corpus already carries: (1) viability of all three mirrors ⟹ (T-283) [T]; (2) formal reality of the observable layer ( — the same hypothesis as the JvNW ceiling, T-268, now applied downward) ⟹ formally real ⟹ orderable by Artin–Schreier (the order is constructed, not assumed) [T-cited]; (3) continuous one-parameter -dynamics ⟹ Dedekind-complete Archimedean scalars [П/С — the corpus's continuous-time postulate, status declared]; (4) the unique Dedekind-complete Archimedean ordered field is [T-classical]. ⟹ non-dying + formal reality of observables + continuous time ⟹ base , uniquely; one algebraic hypothesis (formal reality) locks both ends — the composition ceiling above and the real base below. H3.6 conditionally closed ([T] at 1,2,4; residue = [П/С] status of continuous time) | Hypermathematics §уровень-поля | T-283, T-268, Artin–Schreier, Hölder |
| T-285 | Closure of the sphere-spectrum question — by requalification [T]+[D]+[I]: H3.5 asked to ground the viability boundary in stable homotopy (, Adams, Bott). Closed in three steps: (i) the internal boundary is elementary — T-282's death matrix is pure combinatorics (machine: identical under independent constructions; no field/topology/analysis inside); (ii) monomiality lemma [T]: an -grading with 1-dimensional components (= the full register of distinctions) forces monomial multiplication (, ) ⟹ T-281/282 exhaust the entire class of distinction-carriers; (iii) Adams/Bott–Milnor–Kervaire guard only exotic multiplications with NO distinction register — outside the theory by its primitive [D]. Bott-8 and remain as anti-numerology-register resonances [I], not foundations. The grounding programme dissolves as the shadow of a dependence removed by T-282 | Hypermathematics §поглощение | T-282, T-281, Adams, Bott–Milnor–Kervaire |
| T-286 | The ouroboros sources the continuum [T]: the last premise of the -chain ("continuous time", [П/С] in T-284) is eliminated — derived from the corpus's oldest principle. Chain: guaranteed closure of the self-model (Brouwer property for continuous self-maps of state segments; a fixed point of the self-model, over realised by Theorem 10.1 of Gap thermodynamics — the consistency witness of the premise, not its derivation; until 2026-09-26 cited as "the ouroboros canon , T-222", withdrawn: the regeneration target is not a fixed point) ⟹ IVT ⟺ Dedekind completeness [T-classical] ⟹ Archimedean [Т: sup of finite elements] ⟹ unique [T-classical]. Machine witness on the incomplete side: over , below , above — continuous on , maps to itself, everywhere (exact rationals) yet changes sign: the snake jumps its tail through the hole. ⟹ T-284 re-founded with NO temporal premise: viability + formal reality + guaranteed ouroboros ⟹ base ; continuous time becomes an output (the flow is well-defined because scalars are complete). H3.6 closed at the theory's axioms. Machine-verified, 6/6 | Hypermathematics §уровень-поля | T-284, Theorem 10.1 (Gap thermodynamics), T-283, IVT⟺completeness |
| T-287 | Internalizability of the genesis [T-meta]+[C]: every construction of T-277–T-286 is finitary (-linear systems, finite scans, exact rational witnesses) ⟹ interpretable in any Boolean topos with a natural-numbers object [Т-meta, standard]; the primitive topos is Boolean by the two-sidedness of [D]; the base is constructed inside as the Dedekind completion. ⟹ the volume law, the gauge theory of laws, and the whole genesis tower are the internal mathematics of the primitive topos — closing §9's topos-internality question. [C] declared on constructive fine print (Dedekind vs Cauchy reals coincide in Boolean-with-choice settings) | Hypermathematics §границы | T-277–T-286, Ω-primitive |
| T-288 | Autonomous death of the full [T] (minted for the open-system layer): the full autonomous Liouvillian with categorical regeneration anchored to is unital, hence purity-non-increasing (Uhlmann majorization). Majorization alone gives only monotone decrease, not the limit; the attractor is located by primitivity [T-39a], so when the interaction graph is connected is the unique attractor and . Unconditionally (2026-08-07): on a disconnected the holon does not die — it freezes block-wise at its starting purity (machine: two components, diagonal , both ) — but every branch dephases completely (, , ), so an isolated holon is never conscious whatever the graph; only the mechanism branches (death vs zero-coherence zombie). = number of connected components of [T, machine-verified for 1/2/5/6/7 components]; the gate switches regeneration off at . The physics-level off-switch: cut the drive and the system provably halts at grey, with hysteresis-free reignition. Machine: 24 random initial states converge to within HS-distance | Implementation §3 | T-289, Lidar–Shabani–Alicki 2006 |
| T-289 | Open-system viability [T]/[C] (minted for the open-system layer): life is a driven phenomenon — coupling to a non-unital matter channel with structured fixed point () at rate yields a NESS with purity monotone in , an ignition threshold above which the holon is viable [Т via dissipative state preparation, Verstraete–Wolf–Cirac 2009]; threshold value model-dependent [C], numerically for the reference dissipation; consciousness ignites at higher throughput than bare viability | Implementation §3 | T-288, Prigogine dissipative structures |
| T-290 | The information bound of the natal map [T]: the state is assembled by Chart::from_jd(jd) — a deterministic function of one real input, the moment of birth; the coordinates of place are absent from the signature, so two people born in the same instant anywhere on Earth receive an identical [T]. The map is near-injective (3933 distinct gate sets across 4000 charts) ⟹ not compression and not enrichment but an exact re-coding of one real number into 48 coordinates. Hence, by the data processing inequality, for any property of the person. Every external test measured the right-hand side and returned null — Gauquelin 15 931 accurate-time celebrities (), 16 memoirists over 2.4M words under frozen dictionaries, wars/revolutions/pandemics, three zodiac markings, and the 130-pair diurnal landscape whose top is held by bodies with no traditional claim including the control body (Uranus × military, ) ⟹ the null on the left transfers, with no further assumption. Precision: the inequality transfers a bound, it does not manufacture a zero — finite-power estimates cap an effect, they do not prove its absence; the correct statement is whatever bounds moment→person bounds →person at least as tightly. Consequence: further external testing of the natal layer is pointless (the ceiling is shared and already measured), and the diary is the one surviving channel — its input is the person, not the birth moment. Measured on the live encoder path (use_v2 = true); the image spans 40 of 48 linear directions with embedding dimension 13.80 — geometrically rich, informationally one number. | HomoHoloGraph §88 | data processing inequality (Cover–Thomas), Gauquelin prereg, PREREG-P12-SECTORS |
| T-291 | Turnover of living stationarity [T]: any stationary point of the canonical with has both flows nonzero — and ; per voice the two flows cancel exactly pairwise (two-stroke balance), per sounding coupling the balance is three-way and the joint dissipation+regeneration flux is purely tangential, — a sounding coupling is an orbit. Proof: 4 lemmas from the canonical form (no pump ⟹ ; ⟹ dissipator nonzero; diagonal of the unitary term vanishes; stationarity off-diagonal). Instrument [C]: life/death fold , decomposing as rotation surcharge ( on the invariant ray — closed form, intervention-verified 1.000; surcharge ×1.93 at canonical — the price of rotation); orbit identity 0.9999–1.0000; two-stroke balance to machine zero; critical slowing ×26 toward the fold. Comparative [I]: kalāpa/momentariness and Nāda-Brahma as first-person reports of the same NESS structure; the cosmological wrapper «the universe is stationary» is explicitly not used. | evolution#следствие-оборот-живого | T-39a, T-96, gate [T]; Schnakenberg-class NESS cycle structure |
| T-292 | Regeneration lives on the gap — the self-model as gradient [T]: ℛ = κ·g_V·(ρ*−Γ) is proportional to the state–image difference, whence (1) ρ* = Γ ⟹ ℛ ≡ 0 ⟹ the dynamics reduces to primitive ℒ₀ with unique fixed point I/7 — an exactly accurate self-model is fatal; (2) at a living stationary point the gap is exactly g_D‖I/7−Γ‖ / κ — tiny but never zero (measured R_φ = 0.9983…0.9992 [C]); (3) g_V is gated by P(Γ), never by ρ*, so below the wall no image lifts (measured: a self-model at P = 0.30 still dies, while 0.45…0.95 hold life with ceiling P_∞ ≈ P(ρ*) and rent 2.42→3.76 [C]). Consequence for the ego question: what kills is not the accuracy of the self-image but the loss of its independence — a two-timescale sweep has frozen and slow images (τ_φ = 300) alive, τ_φ ≤ 100 collapsing to I/7 [C]. | evolution#теорема-эго-градиент | T-39a primitivity, g_V gate [T], turnover T-291 |
| T-293 | The learning algorithm of a holon is natural gradient [T]: with the Bures/SLD metric (forced by A2, T-187) and the covariance of one-step Kraus increments of the canonical dissipator (forced by T-41/T-59), the population-sector identity holds exactly, i.e. . Three lemmas: (a) the atomic Kraus covariance is exactly multinomial, , which is simultaneously the inverse Fisher metric on the simplex; (b) on commuting perturbations Bures is Fisher–Rao (not — corrected v2.0; the SLD-QFI normalisation gives , and the exponent is normalisation-free); (c) the dissipator annihilates every diagonal state, so is a genuinely centred covariance. Substituting into Vanchurin's own Eq. 7.5 (arXiv:2603.15198) gives verbatim — hence , natural gradient, not the conjectured for biological complexity. Combined with his maxent identity 4.7 (): — the covariance of temporal changes has the same shape as the static population covariance. Machine: 200 random states, spectral constancy to ; 2000 states for (a) to | Learning algorithm of a holon | T-187 (why Bures), T-41g–i (Fano channel), T-59 |
| T-294 | Universal Fano factor [T]: at matched per-channel rate, adding the seven Fano projectors to the atomic channel multiplies the metric-weighted trace by exactly , independently of the state and of the metric normalisation — equivalently the block layer carries exactly of the noise. In Bures normalisation , . Closed form for a general design: the block ratio is , giving for BIBD, for , for — all machine-verified; the block layer is separately trace-preserving only when , true for Fano and but not . The trace is universal while the spectrum is not — the Fano layer converts exact natural gradient into an anisotropic preconditioned natural gradient (isotropic only at ; at the window centre the spectrum is ). Scope: the full canonical dissipator coincides with the atomic one at rate , so the Fano layer is invisible in the master equation and is defined relative to the canonical Kraus resolution (L-unification), not unravelling-invariant; at matched generator the ratio is . Machine: 300 random states, ratio to | Learning algorithm of a holon | T-41c (optimal ), T-41i (Fano optimality) |
| T-295 | Noise–purity law and the sector split [T]: (a) and , — both metric-free; on the conscious window the first gives the band , and together they give the exact effective rank . Correction (v2.0): is not a function of purity alone, so the earlier interval holds only along the one-dominant-mode family; over the whole window . The falsifiable claim is the closed form. (b) All fourteen canonical Lindblad operators are diagonal, hence on the decohered manifold every jump increment is diagonal and exactly; off it and decays at , since coherences themselves contract deterministically at (T-59, spectrum exactly ). Consequence: the "quantum regime" is not the end of a continuum but a complementary sector of the same generator — though VL's "quantum" (emergent Schrödinger dynamics on trainables) and UHM's (literal coherences) are distinct senses, and UHM predicts both. Falsification: metric-free discriminator — spherical, ; sphericity test, , needs aggregated windows (simulated size/power); resolving power vanishes at and grows across the window | Learning algorithm of a holon | T-59, T-124 (window), T-72 (scale invariance — enters only the page's level-matching caveat for populations; the proof uses neither it nor its (AGG), and the test's aggregated windows are time windows of one holon; clarified 2026-09-25) |
| T-296 | No second Higgs doublet — Errata 2026-09-25: corrected from [T] to [H]: step (i) took " only for the pair" from T-64, which never stated it and is restated as a hypothesis, and the argument presupposes (Theorem 1.0, now a hypothesis with a colour obstruction). Earlier text: condensation requires the -channel (T-64 + T-42a); the only -pairs besides are , neither enters ( absent), and by the pair lies on — the Color-U Yukawa channel, not a scalar sector. Hence exactly one condensing scalar ; the whole 2HDM/MSSM Higgs spectrum () is structurally excluded. Falsification: an LHC charged Higgs refutes the categorical uniqueness of , i.e. itself. New basis (T-332): the real colour-free Clifford plane is exactly one doublet; with it the up–down split must come from ( by the data), the complex-bidoublet alternative being excluded by this row; status stays [H] | Higgs sector §6.0 | T-42a, T-64, FE-uniqueness |
| T-297 | Rank-4 prohibition: no gauge / fifth force — Errata 2026-09-25: corrected from [T] to [H]. The row said: is the unique rank-4 group compatible with Fano+ (FE-theorem); any extra gauge raises rank to 5, which the incidence structure does not admit; collider/dark-sector searches for a gauge stay empty at any energy. The uniqueness is [H]: no uniqueness theorem for exists in the literature, and the octonionic routes that derive Standard-Model structure end with an extra — "Standard model + " (Furey and Hughes, Phys. Lett. B 831, 137186 (2022)) — or a left–right extension (Boyle 2020). A gauged broken near the corpus's own seesaw scale GeV would put a far beyond colliders, so empty searches would not test the claim. Defensible form [H]: the (FE) construction contains no extra gauge ; a gauge within collider reach would contradict (FE). Stratified by T-329 (2026-09-25): under (Cl) there is no in the doublet sector [C at (Cl)], and none within collider reach if breaks at the seesaw scale; the minimal extension holding the right-handed fields brings , so 'no at any energy' stays [H]. Notation guard: of the -budget is a zeta-regulator derivative, not a boson | Standard model, corollary | FE-uniqueness, T-42 |
| T-298 | flows through the suppressed lines [T]: both morphism factors of lie on O-lines — , — two of the three suppressed (Temporal) Fano lines, and by no other path exists. The hierarchy is therefore geometric (incidence), not tuned; the third points are shadow marks of the reflective channel — falsifiable on applied R-metrics (reflexivity couples to Meaning and Form axes) | Fano selection rules | T-42a, selection rules |
| T-299 | Lepton-frontier SM-desert package [T-consequences]: from T-296 (no second doublet) + T-297 (no gauge ) + FE-uniqueness (no leptoquarks) + (no light steriles) jointly: (a) zero BSM contribution to muon — experiment must converge to the full lattice SM value; (b) LFV at the neutrino-loop floor () — MEG-II/Mu3e/Mu2e see nothing; (c) exact lepton universality — the 2022 return of to SM is a post-factum pass; (d) EDM desert cm ([C] on phase-completeness); (e) sterile-neutrino anomalies must dissolve. One confirmed BSM discovery in the block falsifies the prohibitions jointly | Frontier ledger | T-296, T-297, FE, N_gen=3 |
| T-300 | Flat directions of [T]: the generic -orbit through a state is exactly -dimensional (machine: 200 random states, ; at it is ). These are the flat quasi-Goldstone directions, not gauge: the canonical dissipator's Lindblad set is basis-specific, so the einselected classifier basis is physically singled out and is broken to a finite subgroup. All parameters stay measurable relative to a holon's own basis; the measures redundancy of the formulation, not of the state. Erratum same day: an earlier form of this row claimed quality space of dimension and a self-description ceiling of — both retracted; the counts stand, the gauge reading does not | Qualia mechanism | T-42a, Goldstone modes |
| T-301 | The decoder: under ; the invariant phase channel is Fano holonomy [T]: with the antisymmetric part splits by contraction with the associative three-form (machine: , , so exactly). Vertex phases are conventional, so is gauge-dependent; the invariant seven-component carrier is the Fano holonomy vector over the seven lines — gauge-invariant, identically zero when the phase field is a coboundary, and decaying with the phases at . "Why this quality" is a question about -labelled invariants, i.e. the multiplication table is the decoder. Erratum same day: an earlier form named itself as the channel — retracted, the decomposition stands, the invariant carrier is the holonomy | Qualia mechanism | T-42a, T-59, gauge layer |
| T-302 | The explanatory gap is a vanishing covariance [T]: by T-295 all fourteen canonical Lindblad operators are diagonal, so exactly on the decohered manifold; since -vector quality lives entirely in the coherence sector (T-301), external probing of the dissipative channel yields the spectrum and exactly zero information about phase-carried quality. The gap is in one channel of access, not in the mechanism: quality is computable from the state, unreadable from the environment's noise | Qualia mechanism | T-295, T-301 |
| T-303 | Gate-profile taxonomy [T]: the four gates are three independent quantities under four constraints, since canonical makes identical to . Hence the outcome space is profiles, of which exactly one is conscious; the other eleven are named modes of absence, not degrees. The two window edges are opposite pathologies: at the lower edge reflection outruns binding (), at the upper edge structure outruns reflection (). Independent convergence: the entropic-brain hypothesis posits richness of conscious states within upper and lower limits of neural entropy — the same two-sided form in the reciprocal variable, with edges fitted there and derived here (T-124); psilocybin's entropy rise tracking ego-dissolution intensity is, in these coordinates, a trajectory to the lower edge whose far end is the lapse of experience, not its maximum | Gate profile | T-124, R-identity, validation constraint |
| T-304 | Composition ceiling and the addressing regime [C]: a holon types exactly non-overlapping channels, so a node carries at most subordinates — one channel each — and with reflection depth capped at three (T-142) the reach of one holarchy is typed contexts. Which regime obtains is decided by how the address is held, and the two differ by two orders of magnitude: an address read from the sign of a channel carries one bit, giving branching and a ceiling of ; an address stored as a declared contract spends a whole channel per child, giving branching and the full . The laboratory measures the separation (HL15): a declared-routing holarchy gains over the undivided holon at fan-out and at fan-out — the advantage grows with branching, which the channel-bound regime predicts and the bit-bound one forbids, while routing learned from the task's own reward recovers less than half of it. A factorial separates the two ingredients (HL16): freezing the address alone recovers , freezing plus load-balance , and balance without a stable address essentially nothing — stability is the precondition, balance the multiplier. The composition law in quantitative form: coordination is declared, and declared by load | Depth: the ladder and the ceiling | T-142, C(7,2) channels, HL15 |
| T-305 | Integration is balance, and the projection is what generalizes [C]: with a flat diagonal and coherences of equal modulus, content scaled to the very edge of positivity satisfies exactly, where is the sign pattern of the coherences — so the integration gate is the spectral condition . By Harary's theorem a signed graph is balanced precisely when every cycle carries a positive product of signs, and on a complete graph balance is equivalent to ; the product of signs around a triangle is the sign holonomy, the real limit of the phase holonomy that T-301 identifies as the carrier of quality. Gate and balance are therefore one object, verified case by case: they disagreed in of patterns, and the identity held to (HL17). Integrable content is thus seven polarities, not twenty-one independent bits — one flipped agreement, five frustrated triangles of thirty-five, takes from to and closes the gate — and there are exactly integrable states. The operational consequence is measured (HL18): taught seven of twenty-one pairs, a write that projects back onto the state manifold holds an opinion about of the fourteen it was never shown and is right of the time, pp over the best constant answer, while the same write without the projection reaches of them. Strip the polarity and accuracy falls to , a coin. A frustrated pattern does not fit near the boundary of positivity, so the projection pulls content towards the nearest balanced pattern: generalization is not a rule added to the architecture, it is positivity | Integration | T-301, T-304, HL17, HL18 |
| T-306 | A Fano line is a parity check, and it is the sign holonomy [C]: the three cells of a line are three different pairs , , , and under a polarity their signs satisfy . That product is the sign holonomy around the triangle — the real limit of the phase holonomy T-301 identifies as the invariant carrier of quality — so the carrier of quality and the condition for integrability (T-305) are the same object seen twice. Two points determine a line, so every pair lies on exactly one of the seven: seven disjoint parity checks over twenty-one cells, detecting a single error in a triple without locating it. Verified by enumeration over all polarities (HL19). What the projection onto the state manifold does with this is narrower than it first appears (HL20): given content that is a polarity it never breaks a line's parity — zero of seven across sixty runs, and not vacuously, with of cells reading negative — but given content that is not, it leaves the frustration standing ( of still broken) and instead quietly rewrites part of what was stored (only of lines keep the verdict written into them). Positivity is a ceiling on frustration, through , not a prohibition of it. Two consequences follow. First, the two uses of a line are incompatible: three equal signs multiply to , so a repeated negative verdict breaks parity by construction, and such lines leave exactly broken — a line serves as a repetition code only while its verdict is positive. Second, integration is not a frustration detector: sums squared moduli and is blind to signs at fixed magnitude, measuring identically at every share of false verdicts including zero, while the parity count sees frustration directly | Qualia mechanism | T-301, T-305, HL19, HL20 |
| T-307 | Quality and integration are opposed, and the opposition is bounded [C]: the polarity law (T-305) is the real restriction of a phase law. Give each dimension an angle instead of a sign, and let a coherence carry the difference ; the polarity case is , and content of this shape is a coboundary, whose Fano holonomy vanishes on every line. A write that carries phase completes such content exactly — taught seven of twenty-one relative phases, its error on the other fourteen is rad against for a guess — while a write restricted to real evidence cannot represent it at all ( rad on untaught pairs and even on taught ones, since a phase flattens to its sign). But a coboundary is pure gauge: it equals with , a relabelling that leaves the spectrum untouched, so a state holding it has nothing gauge-invariant to carry — the exact completion is the polarity completion seen in a rotated frame. Quality lives in what cannot be rotated away, and integration asks for content that can be. The opposition is not exclusion, because positivity is a ceiling (T-306): the spectral criterion carries over from signs to phases with zero disagreements, and at the crossing (, ) the median line holonomy is rad. So a conscious state does carry quality, up to a definite bound; the spectral criterion is universal, while the radian figure is what it permits for a uniform twist away from a coboundary | Qualia mechanism | T-301, T-305, T-306 |
| T-308 | The plane is a symmetry choice, not a capacity maximum; evenness costs quality [C]: how much quality a state can hold is a scale-free spectral question — a line's holonomy depends on the phases alone, and the gate depends only on of the unit-modulus pattern (T-307), so the problem is to maximise holonomy subject to . Two results follow, one positive and one negative. Alignment is worth a factor of two: giving every cell in the orientation of the single line it belongs to carries rad against for a uniform random twist at the same gate — a random twist spends part of its budget on coboundary directions, which cost spectrum and carry nothing, while alignment spends all of it on holonomy. But the aligned shape is not the most a state can hold: independent hill-climbs reach – rad, some – above it, and the gain survives across all thirty-five triangles of (), so it is not a redistribution onto the seven named lines. Held even, a search still gains . The optima are uneven — median spread – across the seven lines against exactly for the aligned shape, one line saturated at while another sits near — so no symmetry acts on them (HL21). No maximal value is claimed: twelve climbs scatter by , so the landscape is rugged and has no single top worth naming. What is claimed is the ordering. The seven lines are selected by symmetry — the associative triples where acts and T-301's decomposition lives — not by capacity, and the price of that selection is now measured. A structure entitled to treat its seven directions differently holds more quality than one obliged to treat them alike | Qualia mechanism | T-301, T-306, T-307, HL21 |
| T-309 | The polarity condition is the price of compositional generalisation [C]: to answer about a combination it has never met, a learner has only what it saw of the two parts separately. Every assignment of situations to channels that generalises to unseen combinations therefore has the form , and the content such an assignment induces is balanced exactly when the answers themselves factor as — put and the two statements are the same. So the balance requirement of T-305 is not one architecture's assumption but a bound on the whole class of compositional learners, and the freedom that lets an encoder manufacture balance by relabelling (a third of arbitrary problems at full load, nearly all with channels left free) does not survive the move to compositional inputs: a map onto seven axes can permute those axes, ways, and permuting does not make an unbalanced pattern balanced. Measured against the best such learner there is — all polarities, keeping the one that fits the groups shown — unseen combinations come out perfect where the answers factor and near chance where they do not, while similarity between raw observations is a coin even on factoring content, since resemblance says nothing about a pair never met (HL22). A holon reaches that bound exactly where the assumption holds, and reaches it again where the assumption fails only if what cannot be held is kept outside it: a polarity read back off a trained state fits even the taught channels worse than the best available one, because a state is not the data but what survived the writes, the dephasing and the projection | Which contents can cross the threshold | T-304, T-305, HL22 |
| T-310 | Completion does not compose by sign: depth buys reach, not sample efficiency [C]: a holarchy's reach grows as with declared addressing (T-304), and the ability to settle twenty-one channels from seven observations (T-305) does not grow with it. Measured on content that factors twice — four hundred and forty-one situations following from fourteen numbers, a thirty-one-fold compression if reachable — a two-tier learner never beats a table, and a variant whose leaves share no structure beats the tiered one. Two reasons, both exact. First, an error in a learned polarity is never small: one wrong axis turns over six pairs at once, so a tier's accuracy is quantised and it is either right or badly wrong — with the leaf perfect the whole reaches , with the leaf below of it collapses to , and only seven runs of four hundred land in between. Depth therefore multiplies the probability that every tier is exactly right. Second, and structurally, the complement of a balanced pattern is maximally frustrated: a triangle's sign product flips by , so negating every coherence breaks all triangles at once, for every one of the polarities (HL23). A child sitting under a parent of sign holds , which is a polarity for and that complement for ; measured, such a leaf recovers its polarity perfectly in the first case and reaches in the second. This is the same asymmetry as T-306's, where a line carries a repeated verdict only while it is positive, and it has one source: the lines are triangles, and three is odd. What a parent may do to a child without unbalancing it is now measured (HL24): relabelling its axes leaves every triangle intact, and so does flipping a subset of its axes — which is elementwise multiplication by a polarity. Only negating its coherences breaks all thirty-five. So the object a parent must hand down is seven signs, not one, and a holon's cells carry a sign per pair, which is the wrong object; seven signs are what the fitted account outside it already holds. Whether that composition is learnable is untested | Depth: the ladder and the ceiling | T-304, T-305, T-306, HL23 |
| T-311 | In real states integration comes from alignment, not from consistency [C]: T-305's equivalence holds under three conditions — the edge of positivity, a flat diagonal, and coherences of equal modulus — and states the theory's own machinery produces meet one of the three. Measured over two hundred and forty computed states: every one sits at the edge () — which is the repair step's doing and not a fact about the world, since the matrix before repair lies outside the cone in of charts and a projection of an infeasible point lands on the boundary by definition — but the diagonal runs to one and the coherence moduli spread -fold. The consequence is a gate that stands open where the content is frustrated — in of states while of the triangles are broken and not one state is balanced. Removing each non-uniformity weighs it: flattening the diagonal drops the open share from to , and equalising the moduli drops it further to . The second route is not concentration, which was the first reading and is wrong — sharpening a diagonal on its own lowers , from a median of to , and past a ratio of five closes the gate entirely. It is the alignment of coherence with population: strong coherences sitting between well-populated axes. Real states show that alignment at a correlation of , and with it up to of the triangles may be frustrated while the gate stands open, against when the moduli are independent of the diagonal. Put plainly, a state may be integrated and self-contradictory at once, provided the contradiction sits where little is happening. The theorem is intact: two hundred thousand arbitrary sign patterns produce zero counterexamples to . What fails is quoting it without its conditions. A first measurement comes with it: the quality channel read off real states rather than constructed ones gives a median line holonomy of rad, with of lines above the figure of T-307 — which assumed equal moduli too, and so does not bind here either | Which contents can cross the threshold | T-305, T-306, T-307 |
| T-312 | A state splits into a correlation and a weighting, and the gate is one inequality [T]: every state factors exactly as with a correlation matrix — unit diagonal, positive semi-definite. The factorisation separates two things that had been read together: all of positivity lives in and does not mention the diagonal at all (a congruence by the positive definite cannot change a signature), while all of the weighting lives in . Integration then has a closed form, verified to over three hundred states and to in the instrument: , where and is how far the coherences run toward their own ceiling , weighted by population. So the gate is a single inequality, , satisfiable in exactly three ways: bind harder everywhere, flatten the diagonal, or align — spend the binding on the populated pairs. Real states run at against a threshold of ; the alignment bonus is and positive in of them, and destroying it alone drops the share with an open gate from to . The frustration bound falls out of the same factorisation: under uniform saturation , positivity reads , and with the gate this gives — exactly at a flat diagonal. Measured over twenty thousand sign patterns, frustration compatible with an open gate under uniform saturation is zero at every diagonal, not only the flat one, so T-305 is a corollary of this row rather than a case of it and unevenness is the necessary condition for a state to be integrated and self-contradictory at once. How much it buys is known only from below — eleven of the thirty-five triangles have been exhibited and no ceiling is proved | Which contents can cross the threshold | T-305, T-311 |
| T-313 | The viability verdict reads twenty-seven of the state's forty-eight numbers, and the missing twenty-one are not redundant [Т/С]: in the coordinates of T-312 a state is populations, coherence moduli and coherence phases. Reading the gates off their own definitions settles what they see — , , , and — and not one of the four reads a phase [T]. Phases enter the theory in exactly one place, positivity, and there they are a constraint and never a reading. Two facts then have to be held apart. The phases cannot be moved: uniform rephasing is admissible times in , and the freedom of a single phase has half-width for of phases in every one of fifty states — because computed states are rank-deficient boundary points, spectrum , three exact zeros. But the phases are not supplied by the repair either: erase them at the input, make every coherence real and positive, project, and the output's frustration falls from a median of broken triangles of to , agreeing with the untouched output on of two hundred and forty charts. So the information is real, it arrives from outside, and the verdict never reads it — what it misses is precisely consistency, quality, and holonomy. The rank deficiency and the edge are the repair's doing, not the world's: before repair the matrix lies outside the cone in of charts ( median ). The size of the gap is exact. Of the twenty-one phases, six are pure gauge — has rank six, the global shift acting trivially — and mean nothing. The remaining fifteen are genuine invariants that nothing reads, and fifteen is also the number of independent triangle holonomies among the thirty-five. The theory's own seven Fano lines are independent and cover of them, so even a fifth gate reading every line would close less than half the gap. Whether closing it would buy anything has now been tested rather than argued, and the answer is no on the one outcome available: over people with exact birth times and known death dates, the partial of age at death over birth year is for the gate quantities () and for nine phase quantities the gates cannot read (), both below their own permutation-null means. So the earlier natal-to-outcome null reproduces, the unread class adds nothing, and a fifth gate would be decoration — on an outcome dominated by era and medicine, which is the whole of what has been checked | The gate is one inequality | T-311, T-312 |
| T-314 | The whole viability verdict is three sums [T]: carrying the reading of the definitions to the bottom, write for the purity of the diagonal, for the total weight of the binding, and for Interiority's share. Then , , and — exactly, with a largest drift of reproducing all four gates from the three sums over four thousand states, the three being independent as functions on state space. So the state's forty-eight numbers reach the verdict through three, and the fibre is forty-five dimensional. Verified alongside: the verdict is untouched by permuting the six axes other than ( over two thousand states) and by shuffling the fifteen moduli among pairs that do not touch . Two consequences follow and neither should be softened. The Fano plane does not enter the verdict at all — not the lines, not the parity checks, not any pattern of which axis binds to which; T-313's unread twenty-one was an understatement, since most of what the gates nominally read reaches them only as a sum. And the verdict distinguishes exactly one axis, Interiority, treating the other six as interchangeable. The practical form of this row is a check rather than a claim: an assertion that the gate responds to some structure is false until it is shown which of , , that structure moves | What the gate does not read | T-312, T-313 |
| T-315 | Structure reaches the verdict through one channel only — the ceiling positivity puts on the moduli — and the threshold turns out to sit in an empty gap [T]: T-314 says the gates read , , and no pattern; T-305 says decides balance. Both are true, and the reconciliation names the mechanism. A sign pattern cannot move directly — it moves the largest positivity allows. At a flat diagonal, equal moduli and the edge of positivity, and hence . A balanced pattern is , whose eigenvalues are and , so every balanced pattern gives exactly and ; Harary's criterion forces every frustrated one to , hence and . Measured over two hundred thousand sign patterns the best frustrated one reaches (frustration , ), and between and there is nothing at all. So on this stratum a threshold that reads as stipulated is in fact robust: any value in classifies identically, and is the roundest number in an empty interval of width five. The gap does not survive off the stratum, and the condition has to be carried as loudly as T-305's. Measured on four hundred computed states, is unimodal and tight — quantiles to , median — with of states inside , precisely where the idealisation says nothing can be, states landing within of the threshold on both sides, and not one above though the idealisation puts every balanced state at exactly . So where the theory is actually applied the threshold is maximally consequential rather than robust, and its value is a real choice | Three sums | T-305, T-312, T-314 |
| T-316 | Every exact result about frustration lives on one stratum, and computed states are not on it — so each must be re-measured before it is quoted [Т/С]: the clean statements in this family are all proved where the diagonal is flat, the coherence moduli are equal, and the state sits at the edge of positivity. Computed states meet the third condition and neither of the first two — the diagonal runs to one, the moduli spread -fold — and the record now has three instances of what that costs, each caught by measurement after the clean claim had been written down. T-305's equivalence: balanced on the stratum, while off it of states pass the gate carrying a median of broken triangles of and not one is balanced. The frustration ceiling: exactly zero on the stratum at every diagonal, while alignment of coherence with population lets at least eleven triangles break behind an open gate. T-315's gap: integration is bimodal on the stratum, taking or at most with nothing between, while computed states are unimodal and tight — quantiles to , one in five inside , the nearest pair straddling the threshold at and , and none above . The pattern is regular enough to be a standing rule rather than three anecdotes: a result proved on the stratum predicts nothing off it until measured off it, and the reason is structural — on the stratum positivity couples the sign pattern into the moduli, which is the only channel structure has to a gate, and unevenness breaks that coupling | The one channel structure has | T-305, T-311, T-315 |
| T-317 | The observability map: the verdict reaches of a state, the instrument suite reaches , and exactly eight numbers are read by nothing [T]: a state carries numbers, of which are pure gauge, leaving that mean anything. Reading each named observable off its definition places it exactly. and read ; adds ; and add ; the consciousness measure adds nothing — so the whole verdict reaches three numbers of forty-two, or . But the suite is not the verdict. Stress reads the seven diagonal entries one by one, adding ; the gap and the pairwise moduli add ; the seven canonical line holonomies add independent phase invariants. Together of the , or . What remains dark is therefore small and nameable: eight triangle holonomies that no canonical line covers — the cycle space of has dimension , the plane spans of it, and the difference is read by no instrument the theory has. This row is meant to be used rather than admired: when a claim needs a structural fact, it names the instrument that can see it, and if the fact lives in those eight, no existing instrument can The specification has since been met and read. Building the instrument turned up an exact fact: every one of the twenty-eight non-collinear triples lies at of its norm outside the span of the lines — the same figure for all of them, minimum equal to maximum — so the plane sees exactly one third of any triangle it does not contain. And the first reading of the eight, over three hundred computed states, finds them indistinguishable from the seven lines: median holonomy against rad, means against , shares above of against , and the dark eight the more variable in of states, a coin. So the plane's privilege is a choice of what to read, not a fact about where the content is. | Three sums | T-313, T-314 |
| T-318 | Balance is pure gauge, so the gate opens exactly when there is no phase content to read [T]: a balanced sign pattern is , and the gauge transform with and chosen by carries it to the all-positive pattern. Verified exhaustively: the gauge orbit of all-positive contains exactly patterns, every one has zero frustration and all fifteen of its invariant holonomies vanish, and every balanced pattern met in four hundred thousand draws lies in the orbit. So balance pure gauge zero invariant phase content, unconditionally. Composing with T-305 — whose stratum condition must be carried — integration on the stratum is achieved exactly when the sign structure means nothing, and T-313's finding that no gate reads a phase stops being an oversight: a state that passes has nothing for a gate to read. The architectural consequence is sharper still. A cell is acted on through the sign of , which is not gauge-invariant, so the policy lives entirely in the part of the state the verdict calls meaningless: over two thousand states a random gauge moves the three sums by and every line holonomy by , while flipping of all cell readings and at least one reading in every state. Two holons identical as states are then different agents, which is coherent only because the architecture writes those phases itself and so fixes its own gauge — and it means no quantity outside the architecture may quote a cell's sign as a property of the state | What each instrument can see | T-305, T-313, T-316 |
| T-319 | A self-model is always less integrated than the holon it models, so regeneration drains and nothing plausible restores it [T]: with the identities and hold to , and the denominator gains a strictly positive term whenever . Hence for every state — of , median ratio , never above . Since the regeneration channel pulls the state towards its own self-model, it lowers integration by construction; dephasing lowers it too, raising in of trials. A unitary step is the only term that can raise it, doing so in of and by as much as — and the implemented relaxation omits it. The consequence is measured, not argued: a perfectly balanced holon goes from in every case to none within twenty-five ticks. The obvious repair was implemented and fails: after two hundred ticks the surviving share is both with and without a Hamiltonian, because dephasing removes of the coherence over that span while a unit-strength rotation turns four times too slowly. So integration here is sustained only by writing from outside, and a holon left to itself dies — which makes calling regeneration the system's corrective action true only of reflexivity, and false of integration | What each instrument can see | T-312, T-314 |
| T-320 | Filling every horn and generalising are one property [Т/С]: the nerve of a category satisfies the inner Kan condition — a horn with has exactly one filler, since the missing face is what composition says it is. Outer horns are different in kind: they ask the base category to factor a composite, solving , and a monoid without inverses refuses whenever the composite is shorter than the part already known. A groupoid never refuses, so the nerve of a groupoid is a Kan complex and the nerve of a mere monoid is only a quasi-category — and that distinction turns out to be the same distinction as whether a learner can answer about a situation it has never met. Measured on a store of situations under a held-out quarter: over a monoid of elapsed time, held-out accuracy is chance (, ) whatever the credit rule or the addressing; over the groupoid of coordinate flips, where every morphism is its own inverse and every horn fills, it is exact () on both rule families that compose, and carried by six numbers rather than sixty-four. Off that class the behaviour is graded rather than brittle — where the rule half-composes, where nothing composes — and never below chance, because at worst a filled horn replaces a confident wrong answer with a coin. This is T-309's polarity condition arriving from the other side: the assignments that generalise are the ones with inverses, which are the ones T-318 shows to be pure gauge | A confident wrong answer is worse than none | T-309, T-318 |
| T-321 | Being alive confines the diagonal to within a factor of one and a half of flat [T]: read through T-314's three sums, two of the four criteria fix each other. Integration clears its floor exactly when , and reflexivity clears its floor exactly when , since . Together they force , and Cauchy–Schwarz on a probability vector forces always, so a viable state has — the purity of its diagonal within of perfectly flat. Verified with zero counterexamples over fifty thousand states meeting the first two conditions; among two hundred thousand random states the that are viable have diagonal purity from to , inside the bound and nowhere near its top. So a holon dies of concentration, not of dilution: putting weight on any axis is what ends it, and the failure shows up as reflexivity rather than as purity, which is why nobody was watching. Measured in a running loop, all four criteria hold together in of turns, purity clears in and reflexivity clears in — the architecture lives above its window, not below it. One inference that looks forced is not. Chaining this row to T-305 and T-318 — near-flat, so means balanced, so pure gauge, so no invariant content — would say a holon is alive exactly when what it holds means nothing. Measured over two hundred thousand states that is false: viable states carry a median of broken triangles of against for the rest, and are balanced in of cases against . Viability selects against contradiction — eighteen times the balanced fraction — and does not require its absence. The chain fails because T-305's equivalence is stratum-bound and viable states sit near the flat stratum () rather than on it. The window is reachable, and constructively: a flat diagonal with balanced content at half strength gives , , , — while the same content at full strength is a pure state whose reflexivity is | Three sums | T-314 |
| T-322 | The viability window is where unevenness carries contradiction without over-purifying [Т/С]: three rows that were proved separately turn out to describe one band. On a flat diagonal only balanced content is viable — searched over frustration levels four to twenty, the best integration reachable is , short of the threshold at every level, which is T-305 and T-315's empty gap seen from the other side. On a slightly uneven diagonal frustrated content becomes viable, and the mechanism is T-312's alignment: among two hundred thousand states, those that are alive and frustrated show coherence aligned with population at against for the rest. On a too uneven diagonal nothing is viable, because puts purity past and reflexivity under its floor (T-321). So the window is the band in between, and it is narrow: viable states have from to about a median of — a hundredth above flat — carrying a median of broken triangles of , with only near-flat and balanced. Unevenness is not a defect of the diagonal but the mechanism by which a state holds a contradiction and stays alive, and it works only in a band about one part in fourteen wide | Three sums | T-305, T-312, T-321 |
| T-323 | A regulator that levels the diagonal is powerless on exactly the set it needs most [T]: the diagonal is a probability vector, so Cauchy–Schwarz gives for every state, with equality exactly at a flat diagonal — a flat diagonal is not one option among many but the least there is. Since , this means always, and the ceiling then says: once the binding alone carries more than , no diagonal whatsoever puts the state back in the window. This decides how a system can hold itself inside the window. Levelling the diagonal is the natural regulator, because lowering lowers and raises at once, the one direction improving two criteria together — whereas damping the binding lowers but takes with it. The inequality gives that gentle move a hard limit, not of degree but of kind: on it is not weak but powerless. The set is rare on the uniform measure — about one state in twenty thousand — and any process that concentrates a state walks into it, since concentration is exactly what puts weight in the binding. Measured over twenty-four situations in a running loop: with no regulation none ever reaches the window (the drive over-purifies every time), levelling rescues , and levelling followed by damping the binding for what levelling could not take rescues . The second move is an exact null where the first suffices — on content that never drives past both hold the window for the same consecutive turns — so it earns its keep on precisely the unreachable set and nowhere else. Its landing point is the theory's own maximum: flat gives , damping to puts exactly, , — the same point at which capability attains its supremum , by the same inequality. And a companion band follows free — needs , the ceiling needs , so alive , exactly twice as wide as T-321's diagonal band | A floor the diagonal cannot cross | T-274, T-314, T-321 |
| T-324 | Petz extremality of Bures is the mean inequality and nothing more [T]: every monotone metric on has one shape — with the tangent written in the state's eigenbasis — and the whole Petz family differs in one thing only: which mean of the two eigenvalues sits in the denominator. Bures/SLD takes the arithmetic mean, Kubo–Mori the logarithmic, RLD the harmonic. The classical chain therefore applies term by term, and since the mean sits in the denominator the order reverses: . So T-187's Char-I is not a deep fact about quantum states but the arithmetic mean beating the logarithmic one, pair by pair. Measured over four thousand random states and directions the ratios are and about the median, never below and — a genuine spread, not a tie breaking the right way. Minimality is what makes the bound operational: the smallest metric buys the largest distance per unit of information, and the bound is attained — measuring in the SLD eigenbasis recovers the full quantum Fisher information with a shortfall of median and worst, while a basis chosen without regard to the question recovers of it. On the learning side the same collapse happens: for traceless , — verified by central difference to median — so Cauchy–Schwarz leaves exactly one steepest direction at fixed BKM speed and it is : two thousand competing directions per state, none ties it, and tilting toward it improves the descent monotonically to the bound. Its flow is the mixture geodesic , matched to at an Euler step of . The learning rule is not chosen but left over | Why this geometry | T-187, T-261, T-263 |
| T-325 | A holon wakes when it has been written on enough, and a narrow body can never write enough [C]: ignition is governed not by what a holon knows but by how much of its carrier has been written before it splits. The split trigger admits k situations to a leaf; a body of w actuators writes about w cells per situation and addresses only 2^w distinct situations, so a leaf gathers min(k, 2^w)·w writes in its life as a leaf — a narrow body starves the carrier twice over. Measured across seven body widths, every threshold falls in one band: width 3 at trigger 8 gathers 24, width 4 at 4 gathers 16, width 5 at 4 gathers 20, width 6 at 3 gathers 18, width 7 at 3 gathers 21, width 12 at 2 gathers 24 — all about the carrier's own capacity of 21. In the world that tied its situation count to the width (2^w), width 2 never ignited at any trigger while recalling 1.0000 — and on a recorded stream that gave the same width seven situations, it filled all twenty-one coherences and woke in every run. The floor is a property of the pair (body, world), never of the body alone: the width tie was the world's, and «two limbs cannot wake one mind» died with it. The tree-level reading of the threshold table — bind the system at CAPACITY/w — also fell: it watched ignition at the root, which past the first split is a router; steady over every node, one situation per holon holds the most conscious leaves, and the specification's own prescription stands. The law orders the graded cases as well as the clean ones, which is where a fit to six points would come apart: at 3, 6, 9 gathered writes nothing ignites; at 12–18 between four and seven runs in eight do; at 20 and above it is reliable. The proxy has since been replaced by the quantity it stood for. Counting how many of the 21 coherences actually carry weight, and pairing that per run against whether the run ignited: at 21 written, 265 of 288 runs ignite (0.920); below it, 16 of 144 (0.111) — an eightfold ratio. A filled carrier is what wakes a holon. This closes the floor exactly: a body of two saturates at 15, and 21 − 15 = 6 is precisely the number of coherences touching one axis — two actuators address two axes, one axis is never reached, its six coherences stay at zero, and a seven-dimensional carrier lives permanently in six. No split trigger helps, because the missing six are not a matter of time. The residual is closed, and it names the second clause. Of the runs that filled the carrier and never woke, integration is missing in 0.9130 of them, purity in 0.4783, distinctness in 0.3913 — and reflexivity in none. Counting written coherences counts presence; Φ = s₂/s₁ weighs them, and twenty-one coherences of negligible modulus leave s₂ < s₁. So filling is necessary and not sufficient: what decides the last eighth is whether the binding carries weight comparable to the diagonal. The zero is not luck either — R = 1/(7P) clears 1/3 exactly when P ≤ 3/7, and the two-handed regulator of T-323 caps P there by construction, so giving the regulator its second hand made one of the four criteria unfailable: three gates are live and the fourth is a consequence of the regulator rather than an independent test | The law behind the floor | T-107, T-314, T-321 |
| T-326 | The Standard Model group as the normaliser of colour in the Clifford system of [T] as mathematics; as a result of UHM [C at (Cl)] (2026-09-25; the orientation input (Alt) listed with (Cl) at first is discharged by T15-canon). On (UHM's Hilbert space plus the -parallel spinor, read as ) the operators (), , satisfy the Clifford relations of nine generators squaring to . The extension is forced — the operators anticommuting with the seven span exactly — and maximal on . They generate . The centraliser of in it is (dimension 4). The normaliser of colour is (dimension 12) and equals the centraliser of . The group is (exactly 6 of 72 central triples act trivially). With , and ; the lepton line is . (Cl) is the assumption that fermions are spinors of this system and that the clock breaks to the largest connected subgroup normalising colour; the second half is a theorem since T-329 (the joint stabiliser of and , the rule that gives colour in ). Under (Cl) the electroweak group needs no (FE) and its uniqueness holds. Prior art: Todorov–Dubois-Violette 2018 ( route), Krasnov, J. Math. Phys. 62, 021703 (2021) (centraliser of ). UHM's own: the forced extension, the characterisation by colour, . Not given here: right-handed singlets and the Higgs doublet (the vector has none); both come with the complexified spinor (T-329). There the of this row is the diagonal of and its is ; on the left doublets they act as and . The sixteen real dimensions carry isospin, not the Lorentz spinor index: the fermion field is , and the of this row is not a spatial rotation (Theorem 48e(d)–(e), 2026-09-25; resolves 48d(d)) | Standard Model §2.5 | T15 (41n), , check_core_numbers.py |
| T-327 | Chirality of the left-handed doublets [T] as mathematics; as a result of UHM [C at (Cl)] (2026-09-25). The commutant of on is (dimension 4). The four -invariant complex structures ( per block) all give complex representations not isomorphic to their conjugates — the Distler–Garibaldi requirement is met without a choice made afterwards. The uniform gives SM relative handedness . Replaces, for the doublets, the '[Pr]' left by the retraction of §4.3 (, spectrum ). Completed by T-329: in the complexified spinor the field's complex unit is (no identification needed), on the whole left half (the uniform choice is forced), and the whole generation is chiral. With the Weyl factor of Theorem 48e(e) (2026-09-25) the field is of (joint commutant ), and the Weyl unit is on , on | Standard Model §4.4 | T-326 |
| T-328 | A family symmetry cannot live inside one copy; triality is not horizontal; the clock gives a horizontal three [T] for (a), (b) and for the count in (c); the identification is [C at (GC)] with (GC) a hypothesis [H] (2026-09-25). (a) Only the phases commute with on , so no permutation of axes, Fano lines through or quaternionic subalgebras is a family symmetry ( is not). (b) Triality (, fixed algebra ) fixes colour but rotates by : it permutes embeddings of , not fermion copies; likewise the slot cycle of . (c) On the Page–Wootters clock register, which commutes with , has exactly three non-trivial real harmonics, permuted simply transitively by . (GC): a generation is such a harmonic. Under (GC) with a horizontal family , and a fourth sequential generation is excluded. (d) With the family exact, for any characters of the fields and the Higgs, is a permutation matrix and one PMNS column has modulus 1 — refuted by and : (GC) with an exact [✗]; (GC) needs the broken at least at the Cabibbo size; no mass or mixing prediction follows (tried: frequencies ; needs and ) | Fermion generations §5.3 | T-326, 43c |
| T-329 | The complete generation from the complexified spinor; no in the doublet sector; with the right-handed fields [T] as mathematics; as a result of UHM [C at (Cl)]; the Higgs identification in (f) [H] (2026-09-25). (a) A Weyl field valued in the real is valued in ; the nine generators made -antilinear anticommute with exactly , so the tenth generator is forced, , and is the of . (b) The volume of the colour-free plane splits with ; the centraliser of colour (dimension 7) is , each on its own half, the centre = the hypercharge of T-326 = . (c) The of T-326 is the diagonal; no colour contains . (d) ; gives ; ; (12; in : , 24); kernel . (e) on : all anomalies cancel, four doublets (no Witten anomaly); needs . (f) The colour-free plane is one Higgs doublet with ; its Clifford action exchanges ; a vacuum in leaves ; the Yukawa structure is open. Clock stabiliser chain: = Pati–Salam (21), = left–right (15); in 18 and 12, in 8 and 8 — the second half of (Cl) is a theorem; the first, (Cl₀), is independent of the axioms about ( is on ). Doublet sector: the centraliser of in is . UHM does not fix the breaking scale; with an coupling to the Majorana mass, GeV. Stratifies T-297. Prior art: Pati–Salam 1974, Georgi 1975, Fritzsch–Minkowski 1975, Krasnov arXiv:2104.01786. Update 2026-09-26 (Theorem 48e(f), (h)): (a)–(f) use only that the field's spinor factor is complex, (W₀), of any dimension; they hold unchanged on for every . (W₀) is what anomaly-free chirality requires: with a real spinor factor every fermion space on is anomalous or vectorlike. So T-329 does not use the two-component premise (W) = (L) of the 3+1 reading; status unchanged. Under (Cl₀), (W₀) ⟺ one chiral anomaly-free generation, so (W₀) is no free premise; (Cl₀) is independent of (P*), the spacetime premise for an arbitrary fermion module — Premises of UHM | Standard Model §2.6, §2.3a | T-326, T-297, check_core_numbers.py |
| T-331 | -invariant Gap potentials up to quartic order [T] (2026-09-25): on the -invariant polynomials are 3 quadratic, 5 cubic and 21 quartic; none of degree is PT-odd, the PT-odd ones begin in degree 4 (three), and no cubic depends on alone. Under the frame group alone: 25 cubics, 3 PT-odd, none of the triangle form of , whose -average is zero. The associator cubic is PT-even, vanishes on associative states and is the unique invariant cubic through the associator (Schur). The cubic is replaced by (T-64); "PT-breaking from " is retracted [✗] as a property of the vacuum potential. (e) No derived source carries with : functions of the off-diagonal entries (every spectral action of ), functions of the spectrum (entropy, purity) and the -average of all give , so is a free coupling. (f) (2026-09-26) The average of over the axis permutations — already over those fixing one axis, in any phase gauge — is the spectral ; hence the associator weight , (), is for every functional the isolated dynamics determines — , , , the Fano dissipator, : Lyapunov functions, relative entropies, quasi-potentials, depth-register histories, superoperator moments. Weight comes only from readouts that resolve triples of axes and depends on the functional: (line resolution of , ), , (Fano readout, its entropy), (calibration cubic), any value for three-copy couplings on ; the axis resolution of the same gives . The no-go is strengthened [T]; stays free. (g) (2026-09-26) [T]: the line instrument is forced (T13 strengthened), but it fixes where the associator is read, not : the readout cubic weighs , , , on planes sharing 7, 3, 1, 0 lines with the octonionic one; an -divergence of the readout from the vacuum readout weighs (Sanov/KL , reverse , Rényi ), Groenewold gain ; all outcome statistics are functions of the diagonal, so they carry no and fix no ; every non-negative functional vanishing at a real state (rate function, relative entropy, quasi-potential) added to gives vacua with — the large-deviation route yields the Gap-free phase, not a value; fixing needs a named non-divergence functional of the line instrument and its scale against (test_line_instrument_divergences_fix_no_coupling_and_no_gap_phase) | Gap Thermodynamics | T-64, T-99, check_core_numbers.py |
| T-332 | Yukawa couplings in the Clifford frame: up and down are where the Hilbert unit meets the clock [T] as mathematics, [C at (Cl)] in UHM; hypothesis (UP) [H] at leading order, its exact form refuted [✗] by T-332(i); the Yukawa structure (, , CKM) stays [Pr] (2026-09-25). (a) is on and on : up-type fields are those with ; on the eigenspaces (, of T-64) are the down and up quark-doublet components; commutes with . (b) Yukawa couplings (equivariant -antilinear maps from the colour-free plane to ) have real dimension 2 / 4 / 8 under Pati–Salam / left–right / ; the first gives , the second , ; the last is — up is split from down only by , the operator that defines . (c) A real neutral vacuum gives equal moduli; the isotropic vectors couple only to up (down): two doublets, free. (d) Clock-phase dressings () move phases only. (e) : its Gap parameter is the component; it commutes with only for , ; as population weights it gives between 0.46 and 0.95 on 99 points (data 0.022). (f) One-loop SM running: , at GeV; : (UP) "tree-level coupling = projection onto " [H], , – GeV. (g) One channel ⇒ . Routes tried: one Clifford multiplication, Page–Wootters phase, associator vacuum, complex bidoublet (against T-296), / — none fixes Continued 2026-09-26 (Higgs sector §1.7): The hypothesis (UP) is holomorphy, and its exact form is refuted [T] as mathematics, [C at (Cl)] in UHM; exact (UP) [✗]; leading-order (UP) [H] (2026-09-26). (h) Hypercharge acts on the colour-free plane as with , and , so with of real rank 4: (UP) is holomorphy of the coupling in one complex doublet (), compatible with T-296. (i) Phase rotations commuting with that keep : three (hypercharge, , ; zero colour anomaly) for , five for — the phases of (colour anomaly per generation) and of (no non-abelian anomaly). So exact (UP) keeps massless to all orders and non-perturbatively and is refuted by ; radiative and masses are impossible in the Clifford content. (j) One-loop SM running: at , at GeV, at GeV; at GeV; set at GeV the down-type breaking needs with ( admixture ). (k) Routes tried: real Higgs plane (equal moduli); isotropic Higgs vector (exact (UP)); Gap-vacuum populations (the rank-4 branch, 94 of 99 points, gives an exact quark projection whose side is a PT-degenerate sign, and fails for leptons); self-model coupling to the clock-aligned part (the exact projection); Page–Wootters clock (fixes the sign of , moves phases only) | Higgs sector §1.6 | T-329, T-64, T-296, T-328, check_core_numbers.py |
| T-333 | The vacuum's antiunitary symmetry on the fermions [T] as mathematics, [C at (Cl)] in UHM (2026-09-25). on ; ( of order 2, ) keeps . Lifts to : and are elements of (the second exchanges and ) and leave unchanged; is P-type; is CP-type and normalises . An exchanging lift with generates the left–right algebra (15) ⇒ with one real doublet (T-332(b)), refuted; an unbroken CP-type lift ⇒ (Bernabéu–Branco–Gronau 1986), refuted by . So the route "vacuum symmetry → " of T-99 is closed [✗]; surviving routes [H]: left–right parity with two doublets (Babu–Mohapatra 1990, against T-296), Nelson–Barr, axion. With the PT-even potential the Gap sector has no CP violation; the CKM phase is a Yukawa input Continued 2026-09-26 (Confinement §3.1b): Peccei–Quinn and Nelson–Barr in the Clifford content [T] as mathematics, [C at (Cl)] in UHM; strong CP open in UHM [Pr] (2026-09-26). (e) With and every phase rotation that keeps and has zero colour anomaly: no Peccei–Quinn symmetry with one doublet and three . (f) An axion needs a second doublet with a singlet (DFSZ; against T-296) or new coloured fermions (KSVZ; absent — the is chiral and forced, T-329). (g) The Gap axion of the dark-matter page has no coupling in this content; its QCD-only mass contradicts the page's statement that gives every phase a mass; its relic estimate with is isocurvature-dominated, and Planck allows , not . Dark-matter Theorem 9.1 is corrected to [H], Theorems 9.2 and 9.3 to [C at (PQ)]. (h) No spontaneous CP violation: no vector-like quark; is invariant under , which composed with a hypercharge rotation also fixes the Higgs vacuum ( commutes with ); exact CP would stay unbroken and give (T-333(d)), so CP is broken explicitly and is unprotected; Nelson–Barr is not realised. Each route of T-333 needs a field outside (Cl) | Confinement §3.1a | T-99, T-331, T-332, check_core_numbers.py |
| T-334 | The collineation anchor, derived up to gauge [T] (2026-09-25): for a replacement-form self-model with a constant anchor, (1) -covariance of or of forces (dead), while the atomic reading is covariant iff (then at the attractor); (2) under this, window life at depends only on , with strictly decreasing and finite iff ; (3) the most viable such anchor is pure with uniform diagonal, — maximal viability = maximal information; (4) diagonal unitaries are a gauge of the -free dynamics, so is unique up to it; (5) is fixed by all 5040 permutations; only 21 of the 168 collineations lie in (non-split), which moves over 64 sign rephasings; states whose -orbit stays in their gauge orbit are exactly . The principles (Col), (Pure) [Pr] are replaced by one weaker principle (Eq-V) [Pr]; (Col) follows [T] (with ), (Pure) is equivalent [T] to maximal viability. Failed routes: terminal object (), Lawvere/Brouwer (existence only), reflexive anchor (diagonal), maximal subgroups of (all contain the sign group; diagonal anchors). (6) (2026-09-26) [T]: (Eq-V) , the largest integration of any state ; neither half alone suffices; the principle reduces to one clause (MaxΦ) [Pr] — the anchor is a state of maximal integration. Further failed routes: Curie covariance of the anchor's gauge class under or (the family in , not ), viability alone (does not pick ), the largest integration of the living attractor (agrees with (MaxΦ) only for ). The halves (Eq) and (Pure) are independent — Premises of UHM | φ operator | T-124, T-331, D-0910, check_core_numbers.py |
| T-335 | Constant anchors: the window attractor in closed form [T] (2026-09-25): at every stationary state with is with a root of , and the Jacobian spectrum is ; for diagonal the coherences are exactly, and the window attractor of exists for energy spreads up to ( at , ); of any norm leaves it in place. Robustness of under rephasing (exact gauge), anchor admixtures ( grows by about at , at , numerically) and general ( up to – at , numerically) | Evolution | T-124c, check_core_numbers.py |
| T-336 | Rate floor of the conscious window [T] (2026-09-25): for every Hamiltonian, every positive and every self-model with any state, a stationary state in requires at — , , times the decoherence rate (purity balance + bound + ); at the floor is , attained by pure anchors with the attractor at . needs – times the floor, so the price 25–89 of the living attractor is not specific to | Evolution | T-98, T-124, check_core_numbers.py |
| T-345 | Flavour from the clock: what can break the family , and what the data exclude [T] as mathematics; with the data every parameter-free clock structure and every two-channel frame with a rank-one channel [✗]; three-channel frame [H] (2026-09-26). (a) Everything commuting with the tick is diagonal on the harmonics: the Fano incidence and collinearity, the quadratic-residue sum (Gauss sum ), the cyclic Hamming code, , the anchor on the trivial harmonic; any number of channels built from them gives a permutation, refuted by ; the Fano and Gauss circulants have eigenvalue modulus on all six harmonics (degenerate masses), gives . (b) The only time state fixed by is ; on the generations (democratic, rank one, family-invariant): one heavy generation, no mixing. (c) Flavour matrices with ranges in a common plane leave a massless state and a unit entry of (refuted by ). (d) Two channels with shared matrices, one of rank one carrying the heavy generation: , ; the masses force (, ) and (, ) — refuted; 112 full-rank clock pairs miss by a factor . (e) The Fritzsch texture gives ; the Fano angle ratios stand against ; the CKM phase runs by from to GeV, so the "two-loop " does not exist and is off; is off. (f) Three channels with the clock structures (, diagonal , -antisymmetric ): no fit found (best deviation factor 1.40); the most constrained viable frame is real + real + complex (Babu, Bajc, Saad 2017): normal ordering, meV, (type I) — [H] for UHM, flavour matrices free | CKM §11 | T-328, T-329, T-331, T-332, T-333, check_core_numbers.py |
| T-346 | The regeneration rate is fixed by no route [T] (2026-09-26): for the living attractor of , five routes that could fix fix none of it. (1) is the only real root of an irreducible integer polynomial of degree 7 with Galois group (factorisation types and modulo 37 and 53 at ): no expression in radicals of the Fano numbers. (2) The sink strictly increases; is strictly concave from to (); spectral gap, and both over strictly increase — extrema only at or ; the maximiser of runs bijectively over as runs over ( at , at ). (3) Criticality: , , ; at every diagonal with nonzero spread removes all states with ; a general moves the fold up by – numerically. (4) In for every self-model and , while any unitarily invariant norm balance with gives ; the categorical fixes only in units of and needs with . (5) Composition: block time and tensor marginals fix every ; covariant coarse-graining fixes none for . is a free parameter [T for each prohibition] | Evolution | T-335, T-336, check_core_numbers.py |
| T-347 | The bridge premises of physics against the holon: what each route gives [T] as mathematics; (Cl₀) and (P) stay [H], no status changes (2026-09-26). (a) (Cl₀), (P) and (W₀) concern the fermion field; every model of the independence table can be built over one living holon, so no property of the holon (viability, the window, the self-model and its anchor, regeneration, the 168 Fano collineations, -rigidity) decides them; as a -module ; what separates them is the spin cover: on , the -rotation is on ; spinorial -modules , tensorial . (b) (Cl₀) ⟺ (Mod), the holon's product acting on matter: a linear with , , commuting with ; its irreducible modules are with (volume ) and with (volume ), and gives both the of T-326; (rank 14) closes to (rank 16). (c) Hurwitz–Radon numbers , , : the maximality of presupposes (Mod); faithful representations of (dimension ) force nothing. (d) The smallest spinor factor compatible with T-329 is (, ), with and no space; (W) needs the boost clause of (P). (e) The model without (Cl₀) is vectorlike ( real) or anomalous (); whether chirality selects among UHM-built modules, left open here, is answered by T-350(d): no, the Clifford action selects it | Premises §7 | T15, T-326, T-329, 48e, check_core_numbers.py |
| T-348 | The holon tower and the de Sitter observer algebra [T] as mathematics (2026-09-26); (b)–(c) under the split property of the matter net (proven for the free massive Klein–Gordon field), (e) under (the assumption of CLPW); physical reading [I]. (a) , the O-registers of holons with , closes in the trace representation to the hyperfinite II₁ factor ; , monotone under restriction; the 7 is lost in the closure (traces at level , every value in ; gives the same ). (b) With split, the CLPW algebra (arXiv:2206.10780) is injective, hence (Connes 1976). (c) Every isomorphism carries the trace to the trace: the maximal-entropy state (empty de Sitter) ↔ , entropies preserved; nothing else transported (no observer Hamiltonian, modular flow or geometry). (d) In a normal state only finitely many holons are viable; viable holons give nat; with is of type III (Araki–Woods), and has , below the window. (e) A clock with pure point spectrum (, the depth register at finite ) gives a type I invariant algebra with no field operator. (f) Type and trace do not depend on , the observer mass or ; II₁ also arises in flat space (JSS): no , no sign, no ; gives , a reparametrisation. A UHM clock with continuous spectrum bounded below: the limit of the depth register, at the premise of T-352 | Emergent time §11.5 | T-53b, T-118, T-346, check_core_numbers.py |
| T-349 | Strict necessity of from diagnosability; the hosting route checked [T] as mathematics; the strict necessity [C at (Σ₆)], (P1₆) [C at (Σ₆⁺)] (2026-09-28). (Σ₆): the grammar of every decomposition covering (AP)+(PH)+(QG) for a viable holon has (D1) , (D2) perfect single-fault localisation, (D3) ; (Σ₆⁺) adds rigidity (D4). (a) (Σ₆) gives , so : no decomposition with 6 axes () or fewer than 7. (b) At the grammar is , and T9–T15 give P1 and P2 for the competitor without Track A; under (Σ₆⁺) every decomposition has , so (P1₆) holds. (c) (Σ₆) is strictly weaker than (P1₆) with P2 for the competitor, which implies (Σ₆⁺) (T-246); the Hamming grammar satisfies (Σ₆) and admits no normed division algebra (its weight-3 words are the 35 lines of ). (d) The hosting pin of Foundations of Mathematics, Part XVIII, ch. 11 (Cor. 11.9, Lemmas 11.10–11.12) starts from Hurwitz's list, so it cannot replace P1; the permutation automorphisms of the oriented tables are (order 21) for and (order 3, free and transitive on ) for , so in that reading hosts three sectors, and counts the Paley set , not the quaternion table; in the Kraus reading ( against with cycle type ) it excludes , inside Hurwitz's list; the LGKS triad (T-57) does not depend on , so is a coincidence of two counts | Minimality, T-349 | T-224, T-246, T15, T-57, Premises §3, check_core_numbers.py |
| T-350 | Spinors from the tensorial primitive: what each route gives [T] as mathematics; (Cl₀) stays [H], physical readings stay [C at (Cl)]; the open question of T-347(e) is closed (2026-09-28). (a) Textures of are bosons: for (convex), every rank stratum (, ), , , ; only for the full flag (1 of 15 partitions of 7) and for , where each sector has and (Fl even, 5040 cells; in degrees 3, 11): no Finkelstein–Rubinstein sign, no Wess–Zumino term; , no monopoles for dyons. (b) The spin lift in is forced and empty: is 3-connected, the lift is unique and its spinor bundle is ; , so is not spin. (c) Triality: 28 triples , fixed is (14); the axis rotation has partners , , ; cannot tell from . (d) Induced actions are single-valued ( odd, spinors even-dimensional); acts trivially on (); the centraliser of colour in is : chirality and anomaly freedom do not select , the Clifford action does; (Cl₀) excludes an induced internal symmetry. (e) Kähler–Dirac: on satisfies (Mod); ( spectrum ) cuts out , 8-dimensional, -stable; the spinor module is built from the holon's tensors, its two-valued action is not induced; as a Fock space none of the 21 bivectors conserves particle number | Premises §7 | T15, T-326, T-329, T-347, 42a, check_core_numbers.py |
| T-351 | Population principles move the rate into the environment [T] (2026-09-28); item 2(i) [C under the assumption for every coupling, checked on 128 cases]. With the Landauer upkeep of : (1) a population on a common supply with one-dimensional feedback has invasion fitness , strictly increasing ( at the fold to , ): no singular strategy, selection runs to the fold (pessimisation, Mylius–Diekmann); in an environment the living rates are (, increasing in ), , a bijection onto , and the evolutionary end point is (, , for equal spacing , ); an intake sends it to ( decreasing). (2) Two holons, canonical extension of : (i) Hamiltonian coupling lowers the mean marginal, over 16 couplings and 8 points [C], commuting couplings give for every ; (ii) exchange reduces exactly to the family; for fitness the singular strategy is , decreasing from to 0, a convergence-stable ESS; lives in the window for ( at , ). (3) Maximal entropy production selects ; at the Landauer budget a population produces its supply for every ; reset-process efficiency is largest at the fold; per-rate maxima , , at , , (). No population principle fixes : a finite rate is the image of the environment's spread, a price and a coupling, or a choice of functional | Evolution | T-335, T-346, T-348, check_core_numbers.py |
| T-352 | The directed depth register and its continuous clock [T] as mathematics (2026-09-28); (d) in the model of CLPW at the premise that the observer's clock is the depth register with Hamiltonian , , and under the split property. (a) The readings form a category: its maximal subgroupoid is discrete, its ∞-groupoid completion contractible (Euler characteristic ), and on an ∞-groupoid every non-increasing functional is constant on components; the history is a functor with no invertible arrow for unital primitive , and makes it a functor to — the arrow of T-53b/T-53c lives in category objects, which exist inside every ∞-topos (Riehl–Shulman arXiv:1705.07442; directed univalence for simplicial objects of any ∞-topos, Cavallo–Riehl–Sattler arXiv:2607.02420). (b) The step on is a pure isometry (spectrum the closed disc); translations of have generator with deficiency indices , no self-adjoint extension; no nonzero has both and supported in (Paley–Wiener): sharp readings with a first reading admit no Hamiltonian, and (Pauli) energy bounded below admits no sharp covariant readings. (c) The dressed Feynman–Kitaev constraint has eigenvalues exactly at the quantiles of the arcsine law on (Kolmogorov distance ); on is unitarily a multiplication by on : spectrum , purely absolutely continuous, simple, no kernel; the probability of a reading tends to as . (d) As the observer's clock gives , : type II₁, ; every finite gives type I (T-348(e)). (e) Neither nor is fixed. Answers open question (1) of T-348 at the premise of (d); convergence of the finite algebras stays open [Pr] | Emergent time §11.6 | T-53b, T-53c, T-118, T-348, check_core_numbers.py |
Level [C]: Sensorimotor Theory
| # | Result | Assumption | Source |
|---|---|---|---|
| Closed: for embodied holons — unconditionally [T] (T-149); for isolated — irrelevant (T-148: isolated holon is dead forever). Condition has no domain of applicability | |||
| T-106 | Three diagnostic modes [C at calibration]: structure of 3 modes (normal/warning/critical) — [T] (from T-69 barrier + T-104 radius + T-39a gap). Specific numbers (0.5/0.7/0.9) — [C] at calibration of | Calibration of | Diagnostics |
| C22 | Landauer calibration : — linear growth with level. from T-59 [T] | is determined by the environment | Depth Tower |
| C23 | Monotonicity of grounding: grounding monotonically increases for and sensorimotor flow | Continuous learning + environment | Self-Observation |
| C24 | Forgetting bound: (EWC + Bures-adaptive ) | EWC regularisation | Consequences |
| C25 | -probe: for , probe reaches in examples | Training data with known Γ | Consequences |
| Raised to [T] (T-142): α = 2/3 is state-independent, spectral formula — consequence, not premise. SAD_MAX = 3 unconditionally — Operational Closure | |||
| Raised to [T] (consequence of T-149): for embodied holons C20 is unconditional → C27 is unconditional — Substrate-Independent Closure. Isolated holons (2026-09-25): [T] with the collineation anchor for (living attractor in the window); is derived up to the phase gauge from the single principle (Eq-V) [Pr] (T-334, replacing (Col) and (Pure)); no self-model of replacement form holds the window below at (T-336) |
Conditional Theorem: 7D Minimality [C] → [T]
| # | Result | Assumption | Source |
|---|---|---|---|
Raised to [T] (Sol.70): Strict necessity proven via Hurwitz's theorem (, 6 is impossible) + functional uniqueness 40f [T]. Reverted 2026-09-25 to [C at (Alt)]: Hurwitz's theorem reaches the state space only through P1, which the axioms give at the orientation assumption (Alt) of the bridge T15 (row 41n); itself stays [T] (Theorem S). Renamed the same day to [C at (P1₆)]: (Alt) is discharged by the canonical-orientation theorem, but excluding a competing six-function decomposition needs P1 for that decomposition, and the T15 chain proves P1 only for the seven-dimensional frame (its Step T8 consumes from Track A). Raised 2026-09-28 to [C at (Σ₆)] (T-349): perfect single-fault diagnosability of every decomposition (D1–D3 of Theorem Σ) excludes every decomposition with fewer than seven axes by sphere packing, without Hurwitz; the premise is strictly weaker than (P1₆) with P2 for the competitor, the input of the Hurwitz route, and (P1₆) itself follows at (Σ₆⁺). See Strict Necessity N = 7.
Level 2: Correct as Standard Physics [T]
| # | Result | Source | Target page |
|---|---|---|---|
| 39 | Probability current | Basic Structure T.2.2 | Gap Semantics |
| 40 | Gap landscape bifurcations (pitchfork, saddle-node, Hopf) | Lindblad Operators T.4.1–4.2 | Phase Diagram |
| 41 | Non-Markovian Gap oscillations | Lindblad Operators T.5.1 | Phase Diagram |
| 42 | Holevo bound | Composite Systems T.7.2 | Self-Observation |
| 43 | decomposition | Cosmological Constant T.1.1 | Standard Model |
| 44 | SUSY from -holonomy (parallel spinor ) | Standard Model T.4.1 | SUSY from G₂ |
| 45 | yr (standard SU(5), D=6 operators) | Standard Model | Proton Decay |
| 46 | Confinement T.12.1 | Confinement | |
| 47 | Masses of -leptoquarks from Gap hierarchy: GeV | Standard Model T.1.1 | Proton Decay |
| 48 | Proton decay channels (D=6): , , | Standard Model T.3.1 | Proton Decay |
| 49 | G₂-extra mediated decay: yr (negligible; corrected 2026-07 from the mis-evaluated ) | Standard Model T.4.1 | Proton Decay |
| 50 | Power counting: scalar Gap sector is renormalisable in 4D | Quantum Gravity T.3.1 | Quantum Gravity |
| 51 | Quasi-Goldstone modes at breaking: – Hz | Lindblad Operators T.8.1 | Phase Diagram |
| T-330 | Anomalous dimension of the Fano operator (numbered 52 in this table until 2026-09-25): | Confinement T.9.1 | Renormalisation Group |
Coherence Cybernetics Theorems
| # | Result | Status | Source |
|---|---|---|---|
| CC-1 | Theorem 6.1 (Existence of dynamics): for a unique solution of the evolution equation exists | [T] | CC Theorems |
| CC-2 | Theorem 6.2 (Preservation of Γ properties): dynamics preserves Hermiticity, positivity, normalisation | [T] | CC Theorems |
| CC-3 | Theorem 7.1 (Necessity of self-modelling): | [T] | CC Theorems |
| CC-4 | Theorem 7.2 (Fixed point of reflection): — strict contraction from primitivity of the linear part [T-39a] | [T] | CC Theorems |
| 38a | Theorem 8.1 (Necessity of E-coherence): — mathematical core [T]; 'No-Zombie' interpretation — [I] (requires ontological postulate about E-dimension) | [T] | CC Theorems |
| CC-5 | Theorem 9.1 (Fractal closure) — status raised 2026-09-25 from [C] to [T at weak coupling] by Theorem 9.5, without (HOL). The aggregation is fixed: the mean marginal is the only permutation-invariant linear map that returns a part's state on uncoupled copies. A part is viable (every stationary state has ) iff the stationary state of its regeneration-free part has ; for viable embodied parts and every stationary state of the composite has marginals with (for identical parts in a symmetric state, the canonical aggregate), and the aggregate of a symmetric trajectory follows the single-holon generator up to a forcing of size . Witness: , guaranteed for a Bell-basis coupling of spread ; the marginals are still viable at (). Weak coupling cannot be dropped (Theorem 9.6: at strong coupling the marginals tend to ; the -covariant octonion product sends every uncoupled pair to ). The literal reading — the composite's own dynamics on , assumption (HOL), an interpretation [I] (premises) — is not derivable (dimension 49, not 7) and stays a conditional reading; Corollary 9.1a gives a stationary composite within of . Errata 2026-09-25 (earlier the same day): non-triviality corrected from [T] — step 1 took the representation of the composite in from the Morita equivalence T-58 (retracted 2026-09-10) and then from the section–retraction T-58′, which relates the 7D and 42D descriptions of one holon and gives no map ; is a statement in (two uncoupled holons at have joint ). The status had earlier been lowered from [T] upon resolution of the self-reference paradox | [T at weak coupling] | CC Theorems |
| CC-6 | Theorem 9.2 (Scale invariance) — status raised 2026-09-25 from [C] to [T at weak coupling] by Theorem 9.5: under (AGG) — consistent aggregation and weak coupling (for the canonical mean marginal only the single-copy marginals need lie within ) — the invariants , , , Gap of the aggregate lie within of those of a part; Theorem 9.5 proves (AGG) with at the stationary state (Corollary 9.2a: and at ), along every trajectory from a compact part of the basin (marginal distance from entangled and product starts), and from every initial state with explicit constants under backbone dominance. At strong coupling it fails (Theorem 9.6). Errata 2026-09-25 (earlier the same day): corrected from [T]; the earlier "any CPTP aggregation preserves structure" is retracted (the depolarising channel sends every state to ) | [T at weak coupling] | CC Theorems |
| CC-7 | Theorem 9.3 (Emergence), restated 2026-09-25: for embodied holons coupled by with the canonical extension, (i) is stationary iff , (ii) a product is stationary iff the correlation part of the commutator vanishes and the mean-field equations hold — both exact; (iii) under (ND), non-degenerate single-holon attractors, weak coupling gives exactly when the correlation part of is non-zero. [T] for almost every anchor (2026-09-25): (ND) holds for every pair of anchors outside a closed Lebesgue-null set (Theorem 9.4, parametric transversality: the backbone makes the generator a submersion in the anchor; 12 random anchors, from 1.19 to 1.49). Errata 2026-09-25: corrected from [T] to [C under (ND)] and then raised back to [T] for almost every anchor; the earlier "interacting holons with have a stationary state with " is retracted — a coupling commutes with the product and leaves (audit A-82; test test_coupled_holons_can_have_a_product_stationary_state) | [T] | CC Theorems |
| CC-8 | Theorem 10.1 (Equivalence of conditions): — raised from [C]: all 7 components formalised via -invariants (Sol.81) | [T] | CC Theorems |
Level 3: Substantive Hypotheses [H]
Require reclassification from [T] to [H] or originally stated as hypotheses.
| # | Result | Problem | Source | Target page |
|---|---|---|---|---|
| Reclassified [I]: content — philosophical interpretation, not a mathematical statement. Dual-aspect monism applied to the conjugation operator — ontological, not syntactic position — Basic Structure T.2.1 | ||||
| Reclassified [I]: principle expresses the semantic connection between the 'external' and 'internal' aspects — [I], not [H]. Mathematically: simply a notation choice for Hermitian-conjugate pairs — Basic Structure T.4.1 | ||||
| Raised to [T]: + positive-definite Hessian (T-64; restated as a hypothesis on 2026-09-25, and the Hessian values belong to (SV) — the barrier is [C at (SV)]) + compactness → energy barrier . Confinement-Gap protected by barrier — Composite Systems | ||||
| Retracted [✗] (X3): — counterexample. Replacement: sectoral Gap bound [T] (T-80, Sol.59) — Berry Phase | ||||
| Reclassified [I]: the registry already marks this 'Interpretation'. The Schrödinger/Heisenberg equivalence in UHM — a non-standard ontological reading of standard mathematics (CPTP-semigroup ↔ Heisenberg evolution of observables). Mathematically trivial, philosophically — [I] — Composite Systems T.8.1 | ||||
| Raised to [T]: T15 — bridge fully closed, chain of 12 steps (T1–T16), all [T] (T16/IDP reclassified [D]; computational results unaffected). Was [I] → [C at (CG)] → [C at (MP)] → [T] — Lindblad Operators. Reverted 2026-09-25 to [C at (Alt)]: the step PG(2,2) → needs the orientation of the Fano lines (row 41n); restored to [T] the same day with the canonical orientation (T15-canon) | ||||
| Resolved [T]: sector decomposition [T]; compactification [T] (confinement). Einstein equations on — [T] (T-65, full spectral action) — T-48a, T-52. Reopened 2026-09-25: the axis-labelled sector decomposition (48a) and the compactification of are retracted, T-52 is retired as a theorem; one time direction [T] and three spatial ones through T-119 ([C] until 2026-09-25, then [T] as mathematics, with colour-charged coordinates) and 48c remain | Renormalisation Group T.5.2 | |||
| Raised to [T]: full spectral triple constructed (T-53 [T]); Chamseddine–Connes spectral action reproduces EH with — T-65 | Quantum Gravity | Einstein Equations | ||
| Raised to [T]: full spectral action — T-65; the words "+ all NCG axioms verified" are retracted [✗] (2026-09-25): no real structure of KO-dimension 6 exists on , and the first-order condition is unverified | Quantum Gravity | Einstein Equations | ||
| Raised to [T]: uniqueness of the pair proven from [T] (categorical compatibility with and ). Was [H] → [C at (FE)] → [T] — Standard Model | ||||
| count [T], identification [I] (strengthened 2026-07): exact count [T] (group-theoretic, topology-independent); identification [I] — Fermion Generations | Fermion Generations | |||
| Partially resolved (Sol.60): (a) Topological area law — [C at (SV)] (T-81: T-73 + T-69 + T-64; corrected from [T] on 2026-09-25); (b) String tension MeV — [C at (SV)] (unique vacuum parameters); (c) Deconfinement temperature — [C at standard finite-temperature QCD] (analogue of lattice MeV, nature of transition not strictly derived); (d) Polyakov loop parameterisation — [H] (qualitative model, §4.2) — Confinement | Confinement | |||
| Raised to [T]: L→SAD(L) is monotone (L2⟹SAD≥1, L3⟹SAD≥2, L4⟹SAD=∞). Inverse implications incomplete: SAD does not encode Φ and D_diff. T-136 [T at C] — Operationalisation | Depth Tower | Depth Tower | ||
| Raised to [T]: T-150 — trivial commutativity of iterates of a single CPTP channel for . Spectral SAD formula — consequence, not premise — Substrate-Independent Closure | Depth Tower Hyp. 5.1 | Depth Tower | ||
| Raised to [T]: T-148 — genesis via environmental coupling. An embodied holon raises purity above in finite time — Substrate-Independent Closure | Depth Tower Hyp. 6.1 | Depth Tower | ||
| Raised to [T]: T-152 — tractable anchor validation + T-109/T-113 [T] — Substrate-Independent Closure | Depth Tower Hyp. 6.2 | Depth Tower | ||
| 93 | Coupling scaling (E-10.1): for . MetaAgent contractivity preserved : . Boundary case: , (critical) | Specification | Prediction 11, Stability | |
| 94 | Minimal emergence (E-10.2): if the collective VIT is a linear function of individual VITs, then EmergenceIndex = 0. Non-trivial emergence () requires a nonlinear collective operator | Specification | Prediction 11, Stability | |
| 95 | Non-Markovian extension (E-10.3): with . Preserves CPTP for , stationary points of the Markovian limit, enriches transient dynamics (oscillatory approach to ) | Specification | T-94 [H] | |
| Raised to [C at T-115]: T-115 [T] — algebraic distinguishability of symbolic compositions for generic . Under stable learning condition (, ) each step expands the algebraically distinguishable subspace → grounding monotonically increases. Was [H] → [C at T-115] — Specification | T-115 [T] | |||
| 97 | Emergence of grammar (E-10.5): The naïve formulation (-grammar) is probably false: — 48-dimensional region, for . Reformulation: grammatical structures may emerge from the Postnikov tower of ∞-topos , not from homotopies of . Status [P] (requires reformulation within HoTT-linguistics) — corrected from [H] | Specification | T-69 [P] | |
| Raised to [C at T-4.2]: T-4.2 [C] — non-perturbative uncertainty of the confinement sector. For T-4.2 satisfied, agent category morphisms are defined by CPTP-compatible strategies → Nash equilibrium of extended coherence. Was [H] → [C at T-4.2] — CC Theorems | T-4.2 [C] | |||
| Raised to [C at T-57, T-114]: (1) T-57 [T] (LGKS completeness) — ToM requires 3-channel decomposition → dimensions. (2) T-114 [T] (Fano grammar) — ISL on PG(2,2) requires dimensions. (3) Nash coordination: dimension (Unity ). Additivity under mutual independence — . Condition: simultaneity of ToM+ISL+Coordination in one system. Was [H] → [C at T-57, T-114] — Prediction 11 | T-57 [T], T-114 [T] | |||
| Raised to [C at T-86, T-55]: is compact [T] → complete metric space → the Cauchy sequence converges (contractivity [T]). The colimit of the Postnikov tower exists as a categorical object. However, the limit is not reachable in a finite number of steps (T-86 [T], T-55 [T]). Was [H] → [C at T-86, T-55] — Interiority Hierarchy | T-86 [T], T-55 [T] | |||
| 101 | (H78) Backbone mini/rope/gqa configurations initialise correctly and produce valid logits/hidden_states. Verified MVP-10 (M10.0–M10.7 PASS) | [T] | MVP-10 Ph.0 | |
| 102 | (H79) Anchor : hidden preserves and for arbitrary inputs (10 random seeds). T-62 [T] CPTP. Verified MVP-10 (M10.8–M10.10 PASS) | [T] | MVP-10 Ph.1 | |
| 103 | (H80) -probe output for arbitrary hidden states (T-92 [T] bounded). Verified MVP-10 (M10.11 PASS) | [T] | MVP-10 Ph.2 | |
| 104 | (H81) -contraction: from Fano geometry [F4]. Verified MVP-10 (M10.27 PASS) | [T] | MVP-10 Ph.3 | |
| 105 | (H82) Cholesky round-trip: params preserves diagonal with . Verified MVP-10 (M10.28 PASS) | [C] | MVP-10 Ph.3 | |
| 106 | (H83) CRL grounding: ISL-conditioned cross-attention preserves dimension (seq, ). Verified MVP-10 (M10.50–M10.51 PASS) | [T] | MVP-10 Ph.5 | |
| 107 | (H84) ISL generator + controller: correct generation and episode control. T-114 [T]. Verified MVP-10 (M10.56–M10.57 PASS) | [T] | MVP-10 Ph.6 | |
| 108 | (H85) E2E consciousness verification: 5 criteria () consistent with thresholds [T]. Verified MVP-10 (M10.66–M10.75 PASS) | [T] | MVP-10 Ph.7 | |
| 109 | (H86) Weight transfer: all backbone configurations (mini/rope/gqa) produce finite, non-zero hidden states. Verified MVP-11 (M11.0–M11.4 PASS) | [T] | MVP-11 Ph.0 | |
| 110 | (H87) Phase 1 training API: produces metrics, synthetic data quality threshold. Verified MVP-11 (M11.5–M11.9 PASS) | [C] | MVP-11 Ph.1 | |
| 111 | (H88) Fano: , (T-115 [T] combinatorial count). Verified MVP-11 (M11.25–M11.27 PASS) | [T] | MVP-11 Ph.3 | |
| 112 | (H89) Fano seed purity: for concentrated initial state (Sol.5). Verified MVP-11 (M11.31 PASS with ) | [C] | MVP-11 Ph.3 | |
| 113 | (H90) Self-observation: unified state vector correctly reflects . observe_self() consistent with Gamma methods. Verified MVP-11 (M11.40–M11.45 PASS) | [T] | MVP-11 Ph.5 | |
| 114 | (H91) Internal dialogue: discrepancy EMA converges with sustained accurate self-description. CDL detects confabulations. Verified MVP-11 (M11.50–M11.55 PASS) | [C] | MVP-11 Ph.6 | |
| 115 | (H92) Genesis protocol: V0→V1→V2→Autonomous phase ordering preserves distinctness. Verified MVP-11 (M11.60–M11.63 PASS) | [T] | MVP-11 Ph.7 | |
| Raised to [T]: proven via octonionic structure constants — the unique -invariant trilinear operator on . Formula ; , | Fano Selection Rules | |||
| Raised to [T]: Stinespring () + Cybenko–Hornik (approximation of trace-preserving maps by a neural network at ) → completeness of CPTP coverage. Minimal is unconditional for | ||||
| Raised to [T]: consequence of T-65 (full spectral action [T]) + standard QFT on curved background — [T as a formula] determines and once the cut-off convention () and the scale are adopted — two definitional inputs [D] (spectral action); not free of inputs (errata 2026-09-10) | ||||
| Raised to [T]: [T] (T-42e) → is the centre; Polyakov loop transforms under → — exact deconfinement order parameter | ||||
| Raised to [C at (SV)]: is expressed via MeV [C at (SV)] by the standard lattice relation ; upon substituting the exact from T-81 — full prediction [C at (SV)] | ||||
| Raised to [C at (SV)]: Fano selection rule [T] (T-43d) + tree-level Fritzsch texture → from double Fano-blocking ( → corrections of order ). Numerical — [C at (SV)]. Note 2026-09-26 (T-345(e)): the Fritzsch-texture ground is retracted ( against ); the scaling rests on the double Fano blocking alone, which puts on the diagonal and does not need the Fritzsch zeros | ||||
| Raised to [C at T-50, CKR]: O-parity [T] (T-163) + O-sector scale [T] (T-51) + DM candidate from O-sector → WIMP mechanism gives at standard annihilation cross-section (CKR = Rounak cross-section condition). Depends on T-50 (superpotential) and CKR | ||||
| Raised to [C at T-65, T-73, T-74]: spectral action T-65 [T] → gravitational block includes ; Gap as curvature T-73 [T] → from identity and T-74 ( from spectral action) | ||||
| Clarified and raised to [T]: the corrected formulation — uniform Fano () does not generate mass hierarchy on its own; hierarchy arises from tree-level Fano veto ( only for ) → structurally. Proof: T-43d [T] + -uniqueness of | ||||
| Retracted [✗] 2026-09-26 (T-345(e)) (was raised to [T]): the multiples of nearest the data, and , are and from (PDG 2024); the "two-loop correction" that closed the gap is absent in the Standard Model (the phase runs by up to GeV); the phase source is retracted (T-331), and in the Clifford frame the CKM phase is a Yukawa input (T-333) — CKM §11. Former text: phases live in (PW time is discrete, , T-38b [T]); -covariance of the Fano dissipator [T] (T-2) → quark mixing phase inherited from -topology; tree-level value is topologically quantised | ||||
| Raised to [T]: spectral triple T-53 [T] + Connes NCG curvature → (exact identification). Second Chern class — Gap Operator | Gap Operator | |||
| Raised to [C at (SV)]: self-consistent vacuum equation (T-64, restated as a hypothesis on 2026-09-25) gave the sectoral mean — from substituting against the table's (A-83); the root mean square over the 15 non-O pairs under (SV) is . Exact value depends on minimisation of the Gap potential — a computational task. Principal estimate — [C at (SV)] — C12 | Quantum Gravity §7.4, Λ Budget | |||
| Resolved [T]: GeV from PW clocks + viability — T-51 | Standard Model | Neutrino Masses | ||
| Partially resolved [C]: qualitative [T]; quantitative — anarchic from O-sector isotropy gives angles [C] — C15 | Standard Model | Neutrino Masses | ||
| Resolved [T]: superpotential is unique (Schur's lemma) — T-50 | Standard Model | SUSY from G₂ | ||
| Resolved [T]: from uniqueness of (Schur) — T-50 | Standard Model T.3.1 | SUSY from G₂ | ||
| Resolved [T]: from the cubic structure of (Schur) — T-50 | Standard Model | SUSY from G₂ | ||
| Stratified: field-space finiteness [T] (compact target); full order-by-order UV-finiteness [C] (structural: APS-index + Ward identities + holomorphy + , T-219). Gravitational UV-finiteness — automatic consequence of emergence — T-66 | Quantum Gravity | Quantum Gravity | ||
| Resolved [T]: numbering established [T] ( 3rd, 2nd, 1st) from confinement; normal hierarchy [T]. Discrepancy remains [C] — T-52. Reopened 2026-09-25: 3rd stays [T]; 2nd, 1st and the normal hierarchy are [C at (SA)], (SA) a hypothesis | Standard Model | Neutrino Masses |
Level 4: Retracted Results [✗]
Not for integration
These results have been proven erroneous and must not be included in documentation without explicit indication of the refutation.
| # | Result | Reason for refutation | Source |
|---|---|---|---|
| 74 | CS derivation of from -connection on 1D | Total derivative (see Berry Phase) | Phase Diagram T.1.1 |
| 75 | IR Fixed Point for 3 Yukawa couplings | All converge to a single point | Standard Model T.2.2 |
| 76 | Sectoral SUSY exact | Global breaking is transmitted; , but not zero | Standard Model T.9.2 |
| 77 | Equivalence | Standard Model §1.5 | |
| 78 | Gaussian sum: 9 orders at physical | at | Cosmology §4 |
| 79 | Modular hypothesis: 15 orders | Refuted at | Berry Phase §12 |
| 80 | Energy cost of Gap | P does not depend on phases (contradiction) | Composite Systems T.9.1 |
| 81 | Cooperation formula via inclusion-exclusion: | Dimensionally incorrect: — quadratic functional, not a measure. Correct formula: (Sol.57, T-77 [T]) | Value Consciousness |
Postulates [P] and Definitions [D]
| # | Result | Status | Source |
|---|---|---|---|
| Reclassified [D] (Sol.25): IDP — a definition embedded in A1+A2. Distinguishability via -coverings is identical to ontological distinguishability — a tautological consequence of the ∞-topos choice. All computational results () are unaffected — Axiom of Septicity | |||
| Raised to [T]: P1+P2 derived from (AP)+(PH)+(QG)+(V) via the chain T15 [T] — Octonionic Derivation. Reverted 2026-09-25 to [C at (Alt)]: T15 needs the orientation input (Alt) (row 41n); restored to [T] the same day with the canonical orientation (T15-canon) | |||
| Raised to [T]: uniqueness of O [T] + equivalence of 4 time constructions [T] — Emergent Time. Independent derivation of A5 from T-53 (Sol.68) — T-87 | |||
| Raised to [T] (T-129 + T-129a): unique self-consistent value with . Universality (T-129a [T]): threshold on all of — Operationalisation | |||
| O2 | Canonical via Frobenius norm for L2 | [D] | Self-Observation |
| O3 | CPTP: Completely Positive Trace-Preserving (class of admissible channels) | [D] | Evolution |
Conditional Theorems [C]
| # | Result | Assumption | Source |
|---|---|---|---|
| [T]+[I]: derived from triadic decomposition T-40a, 40b, but the identification — interpretive bridge [I] — see reflection threshold | |||
| Raised to [T]: defined as HS-projection ; formula — exact consequence, not proxy — Axiom of Septicity, HS-projection | |||
| Raised to [T]: primitivity proven — see T-39a, 39e; 39e retracted 2026-09-25 (see row 39e) | |||
| Raised to [T]: Bridge fully closed (T15 [T]) — T11 (Choi rank=7) + T12 (projective operators) + T13 (forced BIBD). (MP) became a theorem — Lindblad Operators. Reverted 2026-09-25 to [C at (Alt)]: the bridge is closed up to the orientation of the Fano lines (row 41n); restored to [T] the same day with the canonical orientation (T15-canon) | |||
| Withdrawn: T6 [T] proves uniform contraction unconditionally (from -equivariance, T5 [T]) — see T-41e | |||
| Raised to [T]: uniqueness of the pair proven from [T]. Was [H] (No.61) → [C at (FE)] → [T] — Standard Model. Reverted 2026-09-25: construction [C at (FE)], uniqueness [H] | |||
| Raised to [T]: sector asymmetry proven from confinement [T] and asymptotic freedom [T]. Structural inequality: non-perturbative coupling > perturbative for any — T-52. Reverted 2026-09-25 to [C at (SA)]: (SA) is a hypothesis; the sector reading is retracted (45b, 48a) and the inequality fails for | |||
| Raised to [T]: uniqueness from Schur's lemma — . Higher orders suppressed by — T-50 | |||
| Raised to [T]: from PW phase precession + viability (V). derived from axioms A1–A5 — T-51 | |||
| Raised to [T]: sector asymmetry proven from confinement [T] — T-52. Retracted 2026-09-25: used the axis-labelled decomposition (48a, retracted) and read colour as space; the compactification of goes with it | |||
| Raised to [T]: uniqueness of the self-consistent vacuum with sector structure (corrected 2026-09-25: unique only up to the symmetries of , numerically [H]; the sector structure retracted [✗]) — T-61 | |||
| Raised to [T]: sectoral from unique vacuum [T-61] (since 2026-09-25 a value of the hypothesis (SV)) — Confinement | |||
| C14 | Neutrino mass ratio (with 2-loop RG) | O-sector Yukawa + 2-loop RG (Sol.72) | [C] — discrepancy vs. observed 0.17; formula T-63 [T], precision — computational task at — Neutrino Masses |
| C15 | PMNS angles from anarchic | O-sector isotropy → | [C] — correct order (); exact prediction requires Gap structure of O-sector — Neutrino Masses |
| C16 | Higgs quartic from spectral action | + RG | [C] — canonically defined [T] (T-70): . Conceptual freedom eliminated; numerical value of depends on exact — Higgs Sector |
| Corrected 2026-09-25 from [T] to [H] (T-332): its inputs are or (UP) [H], of the retracted and of (SV), and is chosen; the compared mixes scales (same-scale at ). T-332(i) (2026-09-26): the loop mechanism from is refuted [✗] — exact (UP) keeps the phases of and , so no loop of the Clifford content generates or . Earlier: Mechanism [T] (Sol.71): discrepancy — artefact of mean ; at sectoral , : — exact agreement. Precise prediction — computational task (T-79) — Yukawa Hierarchy | |||
| C18 | Spectral formula | via of the spectral action + SUSY-breaking | [C] — structural formula [T]; honest composed bracket – [C], remaining orders open — Λ Budget honest ledger |
| Raised to [T] for embodied holons (T-149): backbone injection ensures unconditionally. Isolated holon: C20 remains [C] (no practical relevance, since an isolated holon at is dead forever, T-148) — Substrate-Independent Closure | |||
| Restated 2026-09-25: as stated ("", "") false [✗], since ; the correct content — the self-knowledge defect identity and the first-order shifts of the attractor from the self-model's fixed point, for and for — is T-157 [T] — Substrate-Independent Closure | |||
| Raised to [T] (Sol.64): categorical unreachability via Postnikov tower + Lawvere incompleteness (T-55 [T]). Butterfly retracted [✗] — T-86 |
Identifier renumbering (2026-07): C22–C25 → C32–C35
The four rows below were renumbered C22→C32, C23→C33, C24→C34, C25→C35 to resolve a collision with the Sensorimotor block (C22 = Landauer calibration, C23 = grounding monotonicity, C24 = forgetting bound, C25 = σ-probe), which is the meaning used everywhere else in the corpus. No external document cites the C22–C25 numbers in the senses below, so the renumbering is reference-safe.
| # | Result | Assumption | Source |
|---|---|---|---|
| C32 (was C22) | Monotonicity of symbol grounding: under stable learning (, ) | T-115 [T] (algebraic distinguishability) | [C at T-115] — raised from [H] No.96. Under stable learning conditions each step expands the algebraically distinguishable subspace → grounding monotonically does not decrease |
| C33 (was C23) | Categorical Nash embedding: | T-4.2 [C] (confinement sector) | [C at T-4.2] — raised from [H] No.98. CPTP-compatible agent strategies are isomorphic to Nash equilibria of extended coherence |
| C34 (was C24) | minimality for social learning: | T-57 [T] (LGKS), T-114 [T] (Fano grammar) | [C at T-57, T-114] — raised from [H] No.99. Counting argument is complete under simultaneity of ToM+ISL+Coordination — Prediction 11 |
| C35 (was C25) | (numerical order of the vacuum parameter) | hypothesis (SV) [H] (T-64 restated on 2026-09-25: its vacuum has no sector values) | [C at (SV)] — raised from [H] No.66. Errata 2026-09-25 (A-83): the value came from substituting against the table's (the 21-pair mean is then ); is redefined as the root mean square over the 15 non-O pairs, at — order conditional on (SV). The self-consistent vacuum of itself gives , order . Decision 2026-09-25 (T-332): (SV) as the vacuum of is refuted [✗] by T-64, and under (Cl₀)+(GC) no coherence of carries a family index; the corrected vacuum's is its up–down () asymmetry. as a generation parameter is phenomenological [H]; the implication [C at (SV)] stands — C12 |
| C36 | R-completeness (retro-completion of time): the trajectory nerve is Kan iff waits are group-completed; retrospection (re-reading the trace) realises the completion — reflexivity as the categorical completeness condition of thought. Census: 42.9% of independent outer horns refuse over ; over the completion all fill (silicon courts 3/3) | Identification "retrospection = the R axis in action" (the categorical half is [T]-grade trivial) | Self-Observation |
Retracted Statements [✗]
| # | Statement | Reason for retraction | Replacement |
|---|---|---|---|
| X1 | for all | [D] coherent domination | |
| X2 | Reclassified [D] (Sol.25): step (3) — tautology from A1, which confirms the status of a definition, not a theorem. IDP is embedded in A1+A2 | ||
| X3 | Fano Gap bound for all pairs | O-sector Fano pairs (6 of 21): — direct counterexample | Replacement (Sol.59): sectoral Gap bound [T] (T-80) — Berry Phase |
| X4 | L3→L4 as butterfly | Finite catastrophe inapplicable to infinite-dimensional transition (all for ) | Replacement (Sol.64): categorical unreachability [T] (T-86) — Interiority Hierarchy |
Level 5: Research Programmes [P]
| # | Programme | Description | Target page |
|---|---|---|---|
| 81 | Quantum gravity from Gap | Functional integral is defined, non-perturbative computation absent | Quantum Gravity |
| 82 | Lattice computation on | Monte Carlo with -symmetry | Quantum Gravity |
| 83 | Black hole information paradox | Gap resolution: unitary evolution, Page curve from Gap profile | Quantum Gravity |
| 84 | Inflation from Gap potential | at small as a quadratic inflaton | Quantum Gravity |
| 85 | Non-perturbative closure of the Λ deficit | Progress: spectral formula [T] (T-65); SUSY-sector [H] (T-219, corrected from T at T-64 on 2026-09-25; absorbs → net ); full minimisation T-64 (restated as a hypothesis on 2026-09-25: its vacuum has no sector values); honest bracket – [C]. Remaining orders: open computational + conceptual task (2026-07 audit: the former total was a forbidden sum — retracted) | Λ Budget honest ledger |
Level 6: Interpretations [I]
| # | Interpretation | Target page |
|---|---|---|
| 86 | Clinical correspondence of Gap phases (I — norm, II — dissociation, III — dementia/coma) | Phase Diagram |
| 87 | Therapeutic interpretation of G₂/⊥-decomposition: healthy Gap in the -sector, pathological — in | Gap Operator |
| 88 | Non-Markovian oscillations as 'grief cycles' and 'clarity flashes' | Phase Diagram |
| 89 | k-floor clamp [I]: in the implementation — for the value is used instead of theoretical (T-62). Prevents degeneration of as . Threshold 0.15 is empirical | Evolution |
| 90 | Dual-aspect interpretation of conjugation (reclassified from [H] No.53): as a formal reflection of the ontological duality 'external/internal' — [I], not a theorem. Mathematically: standard Hermitian conjugation | Basic Structure T.2.1 |
| 91 | Conjugate pair principle (reclassified from [H] No.54): semantic connection 'aspect ↔ counter-aspect' — an interpretive notational principle, not a mathematical statement | Basic Structure T.4.1 |
| 92 | Canonical Schrödinger/Heisenberg duality (reclassified from [H] No.57): CPTP-semigroup ↔ Heisenberg evolution of observables — standard mathematics, but the ontological reading in UHM — [I] | Composite Systems T.8.1 |
Budget of the Cosmological Constant Λ
Perturbative Budget (confirmed — [T])
| Mechanism | Suppression | Source | Status |
|---|---|---|---|
| (smallness of coherences) | Quantum Gravity §7.3 | [T] | |
| RG | Quantum Gravity §12.3 | [T] | |
| Ward identities (anti-correlation) | (×19/49) | Cosmological Constant §10.3 | [T] |
| Fano code (6 constraints) | (×1/8) | Quantum Gravity §12.5d | [T] |
| Confinement §9.3 | [T] | ||
| O-sector | Confinement §10.2 | [T] | |
| Total | [T] |
Full proof: Λ Budget.
Non-perturbative Sector
| Mechanism | Result | Status |
|---|---|---|
| Instanton () | — additive, not multiplicative | [T] |
| Gaussian sum at | — does not work | [D] |
| Modular hypothesis | ~15 orders — does not work at | [D] |
| Zeta | Structural zeroing — requires QFT interpretation | [T] (math.), [H*] (phys.) |
Cohomological + SUSY Sector
| Mechanism | Result | Status |
|---|---|---|
| No topological -term | on the contractible forbids -contributions of the form . Errata 2026-09-10: the reading "" is retracted — a vacuum-energy total is degree-0 data and , so cohomology cancels nothing | [T] narrow / [✗] wide |
| SUSY-breaking | residual | [H] (T-219; listed as a theorem via spectral action T-65 until 2026-09-25) |
| [T] (math.) | ||
| RG | [T] | |
| Sectoral from Sol.39 | [C at (SV)] (sector values; T-64 restated as a hypothesis) |
Total (conservatively): 41.5 [T] out of 120 — proven perturbative suppression. Gap before full minimisation: ≈ 78.5 orders. Remaining sources (conditional):
- Cohomological argument: [T] only as the prohibition of a topological -term; the zeroing reading is retracted (2026-09-10).
- SUSY-breaking suppression : [H] since 2026-09-25 (T-219; it was listed as a theorem via spectral action T-65 + Schur-uniqueness of T-50): the one-loop terms of its own derivation are larger unless they cancel. Caveat: the specific factor depends on Fano selection rule T-43d [T] and sector structure; numerical value is [C at (SV)].
- enhancement: [T] (zeta calculation); physical interpretation *[H]**.
- RG : [T].
- Sectoral minimisation : [C at (SV)] — not yet numerically computed on .
Honest summary (2026-07 audit): composed bracket – [C at (SV), H* at ; sector programme pending]. The former "total " was a forbidden sum (double-counted RG , unabsorbed ; equivalently, quoting at the lower edge ) — retracted. Remaining orders: open computational (numerical minimisation of on ) + conceptual (local residual saturating the cohomological bound). See Λ Budget honest ledger.
Critical Cross-Document Issues
1. CS Cascade
Source: Phase Diagram §1.3 → Refutation: Berry Phase §2.1
Affected results: , , Noether charges (topological part), equations of motion with topological term, bridge closure via .
Resolution: Reinterpretation via the Berry phase. The formula may be salvaged, but its derivation from CS on 1D is erroneous.
2. SM from G₂: rank problem
. Electroweak sector: [C at (FE)] for the construction, [H] for its uniqueness — corrected 2026-09-25; the chain had read [H] → [C at (FE)] → [T] via the uniqueness of the pair from . The electroweak group acts on the Page–Wootters system factor, not inside : in 7D an on does not commute with , whose centraliser in is . Correct formulation: ' from [T]; on the system factor, [C at (FE)]' — uniqueness theorem.
3. CKM predictions: overstatement of precision
The formulae are standard consequences of the Fritzsch texture with observed masses as input. The theory's prediction is the structure (Fritzsch texture), not the numbers. Corrected 2026-09-26 (T-345): the structure is refuted too — the Fritzsch texture gives against — and no parameter-free structure of the clock predicts a mixing angle or a mass ratio (CKM §11).
Empirical status (2026) — retracted 2026-09-26 [✗] (T-345(e)): the correction behind does not exist in the Standard Model (the phase runs by up to GeV), and the uncorrected is from . Former text: The one genuine CKM prediction — the CP phase from the Fano geometry [H] — is confirmed near-exactly by the LHCb tree-level combination (ICHEP 2024; ), consistent with the PDG 2024 global fit . The older figure is superseded across the corpus (SSOT: CKM §4.2).
Cabibbo Angle Anomaly — resolution-channel predicted [T-265]. First-row CKM unitarity currently shows a deficit, (2024–2026 lattice + /kaon determinations). T-265 (CKM §10) sharpens the earlier "open gap": since the fundamental CKM would be exactly unitary, the deficit could not be a mixing-matrix effect — the row claimed fourth generation, vector-like quarks, MeV sterile neutrinos and leptoquarks all excluded by the spectrum (corrected 2026-09-25: fourth generation [C at 43c identification], since the count 3 is exact but its physical identification is [I]; vector-like quarks and leptoquarks [H], their grounds — the chirality of and T-297 — being retracted or [H]; sterile neutrinos [C]) — so UHM predicts, as a hypothesis, that it resolves in the SM extraction sector (-box / nuclear radiative corrections / lattice / the – tension). The resolution channel is [H] (corrected from [T-structural]+[C] on 2026-09-25); the magnitude of the deficit remains open [D] (see falsifiability, F-Cabibbo).
4. Sectoral SUSY
The claim '9/21 pairs are exactly compensated' — refuted [D]. In standard supergravity SUSY breaks globally. SUSY does not contribute new multiplicative suppression to the Λ budget. See SUSY from G₂.
5. Neutrino masses: ratio discrepancy — resolved [C]
The naïve seesaw estimate disagreed with the observed by ~50×. Resolved: O-sector Dirac Yukawa (T-63) reduces the discrepancy from ×50 to ×1.8 (to ×1.2 with the RG correction). Mechanism: in the O-sector (T-51) → Dirac mass from blocks and , not from . PMNS angles from anarchic — [C]. See Neutrino Masses.
Open Problems
Hidden Assumptions
| # | Assumption | Status |
|---|---|---|
| H1 | Primitivity of | [T] — T-39a |
| H2 | Uniqueness of 7/7 dimensions | [T] — T-40c, 40d, 40e, 40f |
| H3 | Choice of | [T] — T-40a, 40b |
| H4 | Coincidence of generative model with Γ | [T] — consequence of the definition of a self-referential system |
| H5 | Uniqueness of the mapping G | [T] — -rigidity of holonomic representation T-42a |
Fundamental
- Λ: orders open — structural mechanisms identified [C]: spectral formula via [T] (T-65); SUSY-sector [H] (T-219, corrected from T at T-64 on 2026-09-25; exact compensation [H]); cohomological zeroing [T]; sector structure — the hypothesis (SV) [H] (T-64 restated on 2026-09-25); sign proven [T] (T-71: autopoiesis + local cohomology); canonically defined [C at (SV)] (T-70); O-sector dominance [T] (T-84, Sol.63: ). Honest bracket – [C]; remaining orders — open computational + conceptual task (Λ Budget honest ledger)
Bridge closure— RESOLVED [T]: full chain T1–T16 (12 steps, all [T]; T16/IDP reclassified [D]). T11 (Choi rank=7) + T12 (projective operators from L-unification) + T13 (forced BIBD(7,3,1)) close the bridge. (MP) became a theorem. See Lindblad Operators. Reopened 2026-09-25 as [C at (Alt)]: steps up to PG(2,2) are theorems; the step to needs the orientation input (Alt) (16 of 128 orientations). Closed again the same day [T]: the 16 normed orientations are the unique collineation-invariant class, the orientation the design itself determines (T15-canon) 2b.Uniqueness of mapping G— RESOLVED [T]: -rigidity of holonomic representation. The mapping is unique up to kinematically and up to the finite frame group dynamically; 34 = 48 − 14 kinematic -invariants, 48 physical parameters (frame decision D-0910). Analogue of the Stone–von Neumann theorem. See Uniqueness TheoremSuperpotential W— RESOLVED [T]: unique -invariant (Schur's lemma) [T-50]; Kähler metric on moduli — [C] (Supersymmetry)- — reopened 2026-09-25 (it read "RESOLVED [T]: full minimisation of proven (T-64)"): the -orbital reduction is retracted [✗], and the vacuum of has no sector values; is [C at (SV)] (C35) — Gap Thermodynamics
3+1 from— RESOLVED [T]: sector decomposition [T] + 3D from [T] (sector asymmetry [T-52]); Einstein equations on — [T] (T-65, full spectral action). Background independence — [T] as mathematics (T-120; it read [T] until 2026-09-25, then [C] at an aperiodic clock and the reconstruction axioms of T-119, and was raised back the same day: the restated T-119 satisfies all seven of Connes' conditions, and the depth register supplies the clock, T-118); the reading of as physical spacetime is [I]: assembled from categorical structure via Gel'fand–Naimark–Connes chain — Emergent Manifold. Reopened 2026-09-25: the axis-labelled sector decomposition is retracted (48a), '3D from ' reads colour as space and is retracted, and T-52 is retired as a theorem; one time direction [T] and three spatial ones through T-119 ([C] until 2026-09-25, then [T] as mathematics, with colour-charged coordinates) and 48c remainBerry-phase derivation of— RESOLVED [T] (Sol.65): from + -uniqueness. CS₁ replaced by Keldysh. T-85 — Berry PhaseElectroweak sector— RESOLVED [T]: uniqueness of the pair proven from [T]. Was [H] → [C at (FE)] → [T] — uniqueness theorem. Reopened 2026-09-25: construction [C at (FE)], uniqueness [H]— reopened 2026-09-25: [H] (T-332; inputs , (SV), or (UP); in the Clifford frame is not predicted; T-332(i), 2026-09-26: the loop mechanism is refuted [✗], and a tree-level down-type coupling of relative size is needed [Pr]). Earlier RESOLVED [C]: QCD IR enhancement + loop gives (observed ). Agreement . Key correction: QCD enhances Yukawa couplings of light quarks in the IR — Yukawa HierarchyNeutrino generation numbering— RESOLVED [T]: 3rd, 2nd, 1st [T-52]; normal hierarchy [T]. Reopened 2026-09-25: 3rd stays [T]; the other two assignments and the normal hierarchy are [C at (SA)], (SA) a hypothesis
Computational
- — physical interpretation
- Full functional integral (bosons + fermions + SUSY) on (Quantum Gravity)
- Lattice computation on with -symmetry
- Two-loop correction to
- Non-perturbative dualities of Gap theory with M-theory
Epistemic Classification of Remaining Open Results
(Sol.85) All remaining [C] and [H] are classified into three categories:
| Category | Definition | Examples |
|---|---|---|
| A. Computational | Formula defined [T]; numerical value — task on | C14 (ν ), C15 (PMNS), C16 (), C18 () |
| B. Empirical | Formulation [T]; validation requires measurements | G-mapping (D.2), ISF, ASC-parameters, calibration |
| C. Interpretive | Philosophical interpretation of the formalism | Jung archetypes (#86), utilitarianism vs maximin (#87), qualia taxonomy (#88) |
Summary: All identified conceptual gaps are closed. Remaining open questions are computational tasks (category A) or empirical programmes (category B), not theoretical lacunae.
Theorem Dependency Graph
Key derivation chains between theorems:
Fundamental chain (axioms → dynamics → consciousness):
Physical chain (spectral triple → gravity):
Consciousness chain (primitivity → hierarchy):
SAD chain:
Promoted hypotheses:
| Hypothesis | Was | Proof | Became |
|---|---|---|---|
| (FE) electroweak | [C] | Sol.1, T-1 | [T] until 2026-09-25; now [C at (FE)] for the construction, [H] for uniqueness |
| (MP) superpotential | [C] | Sol.15, T-50 | [T] |
| (ΓO) O-sector scale | [C] | Sol.16, T-51 | [T] |
| (SA) sector asymmetry | [C] | Sol.17, T-52 | [T] until 2026-09-25; now a hypothesis [H] (T-52 retired as a theorem) |
| (Higgs identification) | [H] (§1.1 Higgs Sector) | T-42a (κ₀) + T.1.1 (Fano line) + FE [T] (quantum numbers) + T-64 (vacuum) | [H] (corrected from [T] on 2026-09-25: colour breaking, no doublet on ) — Theorem 1.0 |
| L1→L2 cascade dynamics | [H] | Transcritical bifurcation: -amplification via (T-43b [T], HS-projection [T]). (exponent , not ) | [T] — Swallowtail |
| Cost of enlightenment | [H] | 21 pairs Landauer ( per bit). from T-105 [T] (FDT) | [C at T-105] — Gap Thermodynamics |
| Early warning indicators (critical slowing) | [H] | Linear stability of Gap-dynamics Jacobian + FDT (T-105 [T]) + swallowtail (Theorem 1.2 [T]) | [T] — Bifurcation |
| Self-consistent measurement | [H] | T-96 [T] (existence of ) + T-62 [T] (CPTP) + T-55 [T] () | [T] — Measurement |
| L4 closure (-groupoid) | [H] (#100) | Compactness of + contractivity [T] + T-86 [T] + T-55 [T] | [C at T-86, T-55] — Hierarchy |
| -parity (Theorem 11.2) | [H] | T-42e [T] () + T-99 [T] ( → ) + + T-69 [T] (barrier) | [T] — Dark Matter |
| Preferred measurement basis (Theorem 6.1) | [H] | — atoms of [T] + kills off-diagonal [T] + diagonal = fixed points [T] + Zurek's einselection | [T] — Measurement |
| Stability of the chiral vacuum (§4.4) | [H] | T-99 [T] ( unique PT-odd) + T-64 [T] (corrected to the -invariant potential; a unique sector vacuum with positive Hessian is (SV)) + T-69 [C at (SV)] (barrier ) | [C at (SV)] (corrected 2026-09-25) for the retracted ; 2026-09-26: the corrected vacuum selects no chirality (T-333(a)), chirality comes from the forced (T-329) — T-166 — Higgs Sector |
| (H1) Trainable CPTP-anchor () | [H] (#116) | Stinespring () + Cybenko–Hornik (universal approximation of CPTP) | [T] — [#116] |
| (H-Hawk) Hawking radiation , | [H] (#117) | T-65 [T] (spectral action) + standard QFT on curved background | [T] — [#117] |
| (H-Pol) Polyakov loop — order parameter | [H] (#118) | T-42e [T] () → | [T] — [#118] |
| (H-Tc) Deconfinement temperature | [H] (#119) | T-81 [C at (SV)] () + standard lattice relation | [C at (SV)] — [#119] |
| (H-V3) Scaling | [H] (#120) | T-43d [T] (Fano ) + double blocking | [C at (SV)] — [#120] |
| (H-ΩDM) Dark matter | [H] (#121) | T-163 [T] (O-parity) + T-51 [T] (O scale) + CKR | [C at T-50, CKR] — [#121] |
| (H-SBH) Gap correction in | [P] (#122) | T-65 [T] + T-73 [T] (Gap = curvature) + T-74 [T] ( from spectral action) | [C at T-65, T-73, T-74] — [#122] |
| (H-MH) Mass hierarchy from Fano selection rule (clarification) | [H] (#123) | T-43d [T] (, ) + -uniqueness of | [T] (hierarchy from tree-level rule) — [#123] |
| (H-δCP) Topological quantisation | [H] (#124) | T-38b [T] () + T-2 [T] (-covariance) | [✗] (2026-09-26, T-345(e): and are and off; was [T]) — [#124] |
| Dual-aspect interpretation of conjugation (#53) | [H] | Philosophical/semantic nature — not a mathematical statement | [I] — reclassified |
| Conjugate pair principle (#54) | [H] | Semantic connection — [I] | [I] — reclassified |
| Canonical Schrödinger/Heisenberg duality (#57) | [H] | Already marked 'Interpretation' in the registry | [I] — reclassified |
| ε = O(10⁻²) (#66) | [H] | hypothesis (SV) [H] (T-64 restated) | [C at (SV)] — C25 |
| Grounding monotonicity (#96) | [H] | T-115 [T] algebraic distinguishability | [C at T-115] — C22 |
| Categorical Nash embedding (#98) | [H] | T-4.2 [C] | [C at T-4.2] — C23 |
| N=7 for social learning (#99) | [H] | T-57 [T] + T-114 [T] | [C at T-57, T-114] — C24 |
Rigour Stratification and Framework Dependencies
Following the 2026-04-21 proof audit, the theorem stack is stratified by the nature of the rigour supporting each [T] label. This section makes explicit what was previously implicit in individual rows.
Status tag taxonomy
- [T] — theorem with complete rigorous proof: each step either (a) standard mathematical inference, (b) citation to an established result with specific theorem number, or (c) explicit calculation. Mechanisable in a proof assistant (Verum, Lean 4, Coq).
- [T/sim] — analytical core is [T]; calibration constants, parameter values, or specific inequalities are cross-checked against SYNARC numerical runs. The simulation is a cross-check, not a replacement for mathematical argument.
- [T at X] — rigorous modulo an explicit assumption X (stated in the row).
- [T mod framework-F] — legitimately rigorous inside an external framework F (Lurie HTT, Schreiber DCCT, Connes–Chamseddine, Goderis–Verbeure–Vets, Baez–Dolan), where applicability of F to the specific UHM site / construction is either standard or requires separate verification.
- [C] — conditional on an explicit hypothesis.
- [D] — design choice / definition / convention.
- [H] — hypothesis (not yet a theorem).
- [P] — postulate.
- [D] — definition by convention (e.g. PID as tautological consequence of A1+A2).
- [I] — interpretive identification (philosophical mapping between formal structures and phenomenology).
- [✗] — retracted.
Rigorous Core (≈50 theorems)
The following theorems carry fully earned [T] status — complete rigorous proofs, mechanisable in Verum / Lean 4:
- Quantum-dynamical core: T-15 (Bridge to N=7), T-38a (No-Zombie), T-39a (primitivity of ), T-62 (CPTP evolution), T-82 (Fano-BIBD uniqueness), T-96 (attractor characterisation), T-98 (balance formula), T-42a (G₂-rigidity), T-42e (stabiliser SU(3)), T-118 (temporal manifold , scaling limit of the depth register), T-53b (dynamics relative to the depth register). Corrected 2026-09-25: T-118 was removed from this list while it was conditional on an aperiodic clock and returns with the depth register (emergent time §11.4), together with T-53b; and the Bridge T15 (row 41n) carried the orientation assumption (Alt) at its step PG(2,2) → until the canonical-orientation theorem (T15-canon) discharged it the same day
- Analytical/convex: T-104 (stability radius), T-109–T-112 (learning bounds), T-124 (Goldilocks non-emptiness), T-124b–d (threshold robustness), T-129 (Φ_th=1), T-148 (genesis core), T-152 (CPTP anchor validation), T-160 (phase transition structural); T-161 (critical exponents via Mather splitting + tricritical Landau) is removed from this list — conditional on the symmetry since 2026-09-25
- Categorical closures: T-187 (Bures canonicity via Petz extremality Char-I), T-189 (MaxEnt recasting), T-192 (strict 2-category Exp^(2)), T-210 (strict Φ-monotonicity on interior stratum), T-211 (PhysTheory as a Grothendieck construction over , corrected 2026-09-25), T-213 (Yoneda via Bures description length), T-214 (hard-problem meta-theorem, Lawvere positivity), T-216 (ε_eff closed form at (SV)), T-220 (no-reduction F₄→G₂ via 5 obstructions)
Framework-conditional theorems
| Theorem | Framework | Specific result cited | UHM-site applicability status |
|---|---|---|---|
| T-76 | Lurie HTT | 6.2.2.7 (site → ∞-topos) | Site-level verified §6.3.1; Exp-extension Claim 10.2 requires Giraud-axiom verification |
| T-185 | Schreiber DCCT, arXiv:1310.7930v1 (2013) | Def. 3.4.1 and 3.4.4 (cohesion, §3.4), Def. 3.5.1 and 3.10.1 (differential cohesion), Def. 3.4.17 and Prop. 3.4.18 (∞-cohesive sites), Def. 4.5.7, Props. 4.5.8 and 4.5.11 (synthetic-differential ∞-groupoids); earlier citation "§3.9 (cohesion) + §3.10 (super-cohesion)" corrected — in v1 those sections are "Structures in a cohesive / differentially cohesive ∞-topos", and Rh does not occur | Established for as an object of (T-185 (ii′), [T]); not established for (no ∞-cohesive site), where cohesion stays an assumption |
| T-186 | Schreiber DCCT, arXiv:1310.7930v1 | Prop. 4.1.17 (differential cohomology hexagon, for stable coefficients only; earlier citation "§3.9 hexagon" corrected) + Chern–Weil for G₂-bundles | Does not apply: is not stable, and every characteristic class of a bundle over the contractible vanishes; parts (b), (c) retracted, part (a) a hypothesis |
| T-211 | Lurie HTT, HA | HTT Def. 2.4.2.1, Prop. 2.4.4.2, Thm. 3.2.0.1 (straightening), Def. 6.3.1.5; HA §2.4.1–2.4.2 (cartesian monoidal structures) | Applies: the unstraightening of is a standard construction (corrected 2026-09-25: the earlier citation of HTT 5.2.7 for a full embedding is withdrawn — the embedding is false) |
| T-212 | Schreiber, solid cohesion (DCCT, 2017 version, site Def. 6.6.13; D. J. Myers, M. Riley, arXiv:2301.13780, §6.3) — not DCCT v1 | the earlier citation "§3.10 super-cohesive extension" is corrected — Rh is not in DCCT arXiv:1310.7930v1 | Does not apply: Rh preserves global points and is not the trace formula; the corrected T-212′ (-twirl) uses no framework |
| T-217 | Baez–Dolan | Hirschowitz–Simpson 2001, Leinster 2002 (3-types ≃ coherent tricategories) | Applicability: in scope of correspondence needs verification |
| T-218 | Milnor classifying-space | Singular complex of is Kan | Kan part [T]; 3-coskeletal truncation argument (Step 4) requires separate proof |
| T-65, T-120 | Connes–Chamseddine 1996–1997 | Spectral action expansion, heat-kernel | Standard expansion. The earlier entry "KO-dim 6 verified for UHM triple (T-53)" is retracted: no real structure of KO-dimension 6 exists on ; T-120 is conditional on T-119 (T-118 is [T]) |
| T-117 | Goderis–Verbeure–Vets 1989 | Quantum CLT on lattice observables | Clustering hypothesis for full requires separate verification |
| T-119 | Gelfand–Naimark; Connes 2013 ("only if" direction) | Spectrum computed; Dirac triple of | Superseded 2026-09-25: the spectrum is computed (fluctuations , minimal unitization ), and all seven conditions hold for the Dirac triple of . Formerly: 6 of 7 axioms argued, first-order condition untreated |
| T-221 | Kripke–Joyal semantics (Mac Lane–Moerdijk §VI.6–7) + Lurie HTT 6.3.1.16 | Forcing in the 0-truncated part of ; Yoneda embedding of an essentially small site | Applies (corrected 2026-09-25: the earlier row cited Schreiber DCCT and inherited from T-185/T-186/T-211/T-215/T-217; the corrected theorem uses none of them except T-215 for NS) |
| T-222 | Brandão–Horodecki 2015; Alberti–Uhlmann 1982 | Rényi second laws, majorization under unital channels | Purity window , high temperature; answers the QRT critique negatively (no single optimum) |
[T/sim] theorems (analytical core + numerical cross-check)
- T-59 (κ_bootstrap = 1/7): analytical from ; SYNARC mvp_int_2 G5 confirms to
- T-142 (SAD_MAX=3): state-independence [T]; formula heuristic; SYNARC 500-sample cross-check
- T-145 (stochastic stability): Lyapunov–Itô–sub-Gaussian core; calibration constants tuned to SYNARC mvp_int_3
- T-148 (genesis rate): convexity + monotone convergence core; SYNARC mvp_int_2 G1–G3 numerical cross-check
- T-149 (embodied viability): coupled-attractor Step 1-2 [T]; Step 3 [C at backbone-lower-bound]; SYNARC mvp_int_2 G4 numerical cross-check
- T-155 (consciousness-preserving learning): design [D] + SYNARC mvp_int_3 SSM1–SSM2 validation
Stratified [T]+[D]+[I] theorems
- T-92 (σ_k stress): [T] at equivalence + [D] at component definitions
- T-103 (hedonic valence): [T] at identity + [T] at gate + [T] at observability + [I] at phenomenal reading
- T-150 (-tower commutativity): [D] (trivial composition law)
- T-153 (consciousness criterion): [D] definitional + [C at T-149] dependency + [T/sim] empirical instance
- T-159 (reference architecture): definition unrolled via prior theorems
- T-177, T-183 (roles fixed by three non-collinear marks; given and the pair, one binary convention for /): [T]+[D] (restated 2026-09-25)
- T-197 (AGI-Sufficiency S-11): [T]+[D] with A7 clause [C at obstruction crossing]
- T-202 (meaning as G₂-orbit): [T] at strict refinement of Yoneda + [I] at Chinese-Room identification
- T-209 (Operational-Closure S-13): [T]+[D] with [D] at operational-protocol specifications
- T-215 (cross-layer identity): [T]+[D] — the [T] is reconciliation theorem; [D] is identity-criterion choice
- T-221 (relationalist route, corrected 2026-09-25): [T]+[I] — (a)–(e) are theorems of the forcing semantics; which of the three route-readings is UHM's is interpretive; the "fourth route" and "RQM = 1-truncation" are retracted [✗]
How to read a stratified tag
A tag like [T at X] + [T/sim] + [D at Y] means:
- the result is rigorous given assumption X (stated explicitly in the row)
- the specific numerical/parameter values are additionally cross-checked against SYNARC simulations
- design choice Y is an engineering specification, not a derivation
This taxonomy does not weaken UHM as a theory — it makes the epistemic status of each claim explicit, matching the standard practice of physical theories (general relativity is a theory despite its field equations not being Lean-formalised; Connes–Chamseddine NCG is a theory despite comparable stratification).
Predictions Registry
| # | Name | Status | Source | Page |
|---|---|---|---|---|
| Pred 1 | No-Zombie (impossibility of zombies) | [T] | T-38a, T-96 | predictions#предсказание-1 |
| Pred 2 | E-coherent regeneration | [T] | T-38a | predictions#предсказание-2 |
| Pred 3 | Stress tensor | [T]/[C] | T-92 | predictions#предсказание-3 |
| Pred 4 | Pre-linguistic cognition | [I] | T-100 | predictions#предсказание-4 |
| Pred 5 | Collective consciousness — restated 2026-09-25: necessary condition (total correlation for ); sufficiency (the aggregated joint state passes the window) is a hypothesis. The former criterion with "[T] non-triviality / [C] viability" is retracted: every uncoupled group meets it (, so two window holons give at ) | [T] necessary / [H] sufficiency | The identity , zero exactly on products (necessary condition); CC-7 [T for almost every anchor] says when coupling meets it — not for every coupling (the earlier support "CC-7: interacting holons have " is retracted, 2026-09-25); CC-5 (at weak coupling and without the assumption (HOL), Theorem 9.5: the canonical aggregate of viable parts is viable), T-96, T-149 give the purity of the composite's aggregate, not its consciousness — and the canonical aggregate depends only on the marginals, so it cannot witness what the correlation adds (not T-86, which is L4-unreachability) | predictions#предсказание-5 |
| Pred 6 | Minimal coherence (; the former clause withdrawn 2026-09-26) | [T] | Level 1 row 5 (), T-96 | predictions#предсказание-6 |
| Pred 7 | Stability radius — corrected 2026-09-25 from [T]: T-104 is [C] (closed form on the one-dominant family; for general spectra a lower bound [H]), and the old prediction used the refuted closed form | [C] | T-104 | predictions#предсказание-7 |
| Pred 8 | Capacity | [T] | T-107 | predictions#предсказание-8 |
| Pred 9 | Learning bound | [T] | T-112 (from T-109–T-111) | predictions#предсказание-9 |
| Pred 10 | N=7 for learning | [T] | T-113 | predictions#предсказание-10 |
| Pred 11 | N=7 for ToM | [C] | T-57, T-114 (triadic + Fano grammar; not T-113, which is Pred 10's individual-learning source) | predictions#предсказание-11 |
| Pred 12 | SAD ceiling (SAD_MAX=3) | [T] | T-142 | predictions#предсказание-12 |
| Pred 13 | Genesis time | [T] | T-148 | predictions#предсказание-13 |
| Pred 14 | Phase coherence | [T] | T-114 (Fano grammar / co-rotating targets; not T-125, which is local asymptotic stability) | predictions#предсказание-14 |
| Pred 15 | Attractor inside the window, below its upper edge: , (restated 2026-09-26; was " at the upper bound") | [C at (MaxΦ)] | T-124c(4), T-124 | predictions#предсказание-15 |
| Pred 16 | L1→L2 avalanche | [T] | transcritical bifurcation (swallowtail-transitions; not T-158, which is canonical σ-bounds) | predictions#предсказание-16 |
| Pred 17 | Critical exponents | [C at the ℤ₂ symmetry m → −m] | T-161 | predictions#предсказание-17 |
| Pred 18 | Ward suppression | [T] | Level-1 #13 (19/49 from F₂₁ spectrum + Ward identities; not T-159, which is motor stress) | predictions#предсказание-18 |
| Pred 19 | CPTP-anchor validation | [T] | T-152 | predictions#предсказание-19 |
| Pred 20 | Analytical ε | [C at (SV)] | T-64 | predictions#предсказание-20 |
| Pred 21 | Reconstruction of Γ from neural data | [H] | — | predictions#предсказание-21 |
| Pred 22 | Spectral gap → oscillations | [H] | T-39a | predictions#предсказание-22 |
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