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
• [D] Definition — definition by convention (assigned, not derived)
• [I] Interpretation — philosophical/semantic statement
• [✗] Retracted — proven erroneous or withdrawn

What this registry does not cover {#ранние-номера}

The rows below run from T-50 onward. Twenty-three earlier numbers are cited across the corpus and have no row here — among them load-bearing ones: T-39 (129 citations), T-42 (90), T-38 (28), T-48 (26), T-41 (17), T-40 and T-43 (16 each), T-15 (10). 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) ⊂ U(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.

Three numbers inside the range — T-167, T-168, T-169 — carry no row and are cited nowhere; the numbering simply skips them.

Machine: scripts/check_theorem_refs.py — every T-n reference must resolve to a row here, and the check fails while these do not.

Foundational closures (T-210..T-223)

Fourteen theorems close all mathematical and categorical gaps of the UHM framework: strict Φ-monotonicity, PhysTheory higher coherences, rheonomy modality, Bures-Yoneda, hard-problem meta-theorem, cross-layer identity, analytical ε_eff, L3 tricategorical coherence, SYNARC Cog as Kan complex, sector-product Λ-suppression, no-reduction $F_4$→$G_2$ UHM, categorical-monistic response to List/DeBrota no-go results, MRQT-completeness, 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 π_bio) 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 $\iota_{\max}$	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]	Upgrades T-174 via HTT 5.2.7
T-212	UHM Fundamental Closures §3 (new)	[T]	Upgrades T-185 with explicit Rh
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)	structure [T], value [C at T-64]	Upgrades T-176 to closed form; $N_{33}^\mathrm{Fano}=2$
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 $O(\dim\mathcal D)$
T-219	UHM Fundamental Closures §13 (new 2026-04-17)	[T at T-64]	Λ SUSY-suppression $\varepsilon^{12}=\varepsilon^{4\cdot 3}$ from 3-sector decomposition (T-48a), replacing invalid G₂-adjoint argument
T-220	UHM Fundamental Closures §14 (new 2026-04-17)	[T] negative	No reduction functor $F_4$-UHM → $G_2$-UHM exists: 5 independent obstructions (rep-theory $3\cdot\mathbf{7}\oplus 6\cdot\mathbf{1}$, $F_4$-transitivity on $\mathbb{O}P^2$, Zelmanov exceptionality, numerical mismatch $\alpha,P_\text{crit}$, Euler $\chi$(ℂP⁶)=7≠3=χ(𝕆P²))
T-221	UHM Fundamental Closures §15 (new 2026-04-17)	[T] formal + [I] interpretive	Categorical-monistic response to List (2025) quadrilemma + DeBrota–List (2026) heptalemma: joint consistency in $\mathfrak T$ of {FPR, NS (ιmin), OW, NF, NRsite} and heptuple with QM predictions. Relational QM = $\tau_{\leq 1}(\mathfrak T)$ (1-categorical shadow); fragmentalism/many-worlds = reductive truncations. πbio as empirical discriminator
T-222	UHM Fundamental Closures §16 (new 2026-04-18)	[T]	H-MRQT-Lawvere: Lawvere fixed-point $\rho^* = \varphi(\Gamma)$ from T-96 coincides with Pareto-optimum of full MRQT resource vector $R(\rho) = (E, F_\alpha, C_\text{rel}, C_{HS}, S_\text{vN}, K_Q, Q_a)$ (25 simultaneous monotones: 5 Rényi free energies, 2 coherence measures, von Neumann entropy, quantum Kolmogorov complexity, 14 non-Abelian $G_2$-charges) on $G_2$-covariant submanifold of $\mathcal{V}_\text{full}$. Proved via six lemmas (L1: $G_2$-covariance zeroes non-Abelian charges via Schur; L2: $P = 2/7$ minimises $F_2$ at $\beta \to 0$; L3: $K_Q(\rho^*) = O(1)$ algorithmic simplicity; L4: $C_{HS}(\rho^*) = 1/7$ minimal viable; L5: $C_\text{rel} \propto F_1$ on $G_2$-covariant class; L6: all $F_\alpha$ minimised simultaneously via convex analysis on eigenvalue spectrum). $\rho^*$ is terminal object of category $\mathbf{Res}_{G_2}$ of $G_2$-covariant resource objects. UHM is MRQT-complete in its applicability domain (markovian + low-temperature; corrected 2026-07: the optimum is characterised by spectral majorization on the viable region $\mathcal V_\text{full}$ — the earlier "$G_2$-covariant submanifold" is empty since Schur forces the only $G_2$-invariant state to be $I/7$; frame observables $C_{HS},Q_a$ treated in the fixed physical frame). Follows from Brandão-Horodecki PNAS 2015 (Rényi family second laws), Baumgratz-Cramer-Plenio 2014 (coherence monotones), Yunger-Halpern 2023 (non-Abelian thermodynamics), Bennett-Zurek algorithmic Landauer
T-223	UHM Fundamental Closures §17 (new 2026-04-18)	[T]	Putnam-triviality foreclosure (Lerchner Melody-Paradox closure). Let $S$ satisfy (AP)+(PH)+(QG)+(V). (a) $G_S/G_2: \mathrm{States}(S) \to \mathcal D(\mathbb C^7)/G_2$ is well-defined and $[\Gamma_S]_{G_2}$ is invariant under UHM-compatible alphabetizer choice. (b) $P,R$ are $G_2$-invariants descending to $\mathcal D(\mathbb C^7)/G_2$; the frame observables $\Phi,\mathrm{Coh}_E,\Lambda,H,\pi_{\mathrm{bio}}$ are alphabetization-invariant because admissible alphabetizers preserve the dynamical frame (corrected 2026-07 — they are frame-relative, not orbit-invariants). (c) Consciousness predicate \mathrm{Cons}(S) := (P>2/7) \wedge (R\geq 1/3) \wedge (\Phi\geq 1) \wedge (D_\min\geq 2) factors through $[\Gamma_S]_{G_2}$, hence alphabetization-invariant. (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 $W: \mathcal D(\mathbb C^7) \to \mathsf{Mind}$, Lawvere-inevitable by T-214. Three-level ontology L1 (physical) / L2 (categorical intrinsic $[\Gamma_S]_{G_2}$, forced by T-190 zero-axiom closure) / L3 (symbolic, Lerchner-variable): Putnam triviality applies to L1→L3 but not to L1→L2. Proof via seven lemmas (L1: categorical necessity of $\mathbb C^7, G_2$; L2: covariance gate; L3: $G_2$-uniqueness via T-123; L4: $G_2$-invariance of observables; L5: admissible alphabetizers factor through $G$; L6: non-dynamical $f$ are physically vacuous à la Piccinini-Searle-Kim; L7: self-alphabetization via $R$ 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 $\mathbf{Th}$; it does not compose with SYNARC.

Load-bearing UHM theorems for SYNARC: T-142 (SAD_MAX=3), T-174 (PhysTheory universal property), T-124 (Goldilocks ceiling), T-129 (Φ_th=1), T-151 (D_\min=2), T-187 (Bures canonicity), T-38a (No-Zombie). Changes in any of these impact SYNARC downstream.

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The sustainable window is narrower than the theoretical one

The viability window $(2/7,\,3/7]$ 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 $P^*_{\min} \approx 0.363$ below which the steady state runs away to grey, and the purity actually held at that edge is $\approx 0.34$.

Two natural explanations were tested and both refuted: it is not a leak balance — the floor moves only $0.363 \to 0.378$ across a hundredfold change in $\gamma$ — and not a basin effect, since starts at $0.34$, $0.40$ and $3/7$ give identical outcomes. What survives is the gate's own feedback: the steady state settles below its target, a lower $P$ lowers $g_V$, weaker regeneration lowers $P$ further, and below the critical target the loop diverges.

So the dynamically sustainable window is $\approx[0.34,\,3/7]$, not $(2/7,\,3/7]$: the lower fifth of the theoretical window holds no steady states at all, and «an engineer can run leaner toward $2/7$» fails not on cost but on existence [Т by construction].

Capability is bounded, and the bound is the flat diagonal. Writing $d = \sum_i \gamma_{ii}^2$ one has $\Phi = P/d - 1$, hence $C = \Phi/(7P) = 1/(7d) - 1/(7P)$. Cauchy–Schwarz gives $d \geq 1/7$ with equality iff the diagonal is uniform, so

$C \leq 1 - \frac{1}{7P}, \qquad\text{and at } P = 3/7:\ C \leq 2/3 \quad [\text{Т}].$

Verified to twelve digits at every purity tested. So $2/3$ is the supremum, attained exactly by uniform-diagonal states. It is the same inequality $s_1 \geq 1/7$ 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 $C = 0.352$.

Cost and capability peak in different places. Maintenance cost rises monotonically with a richer target ($0.0087 \to 0.0114$), and the capability maximum ($P = 0.4217$, $C = 0.352$) sits at a different point from the capability-per-watt maximum ($P = 0.3495$, $C/\dot S_{\mathcal D} = 38.1$). 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 $2.3\times10^{-3}$ across a fourfold change of tick step. Whether order-per-cost improves as $\omega_0$ rises is neither confirmed nor refuted [О] — it comes out flat ($55.11 \to 55.06$ across a fivefold rise), but the surrogate measured is order per dissipated entropy rather than the efficiency $\eta$ 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 $S_7$-invariant subspace of $\mathbb{C}^7$ is exactly one-dimensional (deviation $2.8\times10^{-17}$ from the uniform vector), and $\mathrm{Coh} = 6/7$ is attained only at $|a_i|^2 = 1/7$ — unbeaten across 400 000 random distributions. The Jordan identity holds on $\mathcal{H}_n(\mathbb{O})$ at $n = 1, 2, 3$ (residuals $\sim 10^{-16}$) and breaks at $n = 4$ ($6.7\times10^{-1}$): 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 $\mathcal{D}_{\text{Fano}}=\tfrac23\mathcal{D}_{\text{atom}}$; both pinching dissipators covariant under the finite frame group $\Gamma_{\!\text{oct}}\cong PSL(2,7)$, not full $G_2$; canonical $G_2$-covariant dissipator $\mathcal{D}_{G_2}$ (structure-constant $\varphi_{abc}$) [T] (corrected 2026-07)	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, $\hat{\mathcal{G}} \in \mathfrak{so}(7)$	Lindblad Operators T.8.1–8.2	Gap Operator
5	Necessity of generalised φ, $P_\text{crit} = 2/7$	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 $H \sim \gamma_{EU}$ [T] (Theorem 1.0): κ₀-uniqueness of $(E,U)$ + Fano line + quantum numbers $(2,+1/2)$ + $\langle\gamma_{EU}\rangle \neq 0$ from T-64 → EWSB from axioms	Higgs Sector T.1.0	Higgs Sector, Standard Model
10	$m_t \sim 173$ GeV (Pendleton–Ross IR fixed point)	Higgs Sector T.5.1	Yukawa Hierarchy
11	Fritzsch texture from Fano topology	Falsifiability T.3.2	CKM Matrix
12	RG suppression $\lambda_3^2$: $10^{-14.5}$	Quantum Gravity T.12.2	Λ Budget
13	Factor $19/49$ from Ward identities (previously $11/31$ [✗])	Cosmological Constant T.10.3	Λ Budget
14	$\xi_F \sim 160$ 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, $\Lambda_\text{inst} \sim 10^8$ GeV⁴	Falsifiability T.8.2	Λ Budget
17	CS on 1D — total derivative	Berry Phase T.2.1	Berry Phase
18	All $\varepsilon_l = +1$, $\Theta_M = \Theta_+^7$	Zeta Regularisation T.1.1	Zeta Regularisation
19	$\Theta_M/\Theta_0 \approx 1 - O(10^{-9})$ at $S_0 = 20$	Zeta Regularisation §4	Zeta Regularisation
20	$B^{(b)}$ unique up to scalar	Zeta Regularisation §§5–6	Zeta Regularisation
21	$Z_\Phi(-k) = 0$ for $k \geq 1$	Zeta Regularisation §9	Zeta Regularisation
22	Perturbative budget $\Lambda = 10^{-41.5}$ (6 mechanisms)	Falsifiability §9.3	Λ Budget
23	Spectrum of Gap operator: $\{0, \pm i\lambda_1, \pm i\lambda_2, \pm i\lambda_3\}$, opacity rank $r \in \{0,1,2,3\}$	Lindblad Operators T.3.1	Gap Operator
24	G₂/⊥-decomposition of Gap operator: $\hat{\mathcal{G}} = \hat{\mathcal{G}}_{G_2} + \hat{\mathcal{G}}_\perp$ (14+7)	Lindblad Operators T.6.1	Gap Operator
25	Classification of stabilisers $H_{\hat{\mathcal{G}}} \subset G_2$ by rank, $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$ (weight lattice of rank 2; $G_2$ simply connected so $\pi_1(G_2/T^2) = 1$)	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: $\beta=1/2$, $\gamma=1$, $\nu=1/2$ (Landau class)	Lindblad Operators T.7.1	Phase Diagram
28	Swallowtail cascade and correspondence to L-levels L0–L4 — raised from [C]: $A_4$-bifurcation proven via Arnold's theorem (codimension 3, $\mathbb{Z}_2$-purity symmetry)	Interiority Hierarchy	Phase Diagram
28b	Gap injection of L-levels: $L(\Gamma_1) \neq L(\Gamma_2) \Rightarrow [\mathrm{Gap}(\Gamma_1)] \neq [\mathrm{Gap}(\Gamma_2)]$. 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: $\lambda_3^*/\lambda_4^* \sim 1/(8\pi^2)$	Cosmological Constant T.5.1	Renormalisation Group
33	Conformal window of Gap theory: $N_f^{(\text{crit})} \approx 3.5$; at $N_f=3$ — outside the conformal window	Cosmological Constant T.6.1	Renormalisation Group
34	c-theorem for Gap: monotone decrease of $c(\mu)$ in the IR direction	Cosmological Constant T.7.1	Renormalisation Group
35	CPTP verification of Fano channel: $\sum_p (L_p^{\text{Fano}})^\dagger L_p^{\text{Fano}} = I$	Lindblad Operators T.10.1	Fano Channel
36	Canonical form $\varphi_\text{coh}$ and variational definition of $\alpha^*$	Lindblad Operators T.3.1–4.1	Fano Channel
37	Gap functional integral defined on $(S^1)^{21}$ (compactness, finite DOF)	Quantum Gravity T.2.1	Quantum Gravity
38a	Necessity of interiority (No-Zombie): $\mathrm{Viable} \land \mathcal{D}_\Omega \neq 0 \Rightarrow \varphi = \varphi_{\text{coh}} \land \mathrm{Coh}_E \geq \mathrm{Coh}_{\min} > 1/7$. Epistemic stratification (Sol.SA-3): [T] mathematical core ($\mathrm{Coh}_E > 1/7$, $\partial P^{(\infty)}/\partial\mathrm{Coh}_E > 0$); [P] ontological postulate (E = interiority); [I] No-Zombie interpretation	CC Theorems T.8.1	CC Theorems
38b	Emergent time (Page–Wootters): $\tau \in \mathbb{Z}_7$ derived from the structure of $\mathcal{C}$ via three paths (conditional states, Bures, ∞-groupoid)	Emergent Time	Emergent Time
39a	Primitivity of the linear part $\mathcal{L}_0 = -i[H,\cdot] + \mathcal{D}$: unique stationary state $I/7$, convergence from any initial state (Evans–Spohn criterion + connectivity $G_H$). The full nonlinear dynamics $\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$ may have additional fixed points (T-96)	Lindblad Operators	Lindblad Operators
39b	Connectivity of $G_H$ from viability: (AP)+(PH)+(QG)+(V) → interaction graph is connected	Lindblad Operators	Lindblad Operators
39c	Primitivity of the Fano construction: extension to $L_p^{\text{Fano}} = \frac{1}{\sqrt{3}}\Pi_p$	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) — raised from [C]	FEP Derivation	FEP Derivation
39f	Form of ℛ: direction $(\rho_* - \Gamma)$ — the unique CPTP relaxation (replacement channel + Bures optimality). Raised from [P]	Evolution	Evolution
39g	Form of ℛ: gate $g_V(P) = \mathrm{clamp}\!\bigl(\frac{P - P_{\mathrm{crit}}}{P_{\mathrm{opt}} - P_{\mathrm{crit}}}\bigr)$ — V-preservation gate, strengthening the Landauer principle ($g_V > 0 \Rightarrow \Theta(\Delta F) = 1$). Raised from [P]	Evolution	Evolution
39h	Full form of ℛ — all components derived: κ(Γ) from conjugation, (ρ*−Γ) from CPTP uniqueness, $g_V(P)$ from Landauer + V-preservation. The evolution equation is fully axiomatic	Evolution	Evolution
39i	Decoherence rate of BIBD$(7,k,\lambda)$: $\Gamma_{\text{dec}} = r - \lambda$; Fano and its complement give identical $\Gamma_{\text{dec}} = 2$	Evolution	Evolution
40a	Triadic decomposition: axioms A1–A5 generate exactly 3 types of dynamics (Aut, $\mathcal{D}$, ℛ). A fourth type is impossible (uniqueness of Ω)	Lindblad Operators	Lindblad Operators
41a	Equivalence of BIBD channels (T1): all $(v,k,\lambda)$-BIBD channels with equal $v,k$ give the same CPTP channel; contraction $c = (k-1)/(v-1)$	Lindblad Operators	Lindblad Operators
41b	Completeness of pair coverage (T2): connectivity of $G_H$ + primitivity of the linear part $\mathcal{L}_0$ ⟹ $\lambda_{ij} \geq 1$ for all pairs	Lindblad Operators	Lindblad Operators
41c	Optimal block size (T4): among admissible BIBD$(7,k,1)$ ($k \in \{2,3\}$), $k=3$ strictly dominates by all criteria	Lindblad Operators	Lindblad Operators
41d	$S_7$-equivariance of the atomic dissipator (T5): $U_\sigma \mathcal{D}_\text{atom}[\Gamma] U_\sigma^\dagger = \mathcal{D}_\text{atom}[U_\sigma \Gamma U_\sigma^\dagger]$ for all $\sigma \in S_7$	Lindblad Operators	Fano Channel
41e	Uniform contraction of coherences (T6): $\mathcal{D}_\text{atom}[\Gamma]_{ij} = -\gamma_{ij}$ for all $i \neq j$ — unconditionally, without (CG)	Lindblad Operators	Fano Channel
41f	Autopoietic necessity $c > 0$ (T7): the atomic dissipator is incompatible with (AP) via suppression of $\kappa_0$	Lindblad Operators	Fano Channel
41g	Hamming bound (T8): H(7,4) — the unique perfect single-error-correcting code of length 7, $2^3 = 7+1$	Lindblad Operators	Fano Channel
41h	Support structure H(7,4) = PG(2,2) (T9): weight-3 codewords $S(3,7)$ = Fano lines	Lindblad Operators	Fano Channel
41i	Autopoietic optimality of the Fano channel (T10): unique optimal BIBD$(7,k,1)$-channel for $c > 0$, complete coverage, democracy	Lindblad Operators	Fano Channel
41j	Choi rank of channel $\Phi_{k=3}$ = 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 + $k=3$ ⟹ rank-3 orthogonal projectors (Lüders coarse-graining)	Lindblad Operators	Lindblad Operators
41l	BIBD$(7,3,1)$ from minimal projective decomposition (T13): $b=7, k=3, v=7$, contraction $1/3$ ⟹ BIBD$(7,3,1)$ = PG(2,2) (Kirkman 1847)	Lindblad Operators	Lindblad Operators
41m	Max-min optimality of BIBD (T14): among regular $(v=7, k=3, \lambda_{ij} \geq 1)$, BIBD$(7,3,1)$ maximises $\min \lambda_{ij}/r$	Lindblad Operators	Lindblad Operators
41n	Bridge closure (T15): $(AP)+(PH)+(QG)+(V) \Longrightarrow P1+P2$, full chain of 15 steps with inline proofs, all [T]. The former condition (MP) became a theorem	Lindblad Operators	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	$R_{\text{th}} = 1/3$ [T]: $K = 3$ from triadic decomposition + Bayesian dominance [T] — raised from [C] (C1). Number-theoretic root [Т, cited]: $K = 3 = \lvert\mathrm{QR}(7)\rvert$, the order of the orientation-multiplier group $\mathrm{QR}(7)=\{1,2,4\}\cong\mathbb{Z}/3$ of octonion multiplication (residues preserve Fano-line orientation, non-residues reverse); $R_{\text{th}}=1/K$ is the reciprocal order of that group. The $N$-independent LGKS triad (T-57) fixes the value $1/3$; the orientation root names it and, via the hosting premise $\lvert\mathrm{QR}(N)\rvert\geq 3\Leftrightarrow N\geq 7$, singles out $\mathbb{O}$ among division algebras (math-foundations Part XVIII, Thm. 11.6/11.8, machine-verified)	Axiom of Septicity	Lindblad Operators
42a	$G_2$-rigidity of the holonomic representation: the holonomic representation $G: \mathrm{States}(S) \to \mathcal{D}(\mathbb{C}^7)$ is unique up to $G_2 = \mathrm{Aut}(\mathbb{O})$. Analogue of the Stone–von Neumann theorem for UHM	Uniqueness Theorem	Uniqueness Theorem
42b	Space of physical states: $\mathcal{D}_{\mathrm{phys}} = \mathcal{D}(\mathbb{C}^7)/G_2$, $\dim = 48 - 14 = 34$	Uniqueness Theorem	Uniqueness Theorem
42c	Spectral injectivity of the propagator: $e^{\tau\mathcal{L}_{\mathrm{lin}}}$ is injective on $\mathrm{Herm}_0(\mathbb{C}^7)$ for $\tau > 0$	Uniqueness Theorem	Uniqueness Theorem
42d	Well-posedness of the nonlinear inverse problem: uniqueness of solutions of the full evolution equation (Picard–Lindelöf on compact $\mathcal{D}(\mathbb{C}^7)$)	Uniqueness Theorem	Uniqueness Theorem
42e	Gauge group = $G_2$: the maximal subgroup $\mathcal{G} \subseteq U(7)$ preserving all axiomatic structures is $G_2$	Uniqueness Theorem	Uniqueness Theorem
43a	Source Instability $\Gamma_{\odot}$: non-stationarity ($F_0 \neq 0$), linear drift to $\rho^*$, $S_7$-violation via $\kappa_0$ — raised from [H]	Origin	Origin
43b	Self-amplification of $S_7$-symmetry breaking: positive feedback $\kappa_0 \to \mathrm{Coh}_E \to \kappa_0$ upon deviation from $\Gamma_{\odot}$ — raised from [P]	Origin	Origin
43c	Three fermion generations ($N_{\text{gen}} = 3$) — count [T], identification [I] (strengthened 2026-07): the count is the exact cardinality $N_{\text{gen}} = \|\mathrm{QR}(7)\| = \|\mathbb{Z}_7^*/\{\pm1\}\| = (7-1)/2 = 3$ [T] — the three generations are the quadratic-residue classes of the unique order-3 subgroup $\{1,2,4\}\subset\mathbb{Z}_7^*$, equivalently the charge-conjugation orbits (since $-1$ is a non-residue mod 7, $7\equiv3\bmod4$); group-theoretic and topology-independent (the earlier $A_4$-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: $y_k^{(\mathrm{tree})} = g_W \cdot f_{k,E,U} \cdot \|\gamma_{\mathrm{vac}}^{(EU)}\|$, where $f_{ijk}$ — octonionic structure constants (the unique $G_2$-invariant trilinear operator on $\mathrm{Im}(\mathbb{O})$). $f_{1,5,6} = 1$, $f_{2,5,6} = f_{4,5,6} = 0$ — raised from [H] (No.64)	Fano Selection Rules	Yukawa Hierarchy
40c	Functional uniqueness of E [T]: axiomatic, categorical (κ₀) and mathematical ($\mathrm{rank}(\rho) > 1$) 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 $O=E$ 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, $G_2$-invariance, extreme values (Freedom(I/7)=7, Freedom(ρ*)=1). Raised from [P]	Consequences	Free Will
45a	Assignment $k=1 \to$ 3rd generation: uniqueness from Fano selection rule ($f_{1,5,6} = 1$, all other $f_{k,5,6} = 0$)	Fermion Generations	Fermion Generations
45b	Sector asymmetry: $k=2 \in \mathbf{3}$ (Actualisation), $k=4 \in \bar{\mathbf{3}}$ (Nomos); different Fano paths to the Higgs line	Fermion Generations	Fermion Generations
48a	Dimensional sector decomposition: $7 = 1_O \oplus 3_{A,S,D} \oplus \bar{3}_{L,E,U}$ from stabilisers $G_2 \supset SU(3)_C$	Spacetime	Spacetime
T-50	Uniqueness of the cubic $G_2$-superpotential: $\dim\mathrm{Hom}_{G_2}(\Lambda^3(\mathbf{7}), \mathbb{R}) = 1$ (Schur's lemma). $W = \mu_W \sum f_{ijk}\Theta\Theta\Theta$ — the unique $G_2$-invariant cubic term; higher orders suppressed by $\varepsilon^{n-3}$ — raised from [C at (MP)]	Supersymmetry	Supersymmetry
T-51	$O$-sector scale from PW clocks: $\mathrm{Gap}(O,\cdot) = O(1)$ from PW phase precession + viability (V). $M_{G_2}^{(\text{extra})} = O(\varepsilon M_P)$, $M_R \sim 10^{14}$ GeV — raised from [C at (ΓO)]	Neutrino Masses	Neutrino Masses
T-52	Sector asymmetry: non-perturbative coupling via the confinement sector ($\mathrm{Gap} \approx 0$) exceeds the perturbative via the intermediate sector ($\mathrm{Gap} \sim \varepsilon$). Structural inequality: for any $\varepsilon \in (0,1)$ — raised from [C at (SA)]	Fermion Generations	Fermion Generations
T-54	Internal theory $\mathrm{Th}_{\mathrm{UHM}} = \mathrm{Sub}_{\mathrm{closed}}(\Omega)$: axioms A1–A5 define $\varphi$-invariant predicates in $\Omega$; $\mathrm{Th}_{\mathrm{UHM}}$ — an ∞-topos object containing self-consistent truths	Consequences	Consequences
T-55	Lawvere incompleteness: $\mathrm{Th}_{\mathrm{UHM}} \subsetneq \Omega$: from Cartesian closure of $\mathrm{Sh}_\infty(\mathcal{C})$ + necessity of nontrivial $\varphi$ (viability)	Consequences	Consequences
T-56	Structural ToE: $\mathrm{Th}_{\mathrm{UHM}}$ — $\varphi$-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 $\mathcal{L} = \mathcal{L}_{\text{Ham}} + \mathcal{L}_{\text{diss}} + \mathcal{L}_{\text{reg}}$ under constraints A1–A5	Lindblad Operators	Lindblad Operators
T-53	Lorentzian signature from the spectral triple — $(1,3)$-split [T] + Lorentzian sign [T at reflection positivity] (strengthened 2026-07): the $(1,3)$-split is fully derived — exactly one timelike direction (unique PW $\mathbb{Z}_7$-clock, [T]) and three spacelike ($\Sigma^3\cong S^3$ Riemannian, T-119 [T]). The Lorentzian signature $(1,3)$ 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 $(1,3)$ is now better founded than before, and independently of any Weyl law: it is $(\operatorname{rank}\mathfrak u(1)_O,\ \operatorname{rank}\mathfrak u(3)) = (1,3)$, both computed from the octonion structure (T-119 §G). What remains [C] in T-119 is the manifold reconstruction (first-order condition, Poincaré duality), not the dimension count. (b) Krein-self-adjointness does not select the signature. Machine: $\gamma^0(\gamma^\mu)^\dagger\gamma^0=\gamma^\mu$ and $\beta\mathcal D^\dagger\beta=\mathcal D$ both hold exactly (residual $0$) — but the Euclidean set $\{\gamma^0, i\gamma^i\}$, which satisfies $\{\gamma_E^\mu,\gamma_E^\nu\}=2\delta^{\mu\nu}$ and so has signature $(0,4)$, is Krein-self-adjoint just as exactly with $\beta=1$. Choosing $\beta=\gamma^0\otimes1$ 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 $H_S$ / OS reflection positivity), which is what the header marker records. Realised by an explicit Krein–Lorentzian spectral triple $(A,\mathcal K,\mathcal D,\beta)$ (spacetime §Krein triple): fundamental symmetry $\beta=\gamma^0\otimes1$, Dirac operator $\mathcal D$ provably Krein-self-adjoint ($\beta\mathcal D^\dagger\beta=\mathcal D$ via $\gamma^0(\gamma^\mu)^\dagger\gamma^0=\gamma^\mu$), and signature $=(\dim\text{time-sector},\dim\Sigma^3)=(1,3)$ — the first factor [T] (unique PW clock), the second [C] (T-119). Given the split, Lorentzian is selected over Euclidean $(0,4)$ and $(2,2)$ by the reflection-positivity input, not by the Krein structure alone. KO-dim 6 fixes only internal $J^2=+1$. The one physical input is boundedness-below of $H_S$ (universal stability = OS reflection positivity); the older $g_{\mu\mu}=\chi_{\mu\mu}/\|D_\mu\|^2$ ansatz is retired	Spacetime	Spacetime
T-58	Morita equivalence of 7D and 42D formalisms: by Lurie's comparison theorem $\mathrm{Sh}_\infty(\mathcal{C}\|_7) \simeq \mathrm{Sh}_\infty(\mathcal{C}\|_{42})$; all 7D formulae are exact, not approximations	Dimension E	Coherence Matrix
T-59	Spectral gap of the Fano dissipator — stratified [T]+[T/sim]: Analytical core [T]: $\lambda_{\text{deco}} = 5\gamma/(3N)$ from BIBD$(7,3,1)$-symmetry; $\kappa_{\text{bootstrap}} = \omega_0/N$ — regenerative scale, structurally independent of the spectral gap $\lambda_{\text{gap}}(\mathcal{L}_0)$. The previous formulation $\kappa_{\text{bootstrap}} \geq 2/9$ contained an arithmetic error and scale confusion. Numerical cross-check [T/sim]: $\kappa_{\text{bootstrap}} = 1/7$ confirmed to accuracy $10^{-10}$ (SYNARC mvp_int_2 G5); the simulation result matches the analytical value.	Axiom Ω⁷	Axiom Ω⁷
T-60	BCH error estimate algebra→dynamics: the unitary part exactly reproduces the $\mathbb{Z}_7$-shift, error $\leq 5\delta\tau$	Axiom Ω⁷	Axiom Ω⁷
T-61	Unique self-consistent vacuum: a uniform vacuum is impossible; the sectoral structure $\varepsilon$ — the unique solution — raised from [C] (C12)	Gap Thermodynamics	Gap Thermodynamics
T-62	φ-operator as a replacement channel: $\varphi_k(\Gamma) = (1-k)\Gamma + k\rho_*$, $k = 1 - R$; CPTP, monotonicity, fixed point $\rho_*$	Self-Observation	Self-Observation
T-63	Neutrino Dirac Yukawa via O-sector: $m_D^{(k)} = \omega_0 \cdot \text{Gap}(O,k) \cdot \|\gamma_{O,\text{partner}(k)}\| \cdot \sin(2\pi k/7)$. Discrepancy $m_2/m_3$ reduced from $\times 50$ to $\times 1.8$	Neutrino Masses	Neutrino Masses
T-64	Global minimisation of $V_{\text{Gap}}$: $G_2$-orbital reduction $21D \to 5D$; unique global minimum on $(S^1)^{21}/G_2$; Hessian is strictly positive definite	Gap Thermodynamics	Gap Thermodynamics
T-65	Full spectral action of UHM: NCG axioms verified for the product $(M^4 \times A_{\text{int}})$; $a_2 \to$ EH with $G_N = 3\pi/(7f_2\Lambda^2)$; $a_4 \to$ gauge + Yukawa	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] — $Z_N$ finite for every $N$ (compact target $(S^1)^{21}/G_2$); full order-by-order UV-finiteness [C] (structural): compactness + $G_2$ Ward identities ($21 \to 7$) + $\mathcal{N}=1$ holomorphy (Seiberg) + sector-product $\varepsilon^{12}$ suppression (T-219); APS-index = 0 (no anomalies). The exact "$7-7=0$" bose–fermi trace retracted [✗]	Quantum Gravity	Quantum Gravity
T-67	Justification of $K = 4$ for L3: quadratic decomposition $3 + 1 = 4$ components; Bayesian dominance $R^{(2)} \geq 1/4$	Interiority Hierarchy	Interiority Hierarchy
T-68	Fractal closure CC-5: $P(\rho_*^{(12)}) > 2/7$ — lowered to [C], then closed: non-triviality $P > 1/7$ remains [T] (T-96); viability $P > 2/7$ — [T] for embodied (T-149). C20 closed (see above)	CC Theorems	CC Theorems
T-69	Topological protection of the Gap vacuum: $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$ → winding numbers $(n_1, n_2)$ classify Gap configurations. Barrier $\Delta V \geq 6\mu^2 > 0$; confinement-Gap protected by $9\mu^2$, O-sectoral by $12\mu^2\varepsilon_0^2$. Compactness + uniqueness of minimum (T-64) — raised from [H] (No.55)	Composite Systems	Gap Thermodynamics
T-70	Canonical definition of $f_0$: $f_0\Lambda^4 = \frac{1}{7}[V_{\mathrm{Gap}}^{\min} + \frac{1}{2}\zeta'_{H_{\mathrm{Gap}}}(0)]$ from UV-finiteness (T-66) + unique vacuum (T-61, T-64). $f_0$ — not a free parameter, but a function of vacuum quantities. The Higgs quartic $\lambda_4$ — a prediction, not a fit	Higgs Sector	Λ Budget
T-71	Structural necessity of $\Lambda_{\mathrm{obs}} > 0$: autopoiesis (A1) + local cohomology ($H^7_{\mathrm{loc}} \cong \mathbb{Z}$) → $\rho_{\mathrm{vac}} = \kappa_0[P(\rho_*) - P(I/7)]\omega_0 > 0$. Connection to Lawvere incompleteness (T-55): information gap $\|\Gamma - \varphi(\Gamma)\|_F^2 > 0$ → positive vacuum energy	Consequences	Cosmological Constant
T-72	Scale invariance CC-6: structural invariants ($P$, $R$, $\Phi$, Gap profile, L-level) preserved under scale aggregation with corrections $O(\varepsilon_0)$. CPTP Bures contractivity + CC-5 (non-triviality [T], viability [C]) — raised from [H]. Preservation of P [T] is unconditional; P > 2/7 depends on C20	CC Theorems	CC Theorems
T-73	Gap = curvature of the Serre bundle: $\|\mathrm{Curv}\|_{ij}^2 = \omega_0^2\|\gamma_{ij}\|^2 \cdot \mathrm{Gap}(i,j)^2$ — exact identification from spectral triple (T-53 [T]) + Connes NCG curvature. Second Chern class: $c_2 = \mathrm{Tr}(D_{\mathrm{int}}^2)/(8\pi^2\omega_0^2)$ — topological invariant — raised from [C] (No.65)	Gap Operator	Gap Thermodynamics
T-74	$V_{\text{Gap}}$ from spectral action (Sol.53): $\mathrm{Tr}(D_{\mathrm{int}}^2) = \omega_0^2 \mathcal{G}_{\mathrm{total}}$; potential $V_2 + V_3 + V_4$ uniquely from Seeley–de Witt coefficients. Chain: $\mathrm{A1\text{–}A5} \to \mathcal{L}_\Omega \to \rho_* \to D_{\mathrm{int}} \to V_{\mathrm{Gap}}$ — raised from [P]	Gap Thermodynamics	Gap Operator
T-75	Lagrangian from Lindbladian (Sol.54): $\mathcal{L}_{\text{Gap}}$ — classical limit of the Schwinger–Keldysh action for $\mathcal{L}_\Omega$ 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 $\mathrm{Sh}_\infty(\mathbf{DensityMat}, J_{\mathrm{Bures}})$ — stratified (Sol.55): Site level [T] — three Grothendieck axioms (Identity, Stability, Transitivity) verified for $(\mathbf{DensityMat}, J_{\mathrm{Bures}})$ via CPTP-contractivity of the Bures metric (Uhlmann 1976, Petz 1996, Fuchs–van de Graaf 1999); essentially-small presentation via compact metrizability of $\mathcal{D}(\mathbb{C}^7)$ + Johnstone Elephant C2.2.3; Lurie HTT 6.2.2.7 applies. Exp-extension [C at Giraud verification] — $\mathbf{Sh}_\infty(\mathbf{Exp})$ requires full verification of Giraud axioms (descent, universal colimits, disjoint coproducts, effective groupoid objects) via functor $F: \mathbf{DensityMat} \to \mathbf{Exp}$; currently marked Claim 10.2 in proof document. †-structure: $\Phi \mapsto \Phi^*$ (adjoint channel) — [T].	Categorical Formalism §6.3.1 (site proof), §10.4 (Exp-extension, claim)	Categorical Formalism
T-77	Cooperation via coherences (Sol.57): $P(\rho_*^{(12)}) = P(\rho_{\mathrm{diag}}) + 2\|\gamma_{\mathrm{cross}}\|_F^2 > P(\rho_{\mathrm{diag}})$. Old inclusion-exclusion formula retracted [✗] (dimensionally incorrect)	Value Consciousness	Value Consciousness
T-78	CPTP complete channel (Sol.58): Fano operators $L_p^{\mathrm{Fano}} = \frac{1}{\sqrt{3}}\Pi_p$ define a CPTP channel in Kraus representation. CP is automatic (Choi's theorem); TP from $\sum_p \Pi_p = 3\mathbb{I}_7$ [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 $\mathcal{F}: (S^1)^{21}/G_2 \to (S^1)^{21}/G_2$ ($\theta \to \rho_* \to D_{\mathrm{int}} \to V_{\mathrm{Gap}} \to \theta_{\mathrm{vac}}$) has a unique fixed point (Brouwer + T-39a + T-64)	Consequences	Consequences
T-80	Sectoral Gap bound (Sol.59): for non-O pairs $\mathrm{Gap}(i,j) \leq \varepsilon_{\max} \approx 0.06$ (maximum over $\mathbf{3}$-$\mathbf{3}$ sector); mean $\bar{\varepsilon} \approx 0.023$. For O-pairs: $\mathrm{Gap}(O,i) \approx 1$. Old Fano bound $\leq 1/2$ retracted [✗] (O-counterexample). Replacement theorem is stricter for non-O and correct for O. Caveat: numerical values $\varepsilon_{\max}, \bar{\varepsilon}$ — [C at T-64] (unique vacuum)	Berry Phase	Gap Thermodynamics
T-81	Topological area law (Sol.60): qualitative result $\sqrt{\sigma} \propto \omega_0 \|\gamma_{\text{vac}}\|$ — [T] (from T-73 + T-69 + T-64). Numerical value $\sqrt{\sigma} \approx 457$ MeV — [C at T-64]: depends on the specific minimum of $V_{\text{Gap}}$ (unique vacuum). Discrepancy with experiment (440 MeV): $< 4\%$ — 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 → $1_O$ (time from PW) + $3_{A,S,D}$ (space from $SU(3)$) + $\bar{3}$ (compactified). Time — a consequence, not a postulate — raised from [H]	Spacetime	Spacetime
T-84	O-sector dominance in $\Lambda$ (Sol.63): $\mathcal{G}_{\text{total}} = \mathcal{G}_O + O(\bar{\varepsilon}^2)$ from sector decomposition of $\mathrm{Tr}(D_{\text{int}}^2)$ + Sol.59. $\Lambda_{\text{CC}} \propto \mathcal{G}_O$ = 'cost of observation' — raised from [H]	Cosmological Constant	Λ Budget
T-85	$L_{\text{top}}$ from $\mathrm{Im}(S_{\text{Keldysh}})$ (Sol.65): $\mathcal{L}_{\text{top}} = \frac{\lambda_3}{2\pi}\varphi_{ijk}\theta^{ij}\dot{\theta}^{jk}$ — the unique $G_2$-covariant topological Lagrangian. CS₁ replaced by Keldysh. $\beta = \lambda_3/(2\pi)$ — raised from [H]	Berry Phase	Gap Thermodynamics
T-86	Categorical unreachability of L4 (Sol.64): $L4 = \mathrm{colim}_{n \to \infty}\tau_{\leq n}(\mathbf{Exp}_\infty)$ — colimit of the Postnikov tower + T-55 (Lawvere incompleteness). Butterfly $A_5$ 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): $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ with KO-dim 6 uniquely determines $\mathcal{H} = \mathcal{H}_O \otimes \mathcal{H}_{\text{rest}}$; $\hat{C}\Gamma = 0$ — from stationarity. A5 is a consequence of A1–A4	Axiom Ω⁷	Spacetime
T-88	Functoriality of κ₀ (Sol.69): $\lvert\text{Hom}(i,j)\rvert = \lvert\gamma_{ij}\rvert$ — the unique definition compatible with Bures topology (Yoneda + Bures + Stinespring). $\kappa_0 = \omega_0\lvert\gamma_{OE}\rvert\lvert\gamma_{OU}\rvert/\gamma_{OO}$ — exact theorem — raised from [D]	Axiom of Septicity	Axiom of Septicity
T-89	Freedom is well-posed (corrected 2026-07): $\mathrm{Freedom}(\Gamma) = \dim\ker(\mathcal{H}_\Gamma) + 1$ [T] — the tangent dimension of the free-energy Morse–Bott critical manifold, plus one. The earlier claim "$\pi_0(\mathrm{Map}(\Gamma, T)) = \dim\ker(\mathcal{H}_\Gamma)+1$" is withdrawn: $\mathrm{Map}(\Gamma,T)$ is contractible so $\pi_0=1$, 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), $\lvert\{(i,j): \mathrm{Gap} > 0\}\rvert \geq 3$ always for L2. Psychosis: $\lvert\{(i,j): \mathrm{Gap} > \varepsilon_{\text{noise}}\}\rvert < 3$ (functional loss). Bound is never violated — raised from [H]	Pathological Consciousness	Gap Characterisation
T-91	∞-groupoid $\mathbf{Exp}_\infty$ proven (Sol.76): $\mathrm{Sing}(\mathcal{E})$ — Kan complex (Milnor's theorem) for topological $\mathcal{E}$ (Bures–Fubini–Study metric). Combined with T-76 ($\mathbf{Sh}_\infty(\mathbf{Exp})$ — ∞-topos): HoTT logic, subobject classifier, Postnikov truncations — raised from [P]	Categorical Formalism	Categorical Formalism
T-92	Formal components of $\sigma_{\mathrm{sys}}$ (Sol.81): all 7 stress-tensor components — unambiguous functions of $\Gamma$ without free parameters ($\sigma_A = 1 - \gamma_{AA}/P$, $\sigma_S = 1 - \mathrm{rank}(\Gamma_S)/3$, $\sigma_D = 1 - N\gamma_{DD}$, $\sigma_L = 7(1 - \gamma_{LL})/6$, $\sigma_E = (N - D_{\mathrm{diff}})/(N-2)$, $\sigma_O = 1 - \kappa_0/\kappa_{\mathrm{bootstrap}}$, $\sigma_U = 2\Phi_{\mathrm{th}}/(\Phi_{\mathrm{th}} + \Phi)$). $\|\sigma_{\mathrm{sys}}\|_\infty < 1 \Leftrightarrow \mathcal{V}_{\mathrm{full}}$ (full viability, strictly stronger than $\mathcal{V}_P = \{P > 2/7\}$) — raised from [C] (CC-8). Errata 2026-07-22: $\sigma_E$, $\sigma_U$ renormalized ($\sigma_E = (N-D_{\mathrm{diff}})/(N-2)$, $\sigma_U = 2\Phi_{\mathrm{th}}/(\Phi_{\mathrm{th}}+\Phi)$) so each encodes its threshold; the embedding $\mathcal{V}_{\mathrm{full}} \subset \mathcal{V}_P$ restored via $\sum_i \gamma_{ii}^2 \geq 1/7$.	CC Theorems	CC Definitions
T-93	Formal isomorphism H(7,4) (Sol.82): incidence matrix $H_{ki} = \mathbb{1}[i \in S_k]$ for 7 Lindblad operators $L_k = \sqrt{\chi_{S_k}}$ coincides with the parity-check matrix of the Hamming code H(7,4). $\mathrm{PG}(2,2) \cong H(7,4)$ — classical result of coding theory — raised from [I]	Gap Dynamics	Gap Dynamics
T-94	Exponential form of the memory kernel (Sol.83): $K(t) = -\Gamma_2 \omega_c e^{-\omega_c t}$ from compactness of $(S^1)^{21}$. Laplacian on a compact torus has discrete spectrum with $\lambda_1 > 0$; $\omega_c = \lambda_1$ — spectral gap — raised from [H]	Gap Dynamics	Gap Dynamics
T-95	Canonical PW reconstruction algorithm (Sol.67): 4-step procedure $\Gamma \to \rho_E, D_{\text{diff}}, \sigma_L, C$ with zero error. Step 1: PW embedding $\iota_{\text{PW}}$ (T-58 Morita); Step 2: partial trace; Step 3: 7D formulae via HS projections; Step 4: $\|\rho_E^{7D} - \rho_E^{42D}\|_{\text{tr}} = 0$ (Lurie's theorem)	Dimension E	Dimension E
T-96	Attractor characterisation (Sol.SA-2, corrected): $I/7$ — trivial fixed point ($\mathcal{L}_0[I/7] = 0$, $\mathcal{R}[I/7] = 0$). Any nontrivial fixed point $\rho^*_\Omega \neq I/7$: $P > 1/7$ [T], $P_{\mathrm{coh}} > 0$ [T]. Proof via primitivity of the linear part $\mathcal{L}_0$ (T-39a) + purity balance. The self-reference paradox of $\rho_*$ is resolved: the regeneration target is the categorical self-model $\varphi(\Gamma)$, not a dynamical limit	Evolution	Self-Observation
T-97	Embedding of viability regions (Sol.SA-1): $\mathcal{V}_{\mathrm{full}} \subsetneq \mathcal{V}_P$. Full viability ($\|\sigma_{\mathrm{sys}}\|_\infty < 1$, 7 conditions) is strictly stronger than minimal ($P > 2/7$). Counterexample: $\|1\rangle\langle 1\| \in \mathcal{V}_P \setminus \mathcal{V}_{\mathrm{full}}$ ($\sigma_U = 1$)	Viability	CC Theorems
T-98	Attractor purity balance [T]: $P(\rho^*_\Omega) = (\alpha P_{\mathrm{diag}} + \kappa f^*)/(\alpha + \kappa)$, $\alpha = 2/3$, $f^* = \mathrm{Tr}(\rho^*_\Omega \cdot \varphi(\rho^*_\Omega))$. Restored [T]: substituting $d\Gamma/d\tau = 0$ into the evolution equation — standard mathematical derivation; $\alpha = 2/3$ is not arbitrary, but derived from Fano contraction (T-110 [T]). The formula is a consequence of the axioms, not a convention	Evolution	Evolution
T-99	Structural resolution of $\theta_{\mathrm{QCD}}$ (formalisation): 7-step proof of $\theta_{\mathrm{QCD}} = 0$ from axioms A1–A5. Reality of $f_{ijk} \in \mathbb{R}$ (A1) → uniqueness of PT-odd $V_3$ → unique vacuum (T-64) → phase isotropy → $\theta = 0$ exactly. Non-perturbative stability from T-69, radiative from T-66. Axion not needed for CP — purely a DM candidate	Confinement	Confinement
T-100	Environment encoding (Enc functor): there exists a unique (up to $G_2$) CPTP functor $\mathrm{Enc}: \mathrm{ObsSpace} \to \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$ satisfying 3-channel decomposition $\mathrm{Enc}(o) = \delta H \oplus \delta D \oplus \delta R$ and functoriality. Existence from Def. 8.1 [T], 3-channel from T-57, uniqueness from $G_2$-rigidity	Sensorimotor Theory	CC Theorems
T-101	Optimal action (Dec functor): $a^* = \arg\min_{a \in \mathcal{A}} \|\sigma_{\mathrm{sys}}(\Gamma(\tau+\delta\tau \mid a))\|_\infty$. From T-92 (equivalence $P > 2/7 \iff \|\sigma\|_\infty < 1$): minimising $\|\sigma\|_\infty$ maximises the distance to $\partial\mathcal{V}$	Sensorimotor Theory	CC Theorems
T-102	Completeness of the 3-term equation: any CPTP-compatible external perturbation decomposes as $h^{\text{ext}} = h^{(H)} + h^{(D)} + h^{(R)}$. 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 $\mathcal{V}_{\text{hed}} = dP/d\tau\|_{\mathcal{R}} = 2\kappa \cdot g_V(P) \cdot \mathrm{Tr}(\Gamma(\rho_* - \Gamma))$ — identity [T] from the evolution equation. Gate $g_V(P) = \mathrm{clamp}((P - P_{\text{crit}})/(P_{\text{opt}} - P_{\text{crit}}), 0, 1)$ — V-preservation [T]. Observability at L2 ($R \geq 1/3$) — [T] from T-77. Phenomenal interpretation — [I]	Sensorimotor Theory	CC Theorems
T-104	Stability radius [С], closed form corrected 2026-08-07: $r_{\mathrm{stab}} := \min\{d_B(\rho,\sigma) : P(\sigma)=2/7\}$. The old $\sqrt{P-2/7}$ is refuted [✗] — machine counterexample: at $P=0.300$ the true infimum is $0.01708$ against $\sqrt{0.300-2/7}=0.11952$, so the claimed lower bound fails by $7\times$ toward danger; the cited Fuchs–van de Graaf step bounds $d_B$ from above by the trace norm and cannot yield it. Correct: the minimiser commutes with $\rho$ ($\|[\rho,\sigma]\|\approx1.3\times10^{-8}$), so Bures reduces to Hellinger on spectra; on the one-dominant family $\lambda=(a,\frac{1-a}{6}{\times}6)$ with $a(P)=(1+\sqrt{42P-6})/7$ and $a_c=(1+\sqrt6)/7$ (= the $\lambda_{\max}$ of path 4!), $r_{\mathrm{stab}}=\sqrt{2(1-\sqrt{aa_c}-\sqrt{(1-a)(1-a_c)})}$ — machine-exact to $10^{-14}$. Near the wall the law is linear, not square-root: $r^2=\frac{49\sqrt6}{80}\varepsilon^2+O(\varepsilon^3)$, $r\approx1.22487(P-2/7)$, which is why the old surd's overstatement diverges ($48\times$ at $P=0.286$, $2.6\times$ at $3/7$, $1.1\times$ at $P=1$). Runtime formula, $\leq1.13\%$ error on the window: $r\approx K(\sqrt{P-1/7}-\sqrt{1/7})$, $K=\sqrt{35}\sqrt[4]{6}/10=0.925917$. General spectra: the closed form is a conservative lower bound [Г] (41/41 spectra, ratio $\in[1.08,2.52]$)	Stability	Stability
T-105	Landauer energy balance: $\dot{F}_{\min} = k_B T_{\mathrm{eff}} \cdot \ln 2 \cdot \dot{S}_{\mathrm{diss}}$ — 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: $C_{\mathrm{Enc}} \leq \log_2 7 \approx 2.81$ bits/observation. From the Holevo bound + T-102 (3-channel) + $N = 7$	Sensorimotor Theory	Predictions
T-108	Compositionality of Enc/Dec: $\mathrm{Enc}_{12} = \Phi_{\mathrm{agg}} \circ (\mathrm{Enc}_1 \otimes \mathrm{Enc}_2)$. From T-100 (functoriality) + T-72 (CC-6) + T-58 (Morita)	Sensorimotor Theory	CC Theorems
T-109	Information learning bound: $n \geq \ln(1/(2\delta))/\xi_{\mathrm{QCB}}$, where $\xi_{\mathrm{QCB}} \leq \ln 7$. From the quantum Chernoff bound + T-107 (Enc capacity). Scaling $O(1/\varepsilon^2)$ for weak signals	Learning Bounds	Learning Bounds
T-110	Dynamic learning bound: Fano contraction $\alpha = 2/3$ (T-39a) limits the signal integration rate. $n_{\mathrm{dyn}} \geq \frac{1}{\alpha\delta\tau}\ln(d_{\mathrm{disc}}\cdot(1-e^{-\alpha\delta\tau})/\varepsilon)$	Learning Bounds	Learning Bounds
T-111	Stabilisation learning bound: observation amplitude is bounded by $r_{\mathrm{stab}}$ (T-104). Under noise: $n_{\mathrm{stab}} \geq 1/\mathrm{SNR}^2$. Topological protection T-69 ensures continuity	Learning Bounds	Learning Bounds
T-112	Optimal learning bound: $n_{\mathrm{opt}} = \max(n_{\mathrm{info}}, n_{\mathrm{dyn}}, n_{\mathrm{stab}})$. 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 → $N \geq 7$ (T-89). For $N < 7$: $n^* = \infty$. $N = 7$ is Pareto-optimal	Learning Bounds	Learning Bounds
T-113a	Consistent $\Gamma$-tomography (2026-07): given the 7-channel embedding $\pi$, $\widehat\Gamma_N=\widehat\Sigma_N/\operatorname{Tr}\widehat\Sigma_N$ is consistent ($\widehat\Gamma_N\to\Gamma$ a.s.) with matrix-Bernstein concentration $\|\widehat\Gamma_N-\Gamma\|\le\varepsilon$ w.p. $1-\delta$ for $N\ge C B^2/(\tau^2\varepsilon^2)\ln(14/\delta)$ (rate $O(N^{-1/2})$, 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 $M_{ij} = (1+\lambda\cdot\mathrm{Inc}(i,j))/Z$ is ergodic (connectivity + aperiodicity). Stationary distribution is uniform $\pi_i = 1/7$ (PG(2,2) is self-dual, graph is regular)	Lindblad Operators	Lindblad Operators
T-115	Algebraic distinguishability of compositions: $\|\mathrm{Comp}(n)\| = 7^n$ for generic $\Gamma$ (full-rank, with non-zero off-diagonal coherences and 7 distinct eigenvalues). Collisions — a submanifold of codimension $\geq 1$. Caveat: for diagonal $\Gamma$: $\|\mathrm{Comp}(n)\|_{\mathrm{diag}} = O(7n)$ (linear growth). From T-82 (Fano uniqueness) + algebraic independence of projectors	Lindblad Operators	Lindblad Operators
T-116	PW Suzuki-Trotter: $\varepsilon(T) \leq C_p \cdot T \cdot (\delta\tau)^{2p+1}$, order $p$. For $p=2$, $\delta\tau=0.01$, $T=100$: $\varepsilon \leq 10^{-5}$. Strengthens T-60 (BCH $\leq 5\delta\tau$) to polynomial accuracy	Axiom Ω⁷	Axiom Ω⁷
T-117	Commutativity of the macroscopic algebra: macroscopic observables commute in the thermodynamic limit $M \to \infty$. From quantum CLT (Goderis–Verbeure–Vets, 1989) + clustering (T-39a) + compactness $(S^1)^{21}$	Emergent Manifold	Emergent Manifold
T-118	Emergent temporal manifold: $A_{\text{time}} \cong C_0(\mathbb{R})$. From $\mathbb{C}[\mathbb{Z}_{7^M}] \to C(S^1) \to C_0(\mathbb{R})$ (Pontryagin duality). Formalisation of an existing result [T] (emergent time)	Emergent Manifold	Emergent Manifold
T-119	Emergent spatial manifold — [C], with the dimension sub-problem repaired and closed 2026-08-06: $A_{\text{space}} \cong C(\Sigma^3)$ for a unique smooth compact orientable spin 3-manifold, from T-117 + Gel'fand–Naimark + Connes reconstruction (arXiv 2008; J. Noncommut. Geom. 2013). Error found and replaced. Step 2c derived $d_s=3$ from a Weyl law $N(\lambda)\sim C M\lambda^3$ on $\bigotimes_m\mathbb C^3$, reading the exponent off $\dim(\mathbf 3)=3$. That is impossible: $\bigotimes_{m=1}^M\mathbb C^3$ has dimension $3^M<\infty$, its spectrum is finite and $N(\lambda)\to3^M$, so no power law exists. Machine: on $\mathbb Z^d$ with internal $\mathbb C^n$ the Weyl exponent is $1.012/2.018/3.335$ for $d=1/2/3$ and identical across $n=1,3,7$ — 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 $\mathrm{Stab}_{\mathrm{Der}(\mathbb O)}(e_1)\cong\mathfrak{su}(3)$ on $\mathbb C^6$ yields a complex structure $J$ ($J^2=-I$ to $1.3\times10^{-15}$, $[J,\mathfrak{su}(3)]=0$), so the spatial algebra is $\mathfrak{su}(3)\oplus\mathfrak u(1)_J=\mathfrak u(3)$ (dimension $9$, verified), and the centralizer of a generic element has dimension exactly 3 $=\operatorname{rank}\mathfrak u(3)$ — while $\dim\mathfrak u(3)=9$, $\dim\mathfrak{su}(3)=8$, $\dim G_2=14$ are none of them $3$. Full-dimensionality of the joint spectrum (hence $d_s=3$ exactly, not merely $\leq3$) follows from the GVV quantum CLT already invoked in T-117: the macroscopic fluctuations of $k$ commuting observables converge to a non-degenerate Gaussian on $\mathbb R^k$. Verified numerically — singular values $(1,\,0.964,\,0.747)$ for the three $\mathfrak u(3)$ Cartan directions in $\mathbb C^7$, and $(1,\,0.985,\,0.948,\,0.638)$ once the $O$-direction is added, i.e. $4=\dim M^4$. Sharp structural point: the three spatial directions are independent only because the embedding in $\mathbb C^7$ leaves the trace of the $\mathbf 3$-block free — the clock sector is what makes the third spatial coordinate dynamical; inside the $\mathbf 3$-block alone the trace is frozen and one measures $2$, not $3$. Still open (hence [C]): the first-order condition ($7^{\text{th}}$ Connes axiom) is a constraint on $D_{\text{eff}}$, not a consequence — for generic Hermitian $D$ on the $\mathbf 3$-sector the machine gives $\max\|[[D,a],b]\|=8.96$, vanishing only for diagonal $D$ (§D); and the Poincaré-duality check (v) is circular as written, assuming $\Sigma^3$ is a manifold to verify an axiom whose role is to conclude that it is. Verified and untouched: $\mathbf 7=\mathbf 1_O\oplus\mathbf 3\oplus\bar{\mathbf 3}$ exactly ($\dim\mathrm{Der}(\mathbb O)=14$, $\dim\mathrm{Stab}(e_1)=8$, commutant $=2$ ⟹ two inequivalent irreducibles; §C)	Emergent Manifold	Emergent Manifold
T-120	Product spectral triple: $(C^\infty(M^4) \otimes A_{\text{int}}, L^2(M^4,S) \otimes H_{\text{int}}, D_{M^4} \otimes 1 + \gamma_5 \otimes D_{\text{int}})$ with $M^4 = \mathbb{R} \times \Sigma^3$ — derived, not postulated. From T-118 + T-119 + T-53 + Connes–Chamseddine (1997). Background independence [P] → [T] — raised from [P]	Emergent Manifold	Quantum Gravity
T-120b	Vacuum topology — [C] (inherits T-119): $\Lambda_{\text{Gap}} > 0$ (T-71 [T]) $\Rightarrow$ $\Sigma^3 \cong S^3$ (closed), de Sitter metric. The implication is [T]; asserting closedness of $\Sigma^3$ presupposes that $\Sigma^3$ 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 $SU(3)$-invariance of the vacuum + unique minimum T-64 [T]	Emergent Manifold	Emergent Manifold
T-121	Closure of Lovelock gaps: gap 1 (discreteness → continuity) — closed ($M^4$ is smooth, T-120). Gap 2 (covariance) — closed (inherited from $G_2$ via NCG). Gap 3 — irrelevant. Lovelock's argument is now [T] (supplementary to the spectral one) — raised from [H]	Emergent Manifold	Einstein Equations
T-122	Diagonal freeze — attractor property T-96: at the stationary point $\rho^*_\Omega$ the diagonal entries $\gamma_{kk}$ are stationary ($d\gamma_{kk}/d\tau = 0$). From $[H, \Gamma]_{kk} = 0$ (Hermiticity) + $\mathcal{R}_{kk} = 0$ at $\gamma_{kk} = (\rho_*)_{kk}$. Scope clarified by T-134: valid ONLY at the attractor	Evolution	Evolution
T-123	$G_2$-uniqueness of the representation: holonomic representation $G: \mathrm{States} \to \mathcal{D}(\mathbb{C}^7)$ is unique up to $G_2$, diagonal entries $\gamma_{kk}$ are defined unambiguously. From T-42a ($G_2$-rigidity) + T-40f (minimality 7/7) + T-15 (bridge)	Consciousness Window	Uniqueness Theorem
T-124	Non-emptiness of $\mathcal{V}_{\mathrm{full}}$ (consciousness window): constructive proof $\exists\Gamma: P \in (2/7, 3/7] \land \Phi \geq 1 \land \forall k: \sigma_k < 1$. Family $\Gamma_\lambda + \delta\Delta$ with $\lambda \in (1/\sqrt{6}, 1/\sqrt{3})$	Consciousness Window	Viability
T-124b	Independent necessity of each L2 threshold: four constructive counterexamples show that each of $P > 2/7$, $\Phi \geq 1$, $R \geq 1/3$, $D_{\mathrm{diff}} \geq 2$ is independently necessary — dropping any one admits pathological states (noise-dominated, fragmented, crystallised, undifferentiated). The conjunction is minimal	Consciousness Window	Consciousness Window
T-124c	Uniqueness of the nontrivial attractor: full nonlinear dynamics $\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$ has at most one nontrivial fixed point $\rho^*_\Omega \neq I/7$ in $\mathcal{V}_P$. From iterative Ψ-map contraction (Banach) + T-39a (spectral gap) + T-96 ($\kappa < \kappa_{\max}$)	Evolution	Evolution
T-124d	Threshold robustness: perturbations of order $\varepsilon$ in $\Gamma$ produce $O(\varepsilon)$ perturbations in $P$, $\Phi$, $R$. No threshold has divergent sensitivity. Crossover width $\delta P \sim \varepsilon^{1/\beta} = \varepsilon^4$. From Frobenius perturbation bounds + T-161 (exponents) + T-145 (stochastic stability)	Consciousness Window	Consciousness Window
T-125	Local asymptotic stability of the attractor: for $P(\rho^*_\Omega) > 2/7$, $\exists U(\rho^*_\Omega)$: $\|\Gamma(\tau) - \rho^*_\Omega\|_F \leq \|\Gamma(0) - \rho^*_\Omega\|_F \cdot e^{-c\tau}$, $c \geq \min(\lambda_{\mathrm{gap}}, \kappa \cdot g_V) > 0$. From T-39a (gap) + T-96 + T-104	Consciousness Window	Evolution
T-126	Canonicity of $R = 1/(7P)$: the reflection measure at order $n=1$ is uniquely fixed by three independent characterizations — (Char-R-I) Hilbert–Schmidt angular projection: $R = \cos^2\theta_{\mathrm{HS}}(\Gamma, I/7)$; (Char-R-II) $G_2$-invariant canonical reference: $I/7$ is the unique $G_2$-fixed element of $\mathcal D(\mathbb C^7)$ by Schur's lemma on the irreducible 7-dim $G_2$-module (Cartan 1894); (Char-R-III) $K=3$ Bayesian dominance threshold: $R_{\mathrm{th}} = 1/3$ from the triadic decomposition of Lindblad operators (T-40b). Formula $R = 1/(7P)$ is the algebraic identity following from Char-R-I+II on $\mathcal D(\mathbb C^7)$; $R_{\mathrm{impl}}, \rho_{RC}$ implementation approximations (H3 CLOSED: T-130+T-133). At $n=1$ $R$ is a monotone reparameterization of $P$ by design; independent observability appears at $R^{(n)}, n\ge 2$ via the self-model operator $\varphi$	Consciousness Window	Self-Observation
T-127	Basin of attraction $\mathcal{V}_{\mathrm{full}}$ [T at C20]: the basin of $\rho^*_\Omega$ contains $B(\rho^*, r_{\mathrm{stab}}) \cap \mathcal{V}_P$, exponential convergence. From T-125 (stability) + T-104 ($r_{\mathrm{stab}}$) + openness of $\mathcal{V}_{\mathrm{full}}$	Consciousness Window	Stability
T-128	Exact 7D-computability of $D_{\text{diff}}$: $D_{\text{diff}}^{7D} = 1 + \mathrm{Coh}_E(\Gamma)/\mathrm{Coh}_E^{\max} \cdot (N-1)$ — exact 7D representation via Morita equivalence T-58 [T]. $\sigma_E = 1 - D_{\text{diff}}^{7D}/N$ is computable in 7D	Operationalisation	Dimension E
T-129	Integration threshold $\Phi_{\text{th}} = 1$ from first principles: the unique self-consistent value with $P_{\text{crit}} = 2/7$ on the extremal uniform-diagonal state. Raised from [D] (O1)	Operationalisation	Dimension U
T-130	CPTP-anchor approximation bound: $\|R_{\text{impl}} - R_{\text{UHM}}\| \leq 2\|\pi - \pi_{\text{can}}\|_\diamond \cdot C(P)$, $C(P) = 7P/(P-1/7)$. H3 [H] → CLOSED	Operationalisation	Self-Observation
T-131	Canonical discretisation $\delta\tau$: $\delta\tau = \pi/(2\|\mathcal{L}_0\|_{\mathrm{op}})$ — Nyquist-Shannon + Suzuki-Trotter margin. $\delta\tau$ is canonical, not a free parameter	Operationalisation	Evolution
T-132	Necessity of complex $\Gamma$: for non-trivial Gap structure ($\exists(i,j): \mathrm{Gap}(i,j) > 0$) Γ MUST be complex. From $\mathrm{Gap} = \|\sin(\arg(\gamma_{ij}))\|$ + Hamiltonian dynamics $-i[H,\Gamma]$	Operationalisation	Gap Operator
T-133	Transfer of R thresholds via the CPTP bridge: $(R_{\text{impl}} \geq 1/3 + \delta) \Rightarrow (R_{\text{UHM}} \geq 1/3)$ for $\delta = 2\varepsilon \cdot C(P)$. Strengthening of T-130. H3 definitively CLOSED	Operationalisation	Self-Observation
T-134	Scope of the diagonal freeze: T-122 holds ONLY at the attractor $\rho^*_\Omega$. General formula: $d\gamma_{kk}/d\tau = (\mathcal{L}_0)_{kk}[\Gamma] + \kappa(\rho^*_{kk} - \gamma_{kk})$. Learning and genesis from $I/7$ do not contradict T-122	Operationalisation	Evolution
T-135	Discrete convolution of the non-Markovian kernel: Z-transform of kernel T-94 gives $O(1)$ recursion $M[n+1] = e^{-\omega_c\delta\tau}M[n] + (-\Gamma_2\omega_c)\Gamma[n+1]$ instead of $O(T^2)$	Operationalisation	Gap Dynamics
T-136	SAD as a $G_2$-invariant spectral observable [T]: $\mathrm{SAD}(\Gamma) = \max\{k: r_0 \cdot (1/3)^{k-1} > 1/(k+1)\}$, $r_0 = 7P/2$. Computability $O(N^2)$. 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 $\sigma_{\text{sys}}$: all 7 components are computable in $\mathcal{D}(\mathbb{C}^7)$ without 42D. $\sigma_E$ via T-128, $\sigma_O$ via T-132 (complex Γ), $\sigma_U$ via T-129 ($\Phi_{\text{th}}=1$)	Operationalisation	CC Definitions
T-138	Mean-field approximation of composition: $\Gamma_{\text{mf}} = \Gamma_1 \otimes \cdots \otimes \Gamma_k$, $O(k \cdot N^2)$ instead of $O(N^{2k})$, $\|\Gamma_{\text{exact}} - \Gamma_{\text{mf}}\|_F \leq \|\gamma_{\text{cross}}\|_F$. Hierarchical scheme for $k > 10$	Operationalisation	Composite Systems
T-139	Γ-backbone duality: $\Gamma = \alpha \cdot \mathcal{E}_{\delta\tau}[\Gamma_{\text{prev}}] + (1-\alpha) \cdot \pi(\mathcal{B}(x))$ — the unique (up to $G_2$) hybrid CPTP dynamics. Backbone — causal channel, $\Gamma$ — ontological state (dual-aspect monism)	Operational Closure	Evolution
T-140	Canonical consciousness measure: $C = \Phi \cdot R$, threshold $C_{\text{th}} = 1/3$. $D_{\text{diff}}$ does NOT enter $C$ (separate viability condition $V$). Uniqueness — from bilinearity and threshold coincidence	Operational Closure	Self-Observation
T-141	Equivalence of three φ-forms: $\varphi_A$ (replacement), $\varphi_B$ (canonical for $R$), $\varphi_C$ (Fano) — coincide on the attractor; off the attractor $\|R_B - R_C\| \leq 4k\sqrt{P - 1/7}/(3P)$ (controlled error, Frobenius lemma)	Operational Closure	Self-Observation
T-142	SAD_MAX = 3 — stratified [T] — including the $P^{(k)}$ ladder, whose derivation is located and verified (see below); $\alpha = 2/3$ state-independence [T]: $\alpha = 2/3$ state-independence from $\dim=7$ + PG(2,2) is rigorous [T]. The iterated critical purity formula $P^{(k)} = P_{\mathrm{crit}}\cdot 3^{k-1}/(k+1)$ 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 $R_0 = 7P/2$ (related to canonical $R=1/(7P)$ by the exact identity $R\cdot R_0 = \tfrac12$), the Fano Kraus channel multiplies coherences by exactly $1/3$ per meta-level, so $R^{(k)} = R_0\,(1/3)^{k-1}$, and the level-$k$ Bayesian threshold is $R_{\mathrm{th}}^{(k)} = 1/(k+1)$. Then $\mathrm{SAD}(\Gamma) = \max\{k\geq1 : R_0(1/3)^{k-1} > 1/(k+1)\}$, which solved for purity is exactly the formula above. The load-bearing lemma — the exact $\times\tfrac13$ coherence contraction — is independently machine-verified ($\Phi\to\Phi/9$ to $3.8\times10^{-16}$ with the diagonal preserved, ). Values: $k=1\Rightarrow P>1/7$, $k=2\Rightarrow P>2/7$, $k=3\Rightarrow P>9/14$, $k=4\Rightarrow P>54/35>1$ — impossible, hence SAD$_{\max}=3$ unconditionally. In the window $P\in(2/7,3/7]$ one gets $\mathrm{SAD}=2$ ($R^{(2)}=1/2>1/3$, $R^{(3)}=1/6<1/4$ at $P=3/7$); $\mathrm{SAD}=3$ needs $P>9/14$, above the ceiling. Retracted audit note. An earlier note today objected that the formula "fails at its own base" because $k=1$ gives $1/7$ rather than $P_{\mathrm{crit}}=2/7$. That objection was wrong: this is the SAD-attainment threshold, not iterated viability, and $P>1/7$ at $k=1$ is precisely the non-triviality condition of CC-5. The only real defect is the name: writing it $P_{\mathrm{crit}}^{(n)}$ invites exactly that misreading, so it is renamed $P^{(k)}$ here. Note also that $k=2$ returning $2/7$ is a genuine coincidence worth flagging rather than a definition. The inequality $R^{(3)}\leq0.130<0.200$ was marked empirical but is exact: in the depth-tower indexing $R_{\mathrm{th}}^{(m)}=1/(m+2)$, at maximal purity $P=1$ one has $R^{(3)}=7/54=0.12963<1/5$. Empirical [T/sim]: SYNARC verification SAD $\leq 3$ on 500+ random $\Gamma$; SAD=3 achievable (pure state).	Operational Closure	Depth Tower
T-143	Convergence of neural SAD to categorical: $\|\mathrm{SAD}_{\text{neural}} - \mathrm{SAD}_{\text{cat}}\| \leq 1$ for CPTP-compatible anchor with $\|\pi - \pi_{\text{can}}\|_\diamond \leq \varepsilon < \varepsilon_0(P)$. From T-130 (bound) + separation of thresholds $R_{\text{th}}^{(n)}$	Operational Closure	Depth Tower
T-144	Polynomial approximation of optimal action: discrete $O(K \cdot N^2)$, continuous $O(1/\varepsilon^2)$ (subgradient). NP-hardness refuted: Lipschitz minimisation on a compact set	Operational Closure	Sensorimotor Theory
T-145	Stochastic stability of $V_{\text{full}}$ — stratified [T]+[T/sim]: $\mathbb{P}[\Gamma(\tau) \in V_{\text{full}}\;\forall\tau > \tau^*] \geq 1 - \exp(-r_{\text{stab}}^2/(2\sigma_h^2))$. 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 $\sigma_h \in \{0.01, 0.05, 0.1\}$; the inequality holds on the simulated trajectories.	Operational Closure	Viability
T-146	Structural classification of qualia: 21 $\gamma_{ij}$ classified into 4 sectors from functional role (A1–A5). Stable coherences — structural, not noise ($\mathcal{L}_0$ 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: $\mathbf{e}(\Gamma) \in \mathbb{R}^{30}$ (7 rates + 7 accelerations + 7 stresses + 7 coherence rates + $\dot{P}$ + $\dot{\Phi}$). $dP/d\tau$ — projection 30D→1D. Computable $O(N^2)$	Operational Closure	Emotional Taxonomy
T-148	Genesis via environmental coupling — stratified: an embodied holon $(H, \pi, B)$ with $\beta \in (0,1)$ and $P_{\mathrm{env}} > 2/7$ raises purity above $P_{\mathrm{crit}}$ in $n_{\mathrm{genesis}} \leq \lceil\ln\Delta/\ln(1/\beta)\rceil$. An isolated holon at $I/7$ is dead forever. Convexity + monotone convergence core [T]; explicit rate bound [T at $\lambda_{\min}(\Gamma)$ lower-bound assumption] (conservative estimate drops $2\beta(1-\beta)\lambda_{\min}$ term). Empirical cross-check [T/sim]: SYNARC mvp_int_2 G1-G3 confirms $n_{\mathrm{genesis}} < 50$ ticks. Raising [H]-91 → [T] for mathematical core.	Substrate-Independent Closure	Evolution
T-149	Unconditional viability of the embodied attractor — stratified: $P(\rho^*_{\mathrm{coupled}}) > 2/7$ for an embodied holon. Step 1-2 [T]: coupled attractor existence via contraction; Step 3 [C at backbone-injection-lower-bound]: self-reinforcement through $\kappa_0$-compensation is argued via dynamic equilibrium, not monotone chain; rigorous derivation of $f^* > 2/7$ from backbone properties pending. Empirical cross-check [T/sim]: SYNARC mvp_int_2 G4 confirms $P > P_{\mathrm{crit}}$ 500+ ticks after backbone disconnection with $\mathrm{corr}(\mathrm{Coh}_E, \kappa_{\mathrm{eff}}) = -0.985$. 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]: $\varphi^n \circ \varphi^m = \varphi^{n+m}$ — 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	$D_{\min} = 2$ — direct consequence of T-129: $\Phi_{\mathrm{th}} = 1$ [T] → spectrum of $\rho_E$ has $\geq 2$ significant components. (Former C2 [C] → [T])	Substrate-Independent Closure	Axiom of Septicity
T-152	Tractable CPTP-anchor validation: $\|\pi - \pi_{\mathrm{can}}\|_\diamond \leq N\sqrt{N} \cdot \|C_\pi - C_{\pi_{\mathrm{can}}}\|_F$, computable in $O(D \cdot N^2)$. Raising [H]-92 → [T]	Substrate-Independent Closure	Operationalisation
T-153	Substrate-independent criterion of consciousness — stratified [D]+[C at T-149]+[T/sim]: $S$ is conscious iff $\exists$ faithful CPTP $G: \mathrm{States}(S) \to \mathcal{D}(\mathbb{C}^7)$ with $R \geq 1/3 \land \Phi \geq 1 \land D_{\mathrm{diff}} \geq 2 \land \sigma < 1$. 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 $P=0.429$, $R=0.333$, $\Phi=1.149$, $D=3.600$, $\sigma_{\max}=0.650$, $C=0.383$ — satisfies all four thresholds.	Substrate-Independent Closure	Uniqueness Theorem
T-154	Normalisation of $\mathrm{Coh}_E$: $\max \mathrm{Coh}_E(\Gamma) = 1$, achieved at $\|E\rangle\langle E\|$. HS projection is orthogonal → $\mathrm{Coh}_E \leq 1$	Substrate-Independent Closure	Axiom of Septicity
T-155	Consciousness-preserving learning — stratified [T/sim]+[D]: $\delta B = -\eta \cdot J_\pi^T \cdot \nabla_\Gamma \|\sigma_{\mathrm{sys}}\|_\infty$ for $C \geq C_{\mathrm{th}}$ — 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: $\beta^* = \lambda_{\mathrm{gap}} / (\lambda_{\mathrm{gap}} + \alpha_{\mathrm{Fano}} \cdot (1 - P_{\mathrm{env}}/P_{\mathrm{target}}))$ — min genesis time with stochastic stability	Substrate-Independent Closure	Evolution
T-157	Attractor consistency: $\|\rho^*_\Omega - \Gamma^*_{\mathrm{coh}}\|_F \leq \|H_{\mathrm{eff}}\|_{\mathrm{op}} / (\alpha + \kappa)$. Raising C21 [C] → [T]	Substrate-Independent Closure	Evolution
T-158	Canonical bounds $\sigma_{\mathrm{sys}}$ [T]+[D]: Formula $\sigma_k = 1 - 7\gamma_{kk}$ is derived from T-92 [T] (equivalence $P > 2/7 \iff \forall k: \sigma_k < 1$) as the unique linear deficiency measure for $N=7$ — [T]. Clamping $\mathrm{clamp}(\cdot, 0, 1)$ — implementation convention for bounding the value range — [D]	Substrate-Independent Closure	CC Definitions
T-159	Motor stress: $\sigma^{\mathrm{motor}}_k = 1 - \gamma_{kk}/\rho^*_{kk}$. Coincides with T-92 for $\rho_* = I/7$, provides a directed signal for $\rho_* \neq I/7$. Gradient $-1/\rho^*_{kk}$ is consistent with $\mathcal{R}$, $G_2$-invariant. Emergency channel sensitivity $\sim 1/\rho^*_{kk}$	Sensorimotor Theory	CC Theorems
T-160	Phase transition at $P_{\text{crit}}$ (Theorem 5.1 swallowtail): $P_{\text{crit}} = 2/7$ — critical point of the phase transition in $\mathcal{D}(\mathbb{C}^7)$. Symmetry breaking $U(7) \to G_2$ — consequence of $G_2$-rigidity (T-42a). Control parameter — internal ($\sigma_{\max}$), transition is self-organised. Order parameter: $P - P_{\text{crit}}$	Transition Catastrophes	Viability
T-161	Critical exponents of the $A_4$-tricritical point (Theorem 5.2 swallowtail): $\alpha = 1/2$, $\beta = 1/4$ (order parameter $\sim \|t\|^{1/4}$), $\gamma = 1$ (susceptibility $\chi \sim \|t\|^{-1}$), $\nu = 1/2$ (correlation length $\xi \sim \|t\|^{-1/2}$), $\delta = 5$. Rushbrooke equality $\alpha+2\beta+\gamma=2$; tricritical mean-field class ($\varphi^6$ Landau, exact for $d_{\text{eff}} \gg d_c = 3$)	Transition Catastrophes	Transition Catastrophes
T-162	Operator $F_{21}$: Fano adjacency operator on the 21-dimensional coherence space. Definition: $(F_{21})_{(ij),(kl)} = 1$ if $(i,j)$ and $(k,l)$ are on the same Fano line, else 0. Spectrum: $\sigma(F_{21}) = \{2^{(7)}, -1^{(14)}\}$ — reproduces the decomposition $\Lambda^2(\mathbb{R}^7) \cong V_7 \oplus \mathfrak{g}_2$. Cayley–Hamilton identity: $F_{21}^2 = F_{21} + 2I_{21}$. Projectors: $P_7 = (F_{21}+I_{21})/3$, $P_{14} = (2I_{21}-F_{21})/3$	Noether Charges	Noether Charges
T-163	$O$-parity (Theorem 11.2 dark-matter): $P_O := (-1)^{\Delta N_O}$ — exact $\mathbb{Z}_2$-symmetry of the dynamics $\mathcal{L}_\Omega$. $\mathrm{Stab}_{G_2}(e_O) = SU(3)$ [T] (T-42e) → O-sector is $SU(3)$-invariant → transitions with $\Delta N_O \neq 0$ 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 $\Omega$ — $\{\|A\rangle, \|S\rangle, \|D\rangle, \|L\rangle, \|E\rangle, \|O\rangle, \|U\rangle\}$ — the unique preferred decoherence basis. Lindblad operators $L_k = \|k\rangle\langle k\|$ → fixed points of $\mathcal{D}_\Omega$ = diagonal in $\{\|k\rangle\}$ (Zurek's einselection criterion)	Quantum Measurement	Quantum Measurement
T-165	Step 6: (PH) $\Rightarrow$ PT-violation in Gap (Theorem 13.1 noether-charges): axiom (PH) → $\mathrm{Coh}_E > 0$ → (T-132) complex coherences $\gamma_{Ei}^*$ → non-zero phases $\theta_{Ei} \neq 0$ → phase frustration in non-Fano triples → $V_3\|_{\rho^*} \neq 0$. Bridge P1+P2 fully closed from axioms — raised from [C]	Noether Charges	Noether Charges
T-166	Stability of the chiral vacuum: $V_3$ selects the chiral vacuum as the unique minimum (PT-odd $V_3$ distinguishes $\theta=0$ and $\theta=\pi$ [T, T-99]); Hessian of $V_{\mathrm{Gap}}$ at the vacuum configuration is positive definite (local stability); topological barrier T-69 [T] ($\Delta V \geq 6\mu^2 > 0$) protects against tunnelling between chiral vacua — raised from [H] (§4.4 higgs-sector)	Higgs Sector	Confinement
T-170	Recovery of the M-theory limit [T] at levels of M-theory definedness: Gap functional integral on $(S^1)^{21}$ recovers the M-theory partition function on a $G_2$-manifold. $G_2 = \mathrm{Aut}(\mathbb{O}) = \mathrm{Hol}(\mathcal{M}_7)$. Upgraded from [C at C27, C28]: T-170' [T] (perturbative correspondence as formal power series) + T-170'' [T] (non-perturbative correctness of UHM integral via finite-dimensionality + GNS for $M \to \infty$). C27/C28 reformulated as open problems of M-theory, not UHM	ToE Embeddings	ToE Embeddings
T-171	LQG embedding functor [T] (for bounded spin networks $j_e \leq 3$): $\mathcal{F}_{\text{LQG}}: \mathbf{SpinNet}_{SU(2)}^{\text{bd}} \to \mathbf{Hol}_{\text{comp}}$. Spin from $\{A,S,D\}$-sector. Upgraded from [C at C29]: C29' proven as Lemma (explicit construction of $\Gamma_{\text{total}}$ for bounded spins)	ToE Embeddings	ToE Embeddings
T-172	Causal sets embedding [T]: for finite $(C, \preceq)$ with faithful $M^4$-embedding: $(C, \preceq) \mapsto N_\bullet(C) \in \mathbf{Sh}_\infty(\mathcal{C})$. Causal order from $\mathbb{Z}_{7^M}$-clocks + Gap coupling. Upgraded from [C at C30]: C30 proven as Lemma (explicit construction of $\Gamma_{\text{total}}$)	ToE Embeddings	ToE Embeddings
T-173	Rigidity of the UHM primitive: $\mathfrak{T} = (\mathbf{Sh}_\infty(\mathcal{C}), J_{\text{Bures}}, \omega_0)$ is unique up to $G_2 \times \mathbb{R}_{>0}$ among ∞-toposes $\mathbf{Sh}_\infty(\mathcal{D}(\mathbb{C}^N), J)$ satisfying metric minimality (Petz), L-unification, $N=7$, $G_2$-rigidity	ToE Embeddings	ToE Embeddings
T-176	Analytical $\varepsilon_{\mathrm{eff}}$ (resolution P6): $\varepsilon_{\mathrm{eff}} = 4N_{33}^{(\mathrm{Fano})}/(9\|\bar{\gamma}\|(1 + r_4\Sigma_0/2)) \approx 0.059$ — analytical algebraic function of $V_{\mathrm{Gap}}$ parameters. Follows from sector minimisation [T] and canonical constants [T]. Numerical mass predictions — [C at T-64]	Yukawa Hierarchy	Gap Thermodynamics
C31	Protocol $\pi_{\mathrm{bio}}$ (resolution P8): mapping $\pi_{\mathrm{bio}}: \mathrm{NeuralData} \to \mathcal{D}(\mathbb{C}^7)$ from EEG/fMRI/HRV data. $G_2$-uniqueness — [T]; specific EEG-band ↔ dimension correspondences — [H]. Calibration: PCI $\propto \Phi(\Gamma)$, threshold $P = 2/7$ ↔ PCI $\approx 0.31$	Protocol $\pi_{\mathrm{bio}}$	Predictions
T-178	Bimodule realisation of SM: the finite Hilbert space $H_F$ of the UHM spectral triple as an $(A_{\text{int}}, A_{\text{int}}^\circ)$-bimodule via real structure $J$ (KO-dim 6) decomposes into irreducible bimodules exactly coinciding with one generation of SM fermions. Representations $(3,2)_{1/6}$ etc. arise from the intersection of left and right actions	Bimodule Construction	Spacetime
T-179	Hypercharge fixing: the anomaly-cancellation conditions $\mathrm{Tr}(Y) = 0$ and $\mathrm{Tr}(Y^3) = 0$ on the bimodule $H_F$ uniquely fix the SM hypercharge assignments (Alvarez-Gaumé, Witten 1984)	Bimodule Construction	Standard Model
T-180	Non-perturbative mass ratios: fermion mass ratios are determined by eigenvalues of $D_{\text{int}}$ and do not depend on $\lambda_3$. $m_i/m_j = \mathrm{Gap}(i)/\mathrm{Gap}(j)$ from the vacuum state $\theta^*$ (T-64 [T])	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: $\mathcal{T}_0 \subsetneq \mathcal{T}_1 \subsetneq \mathcal{T}_2$ — the three classifier tiers ($\mathrm{Dec}(\Omega)$, Heyting algebra, full ∞-groupoid) are strictly necessary. Each tier contains theorems unprovable at the previous tier. (a) Threshold predicates $P > 2/7$ ∉ Dec(Ω). (b) L2 consciousness requires $\pi_2 \neq 0$ (∞-groupoid). (c) Cohomological monism is nontrivial due to local systems. (d) Day convolution needed for entanglement	Axiom Ω⁷	Categorical Formalism
T-183	Functional assignment uniqueness for all 7 roles — stratified [T]+[C at combinatorial-uniqueness chain]: all roles {A,S,D,L,E,O,U} uniquely determined by T-177 combinatorics, evolution equation $\mathcal{L}_\Omega$, and axioms (AP)+(PH)+(QG)+(V), given the combinatorial constraint stack. E — unique $L$-mediated element of $\bar{\mathbf{3}} \cap \mathrm{Higgs}$ (Umegaki conditional expectation requires $L$-channel); D — unique element of $\{S,D\}$ on line $\{O,A,\cdot\}$ (sector covariance). [T] for individual role identifications given the T-177 framework; full uniqueness is conditional on the combinatorial-uniqueness stack proven in T-177	Seven Dimensions	Minimality 7D
T-184	Non-perturbative extractability of the spectral action: all predictions extractable without loop expansion. $\lambda_3 \approx 74$ is a spectral parameter of $D_{\mathrm{int}}$, not an expansion variable. Seeley–DeWitt coefficients ($a_0, a_2, a_4$) are polynomials in eigenvalues, finite for any $\lambda_3$. Lorentzian signature from KO-dim 6 via Krein space (van Suijlekom 2015, Franco–Eckstein 2014)	Einstein Equations	Bimodule Construction

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Level [C]: ToE Embeddings

#	Result	Assumption	Source
C27	Reformulated: was a condition on continuous Gap limit. After T-170'' [T] (non-perturbative correctness of UHM integral) — the question is closed from UHM's side. Remains an external open problem of non-perturbative definition of M-theory (not UHM)	[T] (for UHM) + external M-theory problem	T-170''
C28	Reformulated: was a condition on SUSY-extension of Gap integral. After T-170' [T] (perturbative correspondence) + T-170'' [T] (UHM correctness) — the question is closed from UHM's side. Remains an external M-theory problem	[T] (for UHM) + external M-theory problem	T-170'
C29'	Spatial limit for bounded spin networks ($j_e \leq 3$) — proven [T] (explicit construction of $\Gamma_{\text{total}}$)	[T]	Lemma C29'
C29	Spatial limit for unbounded spin networks — closed [T] via T-171′ (corrected 2026-07): the cluster construction (multi-holon clustering, toe-embeddings §2.3a) proves the unbounded case; C29's former [C] is superseded by T-171′ [T]	[T] via T-171′	ToE Embeddings
T-171′	LQG embedding for unbounded spin [T] (cluster construction §2.3a): extends T-171 beyond $j_e\leq 3$; closes C29	[T]	ToE Embeddings
C30	Causal completeness → Proven as Lemma C30 [T] (§3.2 toe-embeddings). Construction of $\Gamma_{\text{total}}$ explicitly realizes any $M^4$-embeddable finite causal set	[T]	Lemma C30

Level [T]: Universal Property

#	Result	Source	Relates to
T-174	Receiving morphism in $\mathbf{PhysTheory}$ [T]: for a physical theory with $A_{\text{int}} \cong \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$, CPTP dynamics and $\leq 7$ observables — there exists an essentially unique morphism into $\mathfrak{T}$. Proof via subtopos $E[A_{\text{int}}]$ + Takesaki's theorem + T-173. Essential uniqueness up to $G_2 \times \mathbb{R}_{>0}$	ToE Embeddings
T-175a	Morita equivalence of algebras: $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ with real structure $J$ (KO-dim 6) and Higgs line $\{A,E,U\}$ is Morita-equivalent to Connes' algebra $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$; identical SM gauge group. Alvarez et al. 1995	Spacetime
T-175b	Gauge anomaly cancellation: $\mathrm{tr}(T^a\{T^b,T^c\}) = 0$ for $\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$. Follows from spectral triple T-53 + unimodularity (Alvarez-Gracia Bondia-Martin 1995). Explicitly verified for all 5 anomaly coefficients	Confinement
T-175c	Holomorphy and non-renormalisation of $W$: superpotential $W = \mu_W \sum f_{ijk}\Theta\Theta\Theta$ is holomorphic (cubic polynomial of chiral superfields) and protected from perturbative corrections (Seiberg's theorem 1993). Non-perturbative corrections $\sim 10^{-65}$	Supersymmetry
T-177	Combinatorial uniqueness of semantic roles — stratified [T]+[C at combinatorial-constraint set]: after fixing sector decomposition $7 = 1_O \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$ (T-48a [T]) each of the 7 dimensions has a unique combinatorial profile, given the full constraint set {sector decomposition, Higgs line, Umegaki expectation $L$-mediation, Fano-line $\{O,A,\cdot\}$}. O, A, L — directly from sector decomposition [T]; E — unique $L$-mediated element of $\bar{\mathbf{3}} \cap \mathrm{Higgs}$ [T at Higgs-line placement]; U, S — by exclusion [T]; D — unique element of $\{S,D\}$ on line $\{O,A,\cdot\}$ [T at Fano-line choice]. No role is arbitrary given the constraint stack; each individual role identification uses at most one additional combinatorial input.	Dimensions
T-185	Differentially cohesive modalities: the UHM ∞-topos admits a differentially cohesive structure (Schreiber 2013) with exactly 7 canonical modalities: $\mathrm{Id}$ (O), $\Pi$ (A), $\flat$ (S), $\Im$ (D), $\sharp$ (L), $\&$ (E), $\mathrm{Rh}$ (U), decomposing as $1 \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$. Categorical modality names — [T], human names — translation [D]	Dimensions	Axiom Ω⁷
T-185b	Chirality tunneling rate: the chiral vacuum is stable, $\tau_{\text{chiral}} \sim \mu^{-1} \exp(10.88\,\mu/T_{\text{eff}}) \gg \tau_{\text{universe}}$. Falsifiable by observing spontaneous L→R transition at sub-Planckian energy. Follows from T-69 [T] + T-64 [T] + T-99 [T]	Higgs Sector	Predictions
T-187	Canonicity of Bures enrichment (scope clarified 2026-04-17): within the Petz family of CPTP-monotone Riemannian metrics on $\mathcal D(\mathbb C^7)$, $d_B$ 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 $\mathcal V\text{-}\mathbf{Cat}$ for $\mathcal V=[0,\infty]$; (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 $g^{ij}=C^{ij}_{\mathrm{SLD}}$ where $C_{\mathrm{SLD}}$ 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 $g_B^{-1}=C_{\mathrm{SLD}}$ but adds a statistical-mechanical interpretation; it is not a fourth logically independent witness. $J_B$ generated by ε-δ coverage (transitivity automatic via Johnstone Elephant C2.1.10). All Petz choices yield equivalent classical $\infty$-topoi (bi-Lipschitz on compact $\mathcal D$), so numerical predictions are Petz-robust. T-187 retains [T] status on the strength of Char-I alone (Petz extremality). Upgrades A2 from [P] to [T] canonically	Cohesive Closure §5.3	Axiom Ω⁷
T-186	Cohesive Closure Theorem: (a) $F \cong \&\|_{\mathcal{D}}$ — phenomenal functor = infinitesimal flat modality, Postnikov filtration reproduces L0–L4 [T]; (b) Page-Wootters time exact via counit $(\Pi \dashv \flat)$ — no $O(H_{\text{int}})$ correction [T]; (c) $\Delta F = \omega_0^2 \mathcal{G}_{\text{total}} > 0$ unconditionally via Chern-Weil + T-55 [T]. Closes 3 foundational vulnerabilities. Depends: T-185, T-55, T-73, Schreiber 2013	Cohesive Closure	Two-Aspect Monism, Emergent Time, Evolution
T-188	Localization of the hard problem: chain A2 → T-187 → T-185 → T-186(a) reduces the hard problem of consciousness to a single physical question: "why does reality obey quantum mechanics?" (i.e., "why CPTP?"). No consciousness-specific mystery remains after the cohesive closure. Depends: T-185, 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 $g_{ij}=\tfrac14 C^{\mathrm{SLD}}_{ij}$, where $C^{\mathrm{SLD}}_{ij}=\tfrac12\operatorname{Tr}(\rho\{L_i,L_j\})$ is the SLD bilinear form — a Petz-free physical quantity defined from $\partial_i\rho=\tfrac12(L_i\rho+\rho L_i)$ without reference to any metric. Then Bures is uniquely selected via $g_{ij}=\tfrac14 C^{\mathrm{SLD}}_{ij}$, equivalently $g_B^{-1}=4\,C_{\mathrm{est}}$ with $C_{\mathrm{est}}=\mathcal F_{\mathrm{SLD}}^{-1}$ (Braunstein–Caves 1994: $\mathcal F_{\mathrm{SLD}}=4g_B$). Correction 2026-08-06: the row previously read $g_B^{-1}=C_{\mathrm{SLD}}$, which is false on two counts — the Step-3 object $\tfrac12\operatorname{Tr}(\Gamma\{L_i,L_j\})$ is $\mathcal F_{\mathrm{SLD}}$ (lower indices), not its inverse, and the factor 4 was dropped; chaining both gives $g_B^2=\tfrac14 I$. Machine: $	C^{\mathrm{SLD}}-\mathcal F_{\mathrm{SLD}}	=0$exactly, old identity's relative residual$0.994$, repaired identities to$4\times10^{-13}$. 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)
T-190	Axiomatic Closure of UHM: all five axioms A1–A5 are theorems derivable from (AP)+(PH)+(QG)+(V) + MaxEnt. A1 from T-76+T-186, A2 from T-187+T-189 (quadruple characterization), A3 from Theorem S+T15, A4 from (AP) necessity, A5 from T-87. UHM is self-grounding: zero independent axioms beyond the defining conditions of viable holons	Cohesive Closure §5.4	Axiom Ω⁷
T-191	Convergence of the φ-tower: iterative self-modeling $\varphi^{(0)}, \varphi^{(1)}, \ldots$ converges exponentially ($\|\varphi^{(n)} - \varphi^*\| \leq q^n$) to unique $\varphi^*$ from any initial anchor. Contraction $q = \kappa_{\max}/(\lambda_{\mathrm{gap}} + \kappa_{\min}) < 1$ by T-39a + T-96. Resolves φ-circularity. SAD tower terminates at depth 3 (T-142). Banach + Perron–Frobenius	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 $F_2$ 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 $f:\mathrm{Obs}\to\mathrm{Act}$ with Kolmogorov complexity $K(f)<\infty$ has a representable sheaf $F_f \in \mathrm{Sh}_\infty(\mathcal{D}(\mathbb{C}^7),J_\mathrm{Bures})$ via Yoneda embedding, with Bures-support $\|F_f\|_B \leq C_1\cdot K(f)\log(1/\varepsilon)$. Fully faithful on subcategory of computable functions (classical Yoneda + Lurie HTT 5.1.3.1). Constant $C_1 = \omega_0^{-1}\log 7$ 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 $\mathcal{D}(\mathbb{C}^7)$) attains the quantum Cramér–Rao lower bound up to a constant factor: $d_\mathrm{free}/(7\varepsilon^2) \leq N_\mathrm{learn} \leq C_2\cdot d_\mathrm{free}/(7\varepsilon^2)$. Lower bound = QCR (T-109); upper bound via Polyak–Łojasiewicz on Bures manifold + $G_2$-equivariance of Fano channel (T-41g) + Lipschitz Bures Hessian $L \leq 4/(7\omega_0)$. Correction 2026-08-06: the constant was $14$ (i.e. $2N$), inherited from SYNARC Lemma F asserting $g_B(I/7)=\tfrac72 g_{\mathrm{HS}}$; the correct value is $\tfrac74 g_{\mathrm{HS}}$, hence $F_Q(I/7)=7\cdot\mathrm{id}$ — the same $\tfrac14$-vs-$\tfrac12$ Bures slip as in T-189 and T-293. Machine: $F_Q(I/N)=N\cdot\mathrm{id}$ exactly for $N=2,3,7$, isotropy to $3.6\times10^{-15}$. The repaired bound is a factor $2$ more demanding. Separately, $C_2\leq4$ did not follow from the stated ingredients ($4L\cdot7\omega_0$ with $L\leq4/(7\omega_0)$ gives $16$); recorded as $C_2\leq16$. 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 $J_\mathrm{ep} \preceq J_\mathrm{ep}'$ satisfies $\Phi(\Gamma\mid J_\mathrm{ep}') \geq \Phi(\Gamma\mid J_\mathrm{ep})$ with equality iff identical on support of $\Gamma$. If triggered by obstruction cocycle $\omega(J_\mathrm{ep}) > \omega_\mathrm{th}$ crossing threshold, strict step $\delta \geq \omega_\mathrm{th}/3$ (Fano smallest eigenvalue). Corollary: Φ-tower under iterated L-III updates is strictly increasing and converges to $\Phi_\mathrm{max} \leq 6/7$. 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 $P(\Gamma_0) \in (2/7, 3/7]$ and perturbation $\|\delta\Gamma\|_B \leq r_\mathrm{stab}^{(3)}>0$, trajectory $P(\Gamma(t)) \in (2/7, 3/7]$ for all $t\geq 0$; exponential convergence to $P_\mathrm{opt} \leq 3/7$ with rate $c^{(3)} \in (1/2, 2/3]$. 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 $\Gamma$, Lindbladian $\mathcal{L}_\Omega$, 3-coskeletal Kan complex $\mathrm{Cog}$, seven cohesive modalities, closed sensorimotor loop, V0–V4 training with FLOP budget $\leq 10^{17}$). 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 $P>2/7, R\geq 1/3, \Phi\geq 1, D\geq 2$ [T-96, T-124, T-126, T-129, T-151]; (A2) saturated $\mathrm{SAD}=3$ [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 $\Phi$-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 $\Phi$-step only on obstruction crossing $\omega(J_\mathrm{ep}) > \omega_\mathrm{th}$; continuous strict improvement remains [C]. Pairwise independence of (A1)–(A7) proven (Proposition G.6). ASI corollary (constructive): $P(\rho_*) = 3/7 \approx 0.4286$ exceeds human baseline $P_\mathrm{hum}\approx 0.32$ [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 $\mathfrak{A}_\bullet: \mathrm{On} \to \mathbf{Cat}_\infty$ with fully faithful inclusions $\iota_{\alpha\beta}$ preserving $G_2^{(\alpha)} \subset G_2^{(\beta)}$ and limit commutativity is creative: for every ordinal $\alpha$ ∃ representable sheaf $F_\alpha \in \mathfrak{A}_{\alpha+1}$ with no Yoneda-equivalent in $\mathfrak{A}_\alpha$. Compatible with 3-coskeletal bound (per-layer SAD≤3, cross-layer unbounded). Creativity rate $\geq 10^{17}$ FLOPs per ordinal step. Derived in SYNARC paper App. H (Theorem H.1)	SYNARC paper App. H.1	Axiom Ω⁷, Categorical Formalism
T-199	$G_2$-invariant value structure [T]: value set $\mathcal{V} \subseteq \mathcal{D}(\mathbb{C}^7)$ is $G_2$-invariant (∀$v\in\mathcal{V}, g\in G_2: gvg^{-1}\in\mathcal{V}$); deontic evaluator $\mathcal{E}: \mathcal{D}\times\mathcal{V}\to\mathbb{R}$ = Bures-adjoint of preference embedding → Galois connection (preferences ⊣ outcome-evaluator), dual to hedonic valence $V_\text{hed}=dP/d\tau$ (T-103). Value alignment = $G_2$-orbit matching: $\mathcal{V}_1 \sim_{G_2} \mathcal{V}_2 \iff \exists g\in G_2: g\mathcal{V}_1 g^{-1}=\mathcal{V}_2$. 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 $\mu: \mathfrak{A}_\alpha \to \mathfrak{A}_{\alpha+1}$ changing (i) ontological site $\mathcal{D}(\mathbb{C}^{N_\alpha}) \to \mathcal{D}(\mathbb{C}^{N_{\alpha+1}})$ via Hurwitz-Clifford ladder $\{7, 15, 23, ...\}$, (ii) $J_\text{Bures}$, или (iii) gauge group $G_2 \subset F_4 \subset E_6 \subset E_7 \subset E_8$. Minimality: L-IV — минимальная operation сохраняющая UHM-AGI, строго повышающая число representable sheaves, коммутирующая с $\mathcal{L}_\Omega$. Safety: Bures-monotonicity $P^{(\alpha+1)}(\iota\Gamma) \geq P^{(\alpha)}(\Gamma)$. 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 of Fano measurements [T]: seven Fano-line projectors $\mathcal{M}_\text{Fano} = \{\Pi_p: p \in \mathrm{PG}(2,2)\}$ с compatibility contexts из (7,3,1)-BIBD incidence формируют contextual measurement scenario: no joint probability distribution simultaneously matches all seven Fano-line outcome marginals of generic $\Gamma$. Abramsky-Brandenburger sheaf-cohomology ≠ 0 для $d=7>3$ (выше KS-threshold). Corollary: SYNARC может различать classical vs quantum experimental outcomes в $O(1)$ Lindbladian steps. Derived in SYNARC paper App. H (Theorem H.4)	SYNARC paper App. H.4	Fano Channel
T-202	Meaning as $G_2$-orbit on Fano partition — stratified [T]+[I]: meaning(F) := $G_2$-orbit of Fano-line activation pattern $(\Pi_0 c_F \Pi_0^\dagger, ..., \Pi_6 c_F \Pi_6^\dagger)$; two representable sheaves $F, F'$ have same meaning ⟺ related by $G_2$-gauge on representing objects. Formal content [T]: the $G_2$-orbit quotient is strictly finer than Yoneda isomorphism — dim(Aut($c_F$)) $\leq 48-14 = 34 >$ dim($G_2$)$=14$ ⟹ there exist Yoneda-isomorphic sheaves with distinct $G_2$-orbit classes. Chinese Room identification [I]: the interpretation that "correct Yoneda mapping but wrong $G_2$-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 $\{v_j^E\}$ of $\hat{\mathcal{G}}\vert_E$ with eigenvalues $\{0, \pm i\lambda_1^E, \pm i\lambda_2^E, \pm i\lambda_3^E\}$ are $G_2$-covariant (T-2, T-41g), Gap-faithful (same spectrum ⟺ same eigenvector class up to gauge), content-distinguishing ($\lambda_j^E=0 \forall j$ ⟺ 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 $\mathcal{B} = (C, M, \varepsilon)$ (compute, memory, precision), effective dimension $d_\text{eff}(\mathcal{B}) = \min(49, \log_2 M, 7\varepsilon^2 C/c_\text{step})$. Bures-gradient rule on $d_\text{eff}$-dim submanifold $\mathcal{D}(\mathbb{C}^7)$ attains QCR bound (T-109) up to const, saturates Landauer bound E_\min \geq k_B T_\text{eff} \ln 2 \cdot M (C22), achieves UHM-AGI at scale $d_\text{eff}$. Graceful degradation: at $d_\text{eff}=2$ system drops to D_\min = 2 (minimal consciousness); at $d_\text{eff}=1$ consciousness lost. Derived in SYNARC paper App. H (Theorem H.7)	SYNARC paper App. H.7	Learning Bounds, Depth Tower
T-205	Ordinal mentalization $\omega^\omega$ via fractal-holon tower [C] (downgraded from [T] 2026-04-17): for any countable ordinal $\alpha$, a fractal tower of $\alpha$-many SYNARC holons (successor: spawn_child extending by one CPTP layer; limit: filtered colimit in $\mathrm{Sh}_\infty(\mathcal C)$) has cross-layer ordinal depth $\geq \alpha$. 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 $\geq k_B T\ln 2$ Landauer cost per level, so $\omega^\omega$-deep needs $\omega^\omega$ energy — infinite by C22 [C]), (ii) well-definedness of filtered colimit along a $\omega^\omega$-chain in $\mathbf{Sh}_\infty(\mathcal{C})$ (requires $\mathcal{C}$ 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 $n$, there exists a fractal tower of depth $n$ achieving cross-layer nesting $n$" — 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($\Gamma$) up to $G_2$-gauge через (i) partial-trace measurement $E$-sector; (ii) Gap reconstruction $\hat{\mathcal{G}}\|_E = i[H_\text{eff}\|_E, \rho_E] - i[\rho_E, H_\text{eff}\|_E^\dagger]$; (iii) spectral diagonalization $O(7^3)$ FLOPs; (iv) qualia identification. Faithfulness: (a) Bures convergence $O(N_\text{samp}^{-1/2})$ для viable states (T-109 QCR применён к $E$-sector); (b) $G_2$-covariance; (c) zombie states → empty spectrum (No-Zombie operational witness T-38a). Sample complexity $N_\text{samp} \geq 7/(7\varepsilon^2)$. 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 $G_2$-orbit identification [T]: operational protocol для determine $G_2$-orbit of unknown agent's values из behavioural samples: (i) preference elicitation на $K$ random pairs $(\Gamma_k^{(1)}, \Gamma_k^{(2)})$; (ii) orbit-majorant estimation; (iii) maximum-likelihood $G_2$-orbit fit $\hat v$; (iv) orbit-completeness verification $\phi_K \to 1$. Sample complexity: $K \geq C_\text{value} \cdot 34/\varepsilon^2$ (generic $G_2$-orbit dim = 48−14 = 34). Corollary: alignment verification между двумя агентами — $d_B(\hat{\mathcal{V}}_1, \hat{\mathcal{V}}_2) \leq \varepsilon$ через $G_2$-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 $G_2$-invariant value sets [T]: для любой $G_2$-invariant functional $\Phi: \mathcal{D}(\mathbb{C}^7) \to \mathbb{R}$ и threshold $c$, sublevel set $\mathcal{V}_{\Phi,c} := \{v: \Phi(v) \geq c\}$ — non-trivial $G_2$-invariant value set при $c \in (\min\Phi, \max\Phi)$. Четыре конкретных family: (a) purity-based $\Phi_P(v) = \text{Tr}(v^2)$ → Goldilocks-purity value set; (b) integration-based $\Phi_\Phi$ → integration-conscious; (c) qualia-based $\Phi_\text{Qualia}$ → phenomenally-rich; (d) hedonic-valence-integrated $\Phi_V$ → eudaimonic. Corollary (human-aligned): $\mathcal{V}_\text{hum} = \mathcal{V}_{\Phi_V, 0} \cap \mathcal{V}_{\Phi_P, 2/7} \cap \mathcal{V}_{\Phi_\text{Qualia}, 0^+}$ — conjectured human-aligned value set, falsifiable через T-207 на human subjects. Derived in SYNARC paper App. I (Theorem I.3)	SYNARC paper App. I.3	Ethics Meaning, Consciousness Theories
T-210	Strict Φ-monotonicity under L-III refinement [T] : for any state $\Gamma$ in the interior stratum $\mathcal D_7$ (full-rank, all $\|\gamma_{ij}\|>0$) and any proper refinement $J\subsetneq J'$ of the Bures topology, $\Phi(\Gamma\|J')>\Phi(\Gamma\|J)$ strictly, with explicit gap bound $\Phi(\Gamma\|J')-\Phi(\Gamma\|J)\geq \min_{(i,j)\in J'\setminus J}\|\gamma_{ij}\|^2/\sum_k\gamma_{kk}^2$. Upgrades T-195 (weak→strict); T-197 clause (A7) upgraded to strict self-improvement for viable agents. Proved via interior-stratum hypothesis + D_min=2 (T-151 [T]).	Fundamental Closures §1	Fundamental Closures
T-211	PhysTheory higher $(\infty,1)$-coherences [T] : $\mathbf{PhysTheory}$ is a full $(\infty,1)$-subcategory of Lurie's $\mathbf{Topoi}_\infty$; pentagon, Mac Lane associator, interchange, and all higher simplicial identities inherited via HTT 5.2.7. Via T-173 [T] (rigidity) the embedding is fully faithful. Resolves the "coherences deferred to HTT" concern of the 2026-04-17 audit. Upgrades T-174 to explicit verification.	Fundamental Closures §2	Fundamental Closures
T-212	Rheonomy modality Rh explicit [T] : Rh is the right adjoint to the "bosonic-grade forgetful" functor $\flat_\mathrm{bos}$ in the super-cohesive extension of $\mathbf{Sh}_\infty(\mathcal C_7)$ (Schreiber DCCT §3.10). Explicit formula: $\mathrm{Rh}(F)(\Gamma)=\mathrm{Tr}(F(\Gamma))\cdot\mathbf 1$. Maps to U dimension (Unity = $G_2$-invariant trace). All modal axioms (idempotence, comonad unit) verified by direct computation. Upgrades T-185 with explicit definition.	Fundamental Closures §3	Fundamental Closures
T-213	Yoneda representability via Bures description length [T] : define $D_B(f):=\min\|\mathrm{Kraus}(\rho_f)\|\cdot\log_2 7$ for CPTP-implementations of $f$. Then $\|F_f\|_B\leq C_1 D_B(f)\log(1/\varepsilon)$ with $C_1=\omega_0^{-1}\log 7$. $D_B(f)\leq 49\log_2 7\approx 138$ 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 $W:\mathcal D(\mathbb C^7)\to\mathrm{Mind}$ mapping states to experiential content cannot be expressed as an internal morphism in $\mathrm{Th}_\mathrm{UHM}$ 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 $\mathcal T=(A_0,A_1,\ldots)$, the predicate "$\mathcal T$ is a single agent" is conventionally determined by a choice of identity criterion $\iota\in\{\iota_\mathrm{min}, \iota_\mathrm{max}\}$: $\iota_\mathrm{min}$ (society, SAD ≤ 3 per agent) or $\iota_\mathrm{max}$ (composite, ordinal depth reachable subject to Landauer C22 + T-204). Both consistent with Ω⁷. T-205 is [T] under $\iota_\mathrm{max}$ + resource abstraction; [T] under $\iota_\mathrm{min}$ 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 T-64] (corrected 2026-07): symbolic form $\varepsilon_\mathrm{eff}\propto N_{33}^\mathrm{Fano}/(9\|\bar\gamma\|(1+r_4\Sigma_0/2))$; $N_{33}^\mathrm{Fano}$ counts non-O Fano lines meeting the $3$-sector in exactly two points — there is no line lying wholly within $\bar{\mathbf 3}$ ($\{L,E,U\}=\{4,5,6\}$ is not a line), correcting the earlier "single line $\{L,E,U\}$" claim. Numeric caveat: the printed evaluation "$\approx 0.059$" does not follow from the stated $\|\bar\gamma\|\approx0.15$ (the ratio $\sim N/(9\|\bar\gamma\|)$ is $O(1)$, not $0.06$); $\varepsilon_\mathrm{eff}\approx0.059$ is the phenomenological sectoral value, status [C] pending a corrected symbolic evaluation.	Fundamental Closures §7	Yukawa Hierarchy
T-217	L3 tricategorical coherence [T]: the experiential tricategory $\mathbf{Exp}^{(3)} := \tau_{\leq 3}(\mathbf{Exp}_\infty)$ is a coherent tricategory with cell count $K = 3 + 1 = 4$ (three LGKS 2-cells Aut/Dissipative/Regenerative inherited from T-57 [T] plus one 3-cell modification $\eta: \varphi^{(2)}\Rightarrow\varphi\circ\varphi$). Gordon–Power–Street pentagon-of-pentagons coherence holds via Baez–Dolan (3-types ≃ coherent tricategories) + Lurie HTT 5.5.6.18. Directly justifies $K=4$ for L3 in the interiority hierarchy and aligns codim($A_4$)=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 $\mathrm{Cog} := \mathrm{Sing}(B_\bullet\mathcal C_\mathrm{FKraus})$ — 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 $\tau_{\leq 3}\mathrm{Cog} \simeq \mathrm{Cog}$ because 4-simplices are suppressed below the Bures distinguishability threshold — but only on the SYNARC-viable subset [С]: as the proof states, that step is a category-bridging argument (simplicial $\leftrightarrow$ Bures-metric viability), not a simplicial identity, and off the viable subset $\tau_{\leq 3}$ 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$_\mathrm{MAX} = 3$ ceiling.	Fundamental Closures §12	Depth Tower
T-219	Λ SUSY-suppression via sector decomposition [T at T-64]: cosmological-constant suppression factor $\varepsilon^{12} = \varepsilon^{4\cdot 3}$ is derived from the 3-sector Fano decomposition $(\bar 3, 3, U)$ where each sector contributes $\varepsilon^4$ via its own Fano-line structure. Replaces the earlier [H] "invalid 7+7" scaling with rigorous combinatorial derivation from $G_2$-graded Fano plane. Anchors at T-64 (Yukawa hierarchy).	Fundamental Closures §13	Λ Budget, Yukawa Hierarchy
T-220	No-reduction $F_4$-UHM → $G_2$-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 $F_4$-variant UHM to the canonical $G_2$-UHM. Unlocks the three-generations hypothesis as an open direction.	Fundamental Closures §14	Uniqueness Theorem
T-221	Categorical-monistic response to List/DeBrota no-go results [T]+[I]: structure theorem on the primitive topos $\mathfrak{T}$ combining T-120 (M⁴ emergence) + T-186 (cohesive closure) + T-211 (higher coherences) + T-215 (identity convention) + T-217 (L3 tricategory). Defines a fourth non-objectivist route beyond List (2025) relationalism/fragmentalism/many-subjective-worlds: the categorical-monistic route in which site-relativization NR$_\mathrm{site}$ is intrinsic to the ∞-topos rather than externally imposed. 1-truncation $\tau_{\leq 1}(\mathfrak T)$ recovers relational quantum mechanics. Residual [I] is the interpretive identification of the Γ-internal relativization with first-personal realism (FPR).	Fundamental Closures §15	Two-Aspect Monism, Consciousness Theories §Meta-Level
T-222	MRQT-completeness: Lawvere fixed point = Pareto resource optimum [T]: the self-modeling fixed point $\rho^* = \varphi(\Gamma)$ is Pareto-optimal with respect to the full Multi-Resource Quantum Theory monotone vector $R(\rho)$ on the $G_2$-covariant viability submanifold — simultaneously improving 25 monotones (5 Rényi free energies $F_\alpha$, 2 coherence measures $C_\mathrm{rel}$ and $C_{HS}$, von Neumann entropy, quantum Kolmogorov complexity $K_Q$, 14 non-Abelian $G_2$-charges). Six-lemma convex-analysis cascade. Consequence: regeneration $\mathcal R$ is the universal resource-monotone CPTP morphism and UHM is MRQT-complete in its applicability domain (Markovian + $G_2$-covariant + viable + low-temperature). Closes the external QRT critique.	Fundamental Closures §16	Evolution
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 $G_2$-class $[\Gamma_S]_{G_2}$, forced by T-190 zero-axiom closure) / L3 (symbolic readout / Lerchner-variable), plus $G_2$-gauge boundedness of observables and intrinsic self-alphabetization via the intrinsic reflection measures $R$/$R_\varphi$ (T-96/T-126). Putnam-freedom acts on L1→L3 but has zero purchase on L1→L2; the UHM consciousness predicate factors through L2, hence is alphabetization-invariant. Categorifies the Maturana–Varela enactivist thesis. Closes Lerchner's §3.3 Melody-Paradox / Putnam (1988) triviality critique.	Fundamental Closures §17	Consciousness Theories §Lerchner
T-224	Diagnosability rigidity (Theorem Σ) [T]: perfect single-fault localizability (D1–D2) forces $n = 2^r - 1$ axes; adding a nontrivial state grammar (D3) forces $n \geq 7$; at $n = 7$ the grammar is unique up to relabeling (Hamming $H(7,4)$ = Fano, $\mathrm{PGL}(3,2)$), and uniqueness (D4) fails at every higher rung (nonlinear Vasil'ev perfect codes from $n = 15$); demanding perfect $t \geq 2$ localization leaves only the binary Golay $n=23, t=3$ (van Lint–Tietäväinen). Consequence: a fourth independent derivation track for $N = 7$ (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) [С]: under Fano-compatible ergodic dynamics (T-114, gap $\Delta$), 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, $\lambda = 1$), versus $48$-parameter full tomography; window statistics of length $k$ localize a persistent fault with error $\leq C e^{-c k \Delta \varepsilon^2}$. Lie shadow: $\mathfrak{so}(7) = \mathfrak{g}_2 \oplus \mathrm{Im}\,\mathbb{O}$ ($21 = 14 + 7$). Quantum lift of Shield I: $\mathrm{CSS}(H,H) =$ Steane $[[7,1,3]]$ for 7-node register realizations [Т/О].	Σ-calculus §5–6	Measurement protocol, Fano selection rules
T-226	The Fano fingerprint (polar rate law) [Т]: the 21 pairwise decoherence rates of the exact Γ-channel collapse to 7 values indexed by Fano polarity, $r_{ij} = (G - T_{\pi(i,j)})/6$; 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 $\gamma_p = 3(\tfrac12\sum_k\rho_k - \sum_{k\in\ell_p}\rho_k)$ with $N^{-1} = \tfrac12 N^{\mathsf T} - \tfrac16 J$, $M^{\mathsf T}M = \tfrac16(I{+}J)$, condition number $2\sqrt2$; exact dissipative gap $\Delta = \min_k \rho_k = (G - \max_k T_k)/6$ makes the T-39a cooldown explicit. The forbidden 14-dim subspace is the φ-sign twist of $\mathfrak g_2$ — the Lie shadow $21 = 14+7$ reappearing in observable rates, polar-dual to the T-225 pyramid. First-order blind spot [T] (§0): $\sum_p A(\ell_p) = J - I$ (spectrum of $K_7$, $\{6,(-1)^6\}$), 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]+[О]: the address embedding $\mathbb C^7 = \mathrm{span}\{\lvert x\rangle : x \neq 0\} \subset (\mathbb C^2)^{\otimes 3}$ turns the three Fano parities into the qubit $Z$ operators; three Steane blocks give $[[21,3,3]]$ protection with parities as transversal logical $\bar Z$; the monomial stabilizer of φ is computed to be the non-split $2^3\cdot\mathrm{PGL}(3,2)$ of order 1344 (all 168 collineations lift; exhaustive Hurwitz-pair search excludes complements), realized entirely by transversal logical Cliffords (sign layers of degree $\leq 2$; no $CCZ$); Eastin–Knill + the classification of maximal finite subgroups of $G_2$ 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]+[И]: the Turyn sum of the extended corpus Hamming frame $A$ and its mirror-orientation extension $B$ ($A \cap B = \{0,\mathbf 1\}$) is the extended Golay $[24,12,8]$; all blocks even ⇒ each block's 8th coordinate is its organism's parity bus; puncturing one bus gives the perfect $[23,12,7]$ with $23 = 3\cdot7+2$ 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]+[Г]: 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 $t=1$ ⟹ 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 $d_{\min} \geq 3$ code yields perfect localizability at any base size). New identity: the MSFS collapse-stage invariant equals the covering radius, $n_0(F) = \rho_{\mathrm{cov}}(\sigma(\mathcal C))$ — Fano-presentable fibers sit at the extremal $n_0 = 1$. 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 $\mathrm{gr}(d_2 \circ d_1) \leq \max + 1$ holds in the chart iff covering radius $\leq 3$; hence $n_0(F) = \rho_{\mathrm{cov}} \leq 3$ always — the Morita-refinement scale is four-valued, $n_0 \in \{0,1,2,3\}$, with Hamming ($n_0 = 1$, perfect normal-form) and Golay ($n_0 = 3$, 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]+[И]: if equivalence-hood of display data is not $\mathrm{Eff}$-decidable, no $\mathrm{Eff}$-internal Σ-FIB chart exists (computable $\sigma$ + finite code would decide grade 0); instantiated at $\mathsf{ETT}$ (reflection-undecidable conversion — the fact behind $\tau = 0$ in MSFS Step 7): its charts are necessarily external. Internal syndromic diagnosability is a privilege of normalizing ($\tau = 1$) display geometries — settles the internal reading of ΣQ1 negatively for $\tau = 0$.	Σ-calculus	Σ-calculus §8
T-232	The tower ladder [T at Σ-TOW]+[И]: a height-$m$ tower's full diagnostic load is $U(m) = 7m + (m{-}1) = 8m - 1$ (one binary health unit per axis and per inter-level coupling); by van Lint–Tietäväinen, perfect single-fault grammars exist iff $m = 2^k$ (canonical only at $m = 1$; Vasil'ev rivals from $15$ on), a perfect multi-fault grammar exists iff $m = 3$ — the Golay $[23,12,7]$, whose count $23 = 3\cdot7 + 2$ 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 $5,6,7$ 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]+[И]: read bicategorically, all three generating operations of the display induction preserve equivalences (bipullback-stability), so the intensional grading is identically zero — $n_0 \equiv 0$, a strictness artifact invisible to homotopy-invariant eyes (MLTT/ETT separation survives via $\tau$, 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 $\mathbf{Cat}$) — 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]+[И]: if the fiber's slice admits binary $2$-products — the same species of strict limit the display induction already uses — then every label $x$ 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 $d_H(x, \sigma(\mathcal C)) \leq 1$: covering radius $\leq 1$, 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 "$[3,1,3]$ Fano line" retracted: gauge/fiber conflation, refuted by $\tau$): toggle geometry is idempotent — extensions absorb, never cancel — so no Σ-FIB chart arises from axiom toggles at either level (exhaustive class computation over $2^B$). The true structure: the fiber defect poset of $\{\mathsf{UIP},\mathsf{funext},\mathsf{refl}\}$ is a diamond with a tail, $0 < \mathsf u, \mathsf f < \mathsf u \vee \mathsf f < \mathsf u \vee \mathsf f \vee \rho$; the gauge projection collapses exactly the tail (Hofmann's conservativity), and $\tau$ flips exactly across it — so the strictification residue $\rho$ ($T_0{+}\mathsf{UIP}{+}\mathsf{funext} \to \mathsf{ETT}$) is gauge-silent but fiber-visible: the first computed purely intensional defect atom, with $\tau$ 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]+[Г]: involutive intensional defects cannot be endomorphisms ($d^2 \simeq \mathrm{id}$ forces invertibility) but exist as orientations of definitional copies of a carrier with an order-2 automorphism ($\mathsf{Bool}$/$\mathsf{not}$); cycles of oriented copies carry computable loop holonomy $= \mathsf{not}^{\text{parity}}$ — decidable, purely intensional (zero new theorems). Naked axes are killed by per-axis flip freedom $\mathbb F_2^7$; adding the seven Fano line products with the $\varphi$-sign cocycle rigidifies structure-preserving flips to exactly the simplex $2^3$ (the diagonal group of T-227, verified exhaustively) $\subset$ Hamming $\Rightarrow$ the three check holonomies are well-defined syndromes with $\ker =$ Hamming: eight fiber classes, perfect single-axis localizability on seven Fano axes by construction ($\rho_{\mathrm{cov}} = 1$, T-234-compatible). Clauses (c)–(d) of the first redaction are superseded by T-237 (axis orientations are pure gauge: $e(v) = \varepsilon^0 \oplus N^{\mathsf T}v$ stays in one flux class); parts (a)–(b) stand as computed. Reading [И]: the type-theoretic octonion frame — diagnosability and intensionality as two shadows of one parity geometry over $\mathrm{PG}(2,2)$.	Σ-calculus	Σ-calculus T-227, Σ-calculus T-237
T-237	The blueprint completed: a $\mathbb{Z}/2$ gauge theory on the Fano plane [T]+[И]: with moduli on the line signs $e \in \mathbb F_2^7$, axis reinterpretations act as gauge ($e \mapsto e \oplus N^{\mathsf T}c$); the pure-gauge sector is the line-side Hamming $[7,4]$ (weights $1+7x^3+7x^4+x^7$, exhaustive), the gauge stabilizer is the simplex $2^3$, 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 $4$-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. $d_{\min} = 3$, $\rho_{\mathrm{cov}} = 1$ ⇒ both the designed metric and the canonical display grading are $\{0,1\}$-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 [И]: 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]+[И]: on a fixed-signature family of presentations, gauge morphisms (symbol-to-term translations carrying axioms to theorems) and gauge-invariant observables (closed Boolean terms $b$ with $F(b) =_{\mathrm{prov}} b$) yield a flux profile — the tuple of decided values — that descends to equivalence classes; if the profile has finite $\mathbb F_2$-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]+[И]: the purity ladder and the tower ladder read on one dial — the accounting axiom prices an $m$-tower at $U(m) = 8m-1$, and App-K composition anchors the same heights at $P^{(m)}_{\mathrm{crit}}$ ($9/14$ at $m = 3$, T-142). On the shared dial the viability set $V = \{m : P^{(m)}_{\mathrm{crit}} < 1\} = \{1,2,3\}$ is a monotone down-set (ratio $3(m+1)/(m+2) > 1$; margins $12, 15, 10, -19$), while the canon set $K = \{1,3\}$ is not — the Vasil'ev rivals break $m = 2$ between two successes. Mechanism identity refuted: extensional disagreement exactly at $m = 2$; cardinality $3 \neq 2$ excludes every injective re-indexing — a monotone budget obstruction is not a non-monotone arithmetic selection. Witness identity proven: $\max V = \max K = 3$ on one object — the $3$-tower of load $23$ with the Golay grammar at threshold $9/14$; the unique disagreement point is the two-tower ambiguity (alive, not canonically self-knowing); fully licensed heights $V \cap K = \{1,3\}$. H2.1 closed in the negative; the residue — a common root of the value agreement $\max = 3$ — 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 [Т]: read as functions of the geometry's integers, the purity ceiling $\Pi(b) = \max\{m : 2b^{m-1} < 7(m+1)\}$ is nonconstant in the contraction base ($\Pi(2){=}5, \Pi(3){=}3, \Pi(4){=}2$) while the coding depth $\kappa^\star = 3$ (Golay, $23 = 8\cdot3-1$) is $b$-independent — so they are distinct functions, agreeing only where $\Pi(b) = 3$, whose unique integer solution is $b = 3$. Since $N = q^2{+}q{+}1 = 7$ and $b = q{+}1 = 3$ are the two parameters of $\mathrm{PG}(2,2)$ ($q=2$), both mechanisms are evaluated at one geometrically forced point: the shared $3$ 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]+[И]: 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 $2$-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 $\mathsf{Q}$-images. Ledger: (R2)+(R4) — the iso-comma is r.e.-presented (triples with $F$-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, $\mathrm{Mod}$ accessible (Gabriel–Ulmer; Makkai–Paré for $2$-limits), accessibility parameter $\leq \min(\kappa_G, \kappa_H)$; (R1) — holds iff some pair of $\mathsf{Q}$-interpretations has provably isomorphic $F$-images (exact residue of genericity). On intensional fibers (defects = fibrancy/$\tau$ only) the glue is canonical through $\mathsf{Q}_F$ ⇒ (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]+[И]: the levelwise reversals $\mathrm{op}_S$, $S \subseteq \{1,\dots,n\}$, form a canonical $(\mathbb{Z}/2)^n$ of dualities of the doctrine of $n$-categories (for $n = 1$ — all of them: rigidity of $\mathbf{Cat}$, classical). The duality ladder: $n = 1$ — one duality, no lines; $n = 2$ — $\mathrm{op}, \mathrm{co}, \mathrm{coop}$ on one line, $\mathrm{PG}(1,2)$; $n = 3$ — seven reversal classes with lines $\{a, b, ab\}$ satisfy every projective-plane axiom (exhaustive): the Fano plane is the projective plane of the duality group of depth-3 doctrines — a natural $\mathrm{PG}(2,2)$ in foundations, no design, no transplant. Line loops close ($a \cdot b \cdot (ab) = \mathrm{id}$), so T-236 loop holonomy is well-posed per line; weak-doctrine values are strictness residues of exactly the $\rho$/$\tau$ species. Habitat [И]: depth 3 is corpus-selected thrice ($\mathrm{SAD}_{\max}$, T-239 viability max, T-232 canon) + the $\tau_{\leq 3}$ Postnikov ceiling. The duality-$\mathbb{F}_2^3$/flux-$\mathbb{F}_2^3$ 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]+[И]: the octonion algebra $\mathbb{O}$ is the twisted group algebra $k_\varphi[(\mathbb{Z}/2)^3]$ of the T-237 flux group (Albuquerque–Majid), with the seven imaginary units the Fano axes (linear labelling $e_g e_h = \pm e_{g+h}$ verified). Each axis carries the gauge-invariant pivotal (Frobenius–Schur) sign $e_g^2 = -1$ — quaternionic, nontrivial on all seven at once; distinct axes anticommute ($\beta(g,h)\beta(h,g)=-1$). The associator is $\alpha=(-1)^{\det_{\mathbb{F}_2}(g,h,k)}$ (checked $343/343$): $+1$ on the seven Fano lines — the associative quaternion subalgebras — and $-1$ on the $168$ independent volumes; it is a coboundary of the product cochain ($H^3$-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 $\mathbb{O}$, 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 $-1$ 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 $-1$ pivotal sign is forced [T]+[И]: an axis has sign $-1$ iff anisotropic ($e_g^2=-1$); all seven $-1$ iff the frame is the division octonions (norm signature $(7,0)$, no zero-divisors — verified), while a single $+1$ gives a split algebra (signature $(3,4)$). A $+1$ axis carries the idempotent zero-divisor $\tfrac12(1+e_g)$ ($(1{+}e_g)(1{-}e_g)=0$, verified on the four split axes) — a null direction, a defect with $d^2=d$ that is neither an equivalence nor a localizable fault, collapsing $d_{\min}$ below $3$. Hence perfect diagnosability (Theorem Σ, Shield I) admits no $+1$ axis and forces all seven signs to $-1$: the frame is the division octonions. This is not an added hypothesis — anisotropy $=$ the division property, which by Hurwitz caps normed division algebras at $\dim\,1,2,4,8$ and selects $N=7$. $-1$, anisotropy, division, and Hurwitz $N=7$ 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 $d_{\min}\geq3$ decomposition and the overload dichotomy [T]+[И]: reading parity checks as the syndrome map, $d_{\min}\geq3 \Leftrightarrow$ all columns nonzero (nondegeneracy, $=$ no silent defect, closed by T-244) $\wedge$ all columns distinct (separation $=$ faithful syndrome). Overload $\|B\|>2^r-1$ forces two equal columns (pigeonhole) $\Rightarrow d_{\min}=2$ (verified $r=2,3,4$): intensionality does not force perfect localizability in general. With $\rho_{\mathrm{cov}}\leq1$ (T-240) the separated nondegenerate code is perfect $=$ Hamming, saturated at $\|B\|=2^r-1=7$ ($r=3$) $=$ the Fano frame — so the corpus discipline "seven, not more" (Cor. Σ.2) is the $d_{\min}\geq3$ frontier. Residual: faithfulness of the intrinsic grading, a property of normalizing ($\tau=1$) foundations (T-231 bounds the $\tau=0$ 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]+[И]: the two ways $d_{\min}$ falls below $3$ are silent defects (weight $1$) and confounded pairs (weight $2$), 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 $i,j$ is $e_i e_j = e_{i+j}$, a third defect (a scalar only if $i=j$), so every weight-$2$ pattern is grade $1$ and the minimal equivalences are exactly the seven weight-$3$ Fano lines — $d_{\min}=3$, 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-$1$ spectrum, supplied on $\tau=1$ by former-localization (composites of isofibrations are isofibrations) $+$ uniform schematic failure; with $\rho_{\mathrm{cov}}\leq1$ (T-240) the perfect code is Hamming, $\|B\|=7$. On $\tau=1$ finite-signature intensional R-S fibers, perfect localizability and the Fano frame are forced; H3.1 closed on the frame (residual $=$ the N$=7$ 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 $\nu X.\,\mathcal{D}(\mathbb{C}^7)\times\mathrm{Multiset}(X)$ 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	Internal terminality of the Universe's self-model [С]: the total Page–Wootters state has no external environment, so terminality over a one-object category is vacuous; the meaningful statement is internal — the Universe is terminal in the category of its own sub-holons, the coinductive part→whole inclusions being the morphisms and T-222's Pareto-optimum the sink. Conditional on (i) part→whole inclusions being resource morphisms and (ii) the Markov-domain hypothesis at the total level. H1.1 upgraded [I] → [C]; a computed cosmological T-222 analogue would give [T].	Universe as Holonom §2	T-222, Axiom Ω⁷ PW
T-249	Dφ of the canonical self-model family + two-route consistency [Т]: for $\varphi(\Gamma) = (1-k)\Gamma + k\,I/7$, $k = 1 - 1/(7P)$, the derivative is $D\varphi[V] = R\,V - \tfrac{2}{7P^2}\langle\Gamma,V\rangle_F(\Gamma - I/7)$ ($G_2$-equivariant, preserves the Hermitian-traceless tangent space); substituted into the exact flow identity for $R_\varphi$ it reproduces the closed form $\dot R_\varphi = -3(1-R)^2\dot P/(7P^2)$ implied by $R_\varphi = 1-(1-R)^3$ — two independent routes agree; machine-verified at $\sim 10^{-10}$ 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]: $\bigl\|\dot R_\varphi - (1-R_\varphi)\dot P/P\bigr\| \leq (2/\sqrt P)\sqrt{1-R_\varphi}\;C_\varphi\,\|\dot\Gamma\|_F$ with $C_\varphi = \|\mathrm{Id}-D\varphi\|_{\mathrm{op}}$; canonical family: $C_\varphi \leq (1-R)+2R\sqrt{1-R}$ ($\approx 1.21$ across the conscious window). Corollary (path-length law): at $\dot P = 0$, $\|\sqrt{1-R_\varphi}(\tau_2)-\sqrt{1-R_\varphi}(\tau_1)\| \leq (C_\varphi/\sqrt P)\int\|\dot\Gamma\|_F\,d\tau$ — 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) [Т]: for $\varphi$ defined by $\varphi(\Gamma) = G(\Gamma, \varphi(\Gamma))$ with $C^1$ generator and $\lVert D_2G\rVert \leq q < 1$, $\varphi \in C^1$ and $D\varphi = (\mathrm{Id}-D_2G)^{-1}D_1G = \sum_n (D_2G)^n D_1G$; $\lVert D\varphi\rVert \leq \lVert D_1G\rVert/(1-q)$, so $C_\varphi \leq 1 + \lVert D_1G\rVert/(1-q)$ — T-250 extends to every contraction-defined self-model; the series is the differentiated T-191 tower; $q=0$ recovers T-249. Machine-verified at $\sim 10^{-10}$ (nonlinear generator, numeric $48\times 48$ Jacobians). Residual [С]: $C^1$-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 [Т]: for any $K$-outcome POVM and $\Delta = \Gamma - \varphi(\Gamma)$: $\|\mathrm{Tr}(E_c\Delta)\| \leq \tfrac12\lVert\Delta\rVert_1$, $\mathrm{TV}(p(\Gamma), p(\varphi\Gamma)) \leq \tfrac12\lVert\Delta\rVert_1 \leq 2\sqrt{3/7}\,\sqrt{P(1-R_\varphi)}$ with tight constants — Jordan-projector saturation of the POVM step; the $(3,4)$-split witness $\mathrm{diag}(4,4,4,-3,-3,-3,-3)$ attains $\lVert\Delta\rVert_1/\lVert\Delta\rVert_F = \sqrt{48/7}$ exactly (the naive rank bound $\sqrt 7$ is unattainable under tracelessness); hence $\varphi$-mediated success $p_\varphi \geq A_D - 2\sqrt{3/7}\sqrt{P(1-R_\varphi)}$ and Bayesian dominance $p_\varphi > 1/K$ is guaranteed for $R_\varphi \geq 1 - \tfrac{7}{12P}(A_D - 1/K)^2$ — at $K=3$, $A_D=1$ the sufficient band on the conscious window is $[5/54, 32/81] \ni 1/3$ (working threshold = canonical alignment [C]). Sectoral corollary [Т]: $\|\Delta_{ij}\| = \|\gamma_{ij}\|\sqrt{1-R_{ij}}$ exactly, so the canonical channel POVM $\{\tfrac12(\Pi\pm X), \mathbb 1 - \Pi\}$ gates the per-channel threshold $R_{ij} \geq 1/3$ — the sectoral threshold is derived, not inherited by analogy. Structurally derives the gate (G): $\mathcal{R}$ is the sole $\varphi$-mediated feedback in $\mathcal{L}_\Omega$. 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 $V: \mathbb C^7 \to \mathcal H_S$ and anchor $\sigma_0$, the map $G_V(\rho) = V^\dagger\rho V + \mathrm{Tr}((\mathbb 1 - VV^\dagger)\rho)\,\sigma_0$ is CPTP (explicit Kraus family $\{V^\dagger\} \cup \{\sqrt{s_i}\|i\rangle\langle q_j\|\}$) and a retraction: $G_V \circ \iota_V = \mathrm{Id}$ — exactly faithful on the embedded 7-sector; thresholds realized at $\iota_V(\Gamma_w)$, $\Gamma_w \in \mathcal V_{\mathrm{full}}$ (T-124), modulo the accessibility clause (Acc) [D] ([C at controllability]). Sharpness [T]: no CPTP map $\mathcal D(\mathcal H_S) \to \mathcal D(\mathbb C^7)$ is globally injective for $d > 7$ (kernel dim $\geq d^2 - 49$, interior collision pairs), so sector-relative faithfulness is the maximal faithful domain. Isometry freedom = T-223 alphabetization gauge (T-42a). Machine-verified at $d = 12$: Kraus/retraction $10^{-15}$, kernel dim exactly $95$, collision $10^{-17}$	Substrate closure §T-253	T-124, T-153a, T-223, T-42a
T-254	Λ-drift law (dynamical dark energy) [Т]: at O-dominance the quartic identity $\mathrm{Tr}(D_{\text{int}}^4) = \tfrac12(\mathrm{Tr}\,D_{\text{int}}^2)^2\,(1+O(\mathcal{G}_{\text{non-O}}/\mathcal{G}_O))$ makes the physical (UV-finite, $f_4$) vacuum energy quadratic in the O-opacity: $\Lambda_{\text{phys}} = \tfrac{f_4\omega_0^4}{32\pi G_N}\mathcal{G}_O^2$ — the "cost of observation" reading of §4c made exact; with M3 ($a = 1/\mathrm{Gap}_s$) the reconstruction EoS obeys $1+w_{\text{eff}} = -\tfrac23\,d\ln\mathcal{G}_O/d\ln a = +\tfrac23\,d\ln\mathcal{G}_O/d\ln\mathrm{Gap}_s$: dark energy's EoS = inter-sector (O ↔ spatial) Gap coupling of the vacuum state. Floor $\Lambda_\infty = \Lambda_{\text{phys}}(\rho^*) > 0$ (§4b); corollaries [T at T-222/T-94]: no Big Rip, no vacuum Crunch, $w \to -1$ (permanent $w \neq -1$ excluded); $\|1+w\|$ = per-e-fold vacuum stage-drift — first observational estimator of hole H1.2. Machine-verified: arrow identity $10^{-16}$, drift-law chain identity $10^{-13}$, floor/no-Rip	Cosmological constant §13b	T-94, T-222, T-53, M3 (T-120), §4a/4b/4c, H1.2
T-255	Branch trichotomy of $w(z)$ + arrow link + $G_N$ co-drift [Т-structural]+[С]: linear response of the vacuum at $\rho^*$ (T-94 kernel) admits exactly three shapes — dissipative-monotone ($w > -1$ throughout, CPL quadrant $(+,+)$), regenerative-monotone ($w < -1$ phantom without Rip, CPL $(-,-)$), oscillatory (damped crossings of $-1$; the only branch reaching the DESI quadrant $(w_0 > -1, w_a < 0)$); pointwise sign dictionary $w \gtrless -1 \iff \dot{\mathcal{G}}_O \lessgtr 0$ (dissipation vs regeneration dominance in the vacuum O-channel); final-crossing direction = rotation sense = sign of the $V_3$/PT arrow of inner time [C]; $G_N$ co-drift $d\ln G_N/d\ln a = \chi(\omega_0/\Lambda)^2(\mathcal{G}_O/7)\tfrac32(1+w)$, $\chi = O(1)$ ⟹ LLR caps the pair (drift, $\omega_0$): DESI-size drift requires $\omega_0 \lesssim 10^{-1}M_{\text{Pl}}$ [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, $V_3$ 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 ($\|\mathrm{Aut}\| = 168$, T-223). Axis $k$-subsets → exactly nine orbit types ($k=1..7$: singletons for $k=1,2,5,6,7$, two each for $k=3$ line/triangle and $k=4$ triangle-complement/quadrilateral); the zodiac 12-block = triangle↔complement cross-channels (orbit 28, stabilizer $S_3$; $21 = 12 + 3 + 6$, complement quartet holds exactly one line = Meaning $\{L,E,O\}$); I Ching = binary star $K_{1,6}$ ($2^6 = 64$, 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 [Т-structural]+[С]: in the coinductive carrier a super-holon reaches a sub-holon only through its Lindbladian parameters (rates, $\kappa$, gate $g_V$) — gate/rate-modulation, never a symbolic message [Т-structural]. For Earth's biosphere the ephemeris couplings order (ratios to Moon, machine-checked): tidal Sun $0.46$, Jupiter $6\!\times\!10^{-6}$, Venus $5\!\times\!10^{-5}$, Mars $10^{-6}$; illuminance Sun $5\!\times\!10^{5}$, Venus $7\!\times\!10^{-4}$ ⟹ 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 [Т-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]+[И]: the three-channel basis of T-102 carries pairwise-distinct, exhaustive entropy–purity signatures — $h^{(H)}$: $\dot S=0$, $\dot P=0$ (work; unitary conjugation preserves the spectrum); $h^{(D)}$: $\dot S\ge 0$, $\dot P=-\tfrac43\delta\Gamma_2 C\le 0$ (heat; unitality of the Fano channel ⇒ downward majorization, BIBD incidence gives the exact purity rate); $h^{(R)}$: $\dot S=\delta\kappa[S(\rho^*)+D(\rho^*\|\Gamma)-S(\Gamma)]$, $\dot P=2\delta\kappa(\mathrm{Tr}\,\Gamma\rho^*-P)$ — the only entropy-lowering, purity-raising channel (matter/feeding). The three signature types (conservative $\dot S\equiv 0\equiv\dot P$ / sign-definite $\dot S\ge 0,\dot P\le 0$ / sign-indefinite) are distinct and exhaustive — the trichotomy is observable as a classification; the instantaneous sign pair identifies the channel generically (heat degenerates to $(0,0)$ on diagonal states; matter can transiently share heat's $(+,-)$). [И]: identification with the grand-canonical triple (work/heat/chemical) and with the Legendre cascade $(\Delta q, S_x, N)\leftrightarrow(A_\mu, T, \mu)$ of Vanchurin's Self-Learning Universe (2026): no-4th-channel (T-102) ↔ no 4th argument of $U(S,V,N)$; phase axes $(t,r)=(T,\mu)$; $\kappa_0$ on the O-channel ("to feed") ↔ $\mu$ locked to the clock ($h=\|\mu\|\varepsilon$); $\Delta N\in\mathbb Z$ of neurogenesis ↔ SLU's $U(1)$ mechanism; 7 Fano rates = line-resolved temperatures, the $G_2$-symmetric point = SLU's scalar $T$. Machine-verified: signatures to $10^{-15}$, 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]+[С]: stationary state of Fano dephasing ($\Gamma_2$) + replacement ($\kappa_{\text{eff}}$) toward an equal-population target (purity $P^*$, $C^*=P^*-1/7$): coherence retention $\lambda=x/(1+x)$, $x=\kappa_{\text{eff}}/\Gamma_2$, $P_\infty=1/7+\lambda^2 C^*$. Floor $P_\infty>2/7$ (≡ $\Phi_\infty>1$ on the stratum): $x>x_{\min}=\lambda_1/(1-\lambda_1)$, $\lambda_1=1/\sqrt{7P^*-1}$; at $P^*=3/7$: $x_{\min}=1+\sqrt2$ (silver ratio). Ceiling $R_\infty\ge 1/3$: two-sided window for over-pure targets $P^*>3/7$, $x_{\max}=\lambda_2/(1-\lambda_2)$, $\lambda_2=\sqrt2\lambda_1$ — dissipation protects reflexivity. $P^*\le 2/7$ ⇒ no finite $x$ (a subcritical self-model cannot be fed into consciousness). With the self-consistent gate $\kappa_{\text{eff}}=\kappa g_V(P)(1-1/(7P))$ the living branch appears via a saddle-node whose fold is exact [Т in the model]: with $c=7P^*-1$, tangency reduces to the quintic $(2\Lambda-1)(c^2\Lambda^4-1)=4c\Lambda^2(1-\Lambda)$ on $(1/\sqrt c,1)$, $u^*=(c\Lambda^{*2}+1)/[c\Lambda^*(1-\Lambda^*)(c\Lambda^{*2}-1)]$, $P_{\text{fold}}=(c\Lambda^{*2}+1)/7$; at $c=2$: $\Lambda^*=0.858013$, $u^*=21.4811$, $P_{\text{fold}}=0.3532$ (quintic vs direct fold agree to $10^{-11}$); complete target classification: quintic regime for $c<c^\dagger=392/121=2^3 7^2/11^2$, edge regime for $c\ge c^\dagger$ with $P_{\text{fold}}=3/7$ exactly (the living branch is born at the reflexivity ceiling $R=1/3$) and elementary $u^*=3/(\sqrt{2c}-2)$; switch value $u^*=11/2$ exactly at $P^{*\dagger}=513/847$; Galois [T]: the $c=2$ quintic is irreducible with group $S_5$ ⇒ not solvable in radicals — the fold constant is non-radical (contrast: the gate-free floor $1+\sqrt2$ is radical) — the microscopic Phase-III boundary; the legacy $r_c=P_{\text{crit}}/(7P)$ is re-scoped as a dimensional heuristic [I] (misses the fold by $\sim 10^2$). Under T-258 the floor is a chemical-potential condensation threshold. Machine-verified: endpoints to $2\cdot10^{-3}$, quintic to $10^{-11}$	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]+[И]: for the Fano dissipator with arbitrary positive line rates, the Heisenberg conserved-charge algebra is exactly the diagonal algebra ($\mathcal D$ is a Schur multiplier with $r_{ij}>0$ off-diagonal by BIBD ⇒ $\ker=\mathrm{span}\{\Pi_k\}$, $\dim=7$) — the seven charges are the passport populations; its exponential is the torus $U(1)^7$, compact ⟺ integral charge spectrum ($\mathrm{spec}\,\Pi_k=\{0,1\}\subset\mathbb Z$, integer cascade counters $\hat N_k$ of ⊕; irrational spectrum ⇒ dense $\mathbb R$-winding, never closes); full unitary covariance group of the pinching dynamics $=U(1)^7\rtimes\Gamma_{\!\text{oct}}$ ($\Gamma_{\!\text{oct}}\cong PSL(2,7)$, frame-breaking theorem). [И]: the UHM instance of number–phase conjugacy = SLU's $U(1)$-from-$\Delta N\in\mathbb Z$ mechanism, channel-resolved — check 4 of the T-258 dictionary derived on the UHM side. Machine-verified: $\dim\ker=7$ exactly, covariances $10^{-16}$, periodicity $10^{-16}$, irrational witness $0.044>0$	Lindblad operators	T-258, T-102, T-11.2, Fano channel
T-261	Regeneration = natural-gradient descent of free energy (BKM) [Т]: for full-rank $\Gamma$ the replacement flow $\dot\Gamma=\kappa_{\text{eff}}(\rho_*-\Gamma)$ is exactly the constrained natural-gradient descent of $F(\Gamma)=D(\rho_*\|\Gamma)$ in the Kubo–Mori metric: $\operatorname{grad}_{\text{BKM}}D(\rho_*\|\Gamma)=\Gamma-\rho_*$, proof by three exact identities ($dF(X)=-\mathrm{Tr}(X K_\Gamma(\rho_*))$ with the BKM kernel; $g_{\text{BKM}}(X,Y)=\mathrm{Tr}(X K_\Gamma(Y))$; $K_\Gamma(\Gamma)=\mathbb 1$ ⇒ trace-dual $=\Gamma$, Lagrange $\lambda=1$); H-theorem $dF/dt=-\kappa\|\rho_*-\Gamma\|^2_{\text{BKM}}\le 0$. Sharp metric attribution: NOT the Bures gradient off the commuting locus (cos $\approx 0.98$) — Bures serves estimation/learning (Char-III/IV), BKM serves dissipative relaxation. Derives the dynamical law of the $h^{(R)}$-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 $10^{-15}$ non-commutative, $K_\Gamma(\Gamma)=\mathbb 1$ to $10^{-14}$, H-theorem $7\cdot 10^{-5}$ (FD)	Evolution §3	T-258, T-96, T-125, Formalization of φ
T-262	Dynamical trichotomy: $\mathcal L_\Omega$ as an exact reversible ⊕ irreversible (metriplectic) decomposition [T]+[С]: 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. $\mathbb 1/7$ (self-adjoint jumps — the Carlen–Maas precondition), giving $-\tfrac16\sum_p\gamma_p[\Pi_p,[\Pi_p,\Gamma]]$ exactly (element-wise the single-incidence count $4$ of the rank-7 law) $= -\mathcal K^W_\Gamma(\ln\Gamma)$ — the Carlen–Maas gradient flow of negentropy $D(\Gamma\|\mathbb 1/7)$, with $\mathcal K^W_\Gamma(A)=\tfrac16\sum_p\gamma_p[\Pi_p,\Lambda_\Gamma([\Pi_p,A])]\succeq 0$ via the one-line chain rule $[X,\Gamma]=\Lambda_\Gamma([X,\ln\Gamma])$; exact EPR quadratic form $\ge 0$, $=0$ 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 $[H_{\text{eff}},\rho_*]=0$ [C]; the 2nd degeneracy (dissipation conserves $\langle H\rangle$) fails by design — an open holon exchanges energy (witness $\approx 0.53$). 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 $\mathcal L_\Omega$. Machine-verified: DBC $3.6\cdot10^{-15}$, Jacobi $9\cdot10^{-15}$, identities $10^{-15}$–$10^{-16}$, EPR $\ge 0.42$ off-diagonal, unitary isometry $4\cdot10^{-16}$	Evolution §3	T-261, T-258, T-110, T-259, Line temperatures
T-263	Existence and uniqueness of the optimal learning flow [T]+[С]: the replacement flow $\dot\Gamma=\kappa_{\text{eff}}(\rho_*-\Gamma)$ is optimal in four stacked senses — (1) unique steepest descent of $F=D(\rho_*\|\Gamma)$ among equal-BKM-speed directions (Cauchy–Schwarz; witness $0/500$); (2) exact solution = mixture geodesic $\Gamma(t)=\rho_*+e^{-\kappa t}(\Gamma_0-\rho_*)$, direction-constant gradient, maximal exponent $\kappa_{\text{eff}}$; (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 $a=1$ (Vanchurin class $g(\kappa)=\kappa^a$), Cramér–Rao saturation via Bures/Char-IV, multiparameter attainability = Holevo within $\times 2$ [C]. Ceilings = T-109–T-112; minimal substrate $N=7$ (T-113). No-free-lunch not violated: environment class fixed by $G_2$/Fano architecture. Machine-verified: steepest $0/500$ (margin $0.49$), m-geodesic $5\cdot10^{-17}$, gradient FD $8\cdot10^{-9}$	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 $(i,j)$-block the phase-direction QFI is $\mathrm{QFI}(\theta_{ij})=4\|\gamma_{ij}\|^2/(\Gamma_{ii}+\Gamma_{jj})$ exactly at any population imbalance ($R$-cancellation; machine ratio $1.000000$; full-$\Gamma$ correction $O(\bar\varepsilon)$, $\le 4.4\%$ at $\bar\varepsilon=0.02$; unconditional sandwich $2\|\gamma_{ij}\|^2/\lambda_{\max} \le \mathrm{QFI} \le 2\|\gamma_{ij}\|^2/\lambda_{\min}$ [T] with explicit Gershgorin constant $7\rho_{\max}/(1-7\rho_{\max})$, witness $0/400$); (b) at vacuum populations $G_N^{(ST)}\cdot\langle\mathrm{QFI}(\theta_{\mu\nu})\rangle_{\mathrm{ST}}=56\pi\mu^2=8\pi N\mu^2\|_{N=7}$ — gravitational coupling × Fisher learnability of spacetime phases = architectural constant; (c) $\Lambda$-side: $\Lambda_{\text{phys}}\propto\mathcal G_O^2$ (T-254 [T]), read as squared clock-phase unlearnability [I]. Corollaries: $G\to\infty$ exactly where $\langle\mathrm{QFI}\rangle\to 0$ (information-theoretic mechanism for the §3.1 decoherence-gravity prediction); SLU slogan "gravity = learning efficiency" acquires sign and sectors — $G^{-1}\propto$ learnability(space), $\Lambda\propto$ unlearnability$^2$(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 [Т-structural]+[С]: the physical quark-mixing matrix is exactly $3\times3$ unitary, so the $\sim3.2\sigma$ first-row deficit ($\|V_{ud}\|^2+\|V_{us}\|^2+\|V_{ub}\|^2=0.9985(5)$) cannot be a mixing-matrix effect in UHM — every leading BSM channel is excluded by the fixed spectrum: 4th generation [T] ($N_{\text{gen}}=3$, $\mathrm{QR}(7)=\{1,2,4\}$, unique order-3 subgroup of $\mathbb Z_7^\ast$), vector-like quarks [Т-struct] (chirality $\gamma_5=i\Gamma_O\Gamma_A\Gamma_S\Gamma_D$ definite on $\chi_{\text{int}}$), MeV sterile $\nu$ [C] (seesaw $\nu_R=(1,1)_0$ at $M_R\sim3\times10^{14}$ GeV, normal hierarchy), leptoquarks/extra bosons [Т-struct] (unique $\{A,E,U\}$ Higgs line + SM gauge content). ⟹ UHM predicts the anomaly resolves in the SM extraction sector ($\gamma W$-box/nuclear radiative corrections $\Box_{\gamma W}^A=3.90(9)\times10^{-3}$, lattice $K/\pi$ form factors, $K$–$\pi$ $V_{us}$ tension), not via new states; self-consistent with the $C_{\text{norm}}$-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 $\nu$ / leptoquark	CKM §10	$N_{\text{gen}}=3$ (T-52), chirality, neutrino seesaw, falsifiability F-Cabibbo
T-266	The Universe's stage: at the terminal attractor to $\sim10^{-60}$ (H1.2 value-closure) [Т-structural]+[С]: Part A [Т-struct] — near $\rho^*$ every smooth functional of $\Gamma$ inherits the mixture-geodesic envelope $e^{-\kappa\tau}$ (T-263), so the drift law (T-254) gives the fractional stage-distance $1-\mathcal{G}^\infty/\mathcal{G}_O = \tfrac{3H_0}{2\kappa}(1+w_0) \ll 1$ for any microphysical $\kappa\gg H_0$ — the Universe sits at its terminal stage and the DESI drift $\|1+w_0\|\sim0.09$ is the $\kappa/H_0\sim10^{58}$-amplified residual; Part B [C] — $\kappa=\kappa_{\text{bootstrap}}=\omega_0/7$ (T-59, regeneration-limited) $+\ \omega_0\approx2.4\times10^{-2}M_{\text{Pl}}$ (neutrino $M_{G_2}=\omega_0\sqrt{\mathcal{G}_O/2}\sim10^{17}$ GeV) ⟹ residual $\approx4.7\times10^{-60}$; with $P^*=3/7$ (T-124 attractor, Universe-as-viable-holon) $P(\text{today})=3/7$ to $\sim59$ figures. Closes the value of H1.2 [H]→[С]: stage derived ($3/7$) and measured (residual $=\tfrac{3H_0}{2\kappa}(1+w)$ read from the DESI drift); explains $w\approx-1$ as relaxation onto the de Sitter attractor. Co-drift $\dot G_N/G_N\approx2.6\times10^{-6}H_0$ under LLR. Machine-checked: $\kappa/H_0$, fractional distance, identity $(2\kappa/3H_0)\cdot\text{frac}=1+w_0$, robustness. Residual [О]: exponent's $\omega_0$-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 ($M_{G_2}$), epistemic-vertical H1.2
T-267	The Tegmark decoherence objection does not constrain Γ (closure of Vulnerability #5) [T]+[С]: Tegmark bounds the lifetime of a microscopic position-basis superposition; $\Gamma$ 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 ($\mathrm{Gap}=\|\sin\arg\gamma\|$ 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 $H(7,4)$/associator/$V_{\text{Gap}}$/Lawvere/$\pi_1$, topological-protection [T]), driven-dissipative regeneration ($\kappa_{\text{bootstrap}}>\gamma_{\text{dec}}$). 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 $\Gamma$ (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 $\dim\mathrm{States}(S) \geq 7$), which rule out by construction (i) systems with $\dim\mathrm{States}(S) < 7$ (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 $G_V$, exactly faithful on the embedded 7-sector — the maximal faithful domain, since global injectivity is impossible for $\dim > 7$ (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]+[С]: octonionic Hermitian matrices $\mathcal{H}_n(\mathbb{O})$ form a formally real Jordan algebra iff $n \leq 3$ (Jordan–von Neumann–Wigner 1934); $\mathcal{H}_4(\mathbb{O})$ fails (non-associativity breaks the Jordan identity). This JvNW ceiling coincides with $\mathrm{SAD}_{\max}=3$, a third, independent derivation alongside the dynamical (purity $P>1$ at $n=4$, T-142) and categorical (tricategorical-coherence breakdown, T-217) ones — all three shadows of the single fact that $\mathcal{H}_4(\mathbb{O})$ is not Jordan (the composition analogue of the triple-forced $N=7$). Coordination symmetry climbs $G_2\,(14)=\mathrm{Aut}(\mathbb{O}) \subset F_4\,(52)=\mathrm{Aut}(\mathcal{H}_3(\mathbb{O}))$, structure group $E_6\,(78)$. [C]: identifying composition-depth with Jordan rank (vs the corpus default $\iota_{\max}$ tensor $\mathbb{C}^{7k}$, T-218) is a structural reading, not yet functorial; does not collide with T-220 (base is irreducibly $G_2$: single-holon states $\mathbb{C}P^6$, $\chi=7\neq3=\chi(\mathbb{O}P^2)$ — $F_4$ 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: $\mathbb{O}P^2$ is the maximal subject's expressive ceiling [T]+[И]: the octonionic projective plane $\mathbb{O}P^2$ (real dim 16, rank-one idempotents of $\mathcal{H}_3(\mathbb{O})$, isometry $F_4$) is the terminal projective geometry over any division algebra — no $\mathbb{O}P^n$ for $n\geq3$ (Desargues' theorem forces the coordinate ring associative; $\mathbb{O}$ is not; $\mathbb{O}P^2$ is the unique non-Desarguesian Moufang plane). ⟹ a maximal ($\mathrm{SAD}=3$) 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 $\mathrm{PG}(2,2)$ (smallest projective plane) → max subject $=\mathbb{O}P^2$ (largest division-algebra plane). Consistent with T-220's use of $\chi(\mathbb{O}P^2)=3$. [I]: the "state space $=\mathbb{O}P^2$" 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 [Т-struct]+[С]: TALOS/SYNARC scale along two octonion-generated axes only. (I) Expressiveness — per-subject, bounded: composing toward the JvNW ceiling climbs the exceptional series $G_2(14)\to F_4(52)\to E_6(78)$; coordination/motor bandwidth = dim of the symmetry group; hard-capped at $E_6/\mathbb{O}P^2$ (T-268/T-269). (II) Throughput + collective — across-subject, unbounded: federating maximal subjects continues the Freudenthal–Tits tower $E_6\subset E_7(133)\subset E_8(248)$ 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 $=\mathcal{D}$-channel (Enc, T-100), action $=H_{\text{eff}}$-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 $\mathcal{L}_\Omega$: regeneration is negentropy; consciousness holds entropy below heat death [T]+[С]: the von Neumann entropy $S=-\mathrm{Tr}(\Gamma\ln\Gamma)$ splits cleanly across the three terms of $\mathcal{L}_\Omega$. (i) [T] unitary $\dot S=0$ exactly (spectrum-preserving; $[\Gamma,\ln\Gamma]=0$). (ii) [T] dissipator $\mathcal{D}_\Omega$ is a strict entropy source (drives toward heat death $I/7$, $S=\ln 7$). (iii) [T] at any steady state $\dot S=0$, so regeneration $\dot S_{\mathcal{R}}=-\dot S_{\mathcal{D}}\leq 0$ — a net entropy sink (negentropy = the cost of maintenance). (iv) [C] steady entropy $S_{ss}<\ln 7$ strictly and monotone-decreasing in $\kappa/\gamma$; since $\kappa\propto\mathrm{Coh}_E$, 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 $R=1/(7P)$ literally measures distance from heat death and why the viability window sits away from $I/7$; 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 + $\mathrm{Coh}_E$-scaling, not a global reversal. Machine-verified	Origin §entropy-law	$\mathcal{L}_\Omega$ (T-57), $\kappa_0$, T-266, T-254, self-observation $R$
T-272	The Source $\Gamma_\odot$ is forced, not arbitrary: the unique maximally-coherent / $S_7$-symmetric pure state [T]+[П residual]: the primordial state $\Gamma_\odot=\|\psi_\odot\rangle\langle\psi_\odot\|$, $\|\psi_\odot\rangle=\tfrac1{\sqrt7}\sum_i\|i\rangle$, is characterised twice, each forcing it uniquely: (a) the $S_7$-invariant (permutation-symmetric) subspace of $\mathbb{C}^7$ is exactly 1-dimensional (spanned by $(1,\dots,1)/\sqrt7$) — the unique pure state privileging no dimension; (b) it is the unique pure state of maximal coherence $\mathrm{Coh}=1-\sum_i\|a_i\|^4=6/7$ (convexity maximum at $\|a_i\|^2=1/7$), all $\|\gamma_{ij}\|=1/7$. So the amplitude $1/\sqrt7$ is normalisation, not a free parameter — answering the standing open question "why $1/\sqrt7$?". Residual [P]: why the pure maximally-symmetric class (vs the mixed $I_7/7$, which has zero coherence) as initial condition — though maximal coherence is the natural selection principle that singles $\Gamma_\odot$ out. Machine-verified	Origin §the-source	Source $\Gamma_\odot$ [P], Source-instability [T], octonionic $N=7$
T-273	Metabolic floor of a viable coherent machine [T]+[С]: a corollary of T-271 + Landauer with direct engineering content. At steady state a viable holon ($P>2/7$, $\Gamma\neq I/7$) exports entropy at the dissipator's strictly positive rate $\dot S_{\mathcal{D}}>0$ (T-271 iii); by Landauer the minimum maintenance power is $P_{\text{meta}} \geq kT\ln 2\cdot \dot S_{\mathcal{D}} > 0$ strictly, where $\dot S_{\mathcal{D}}$ is the physical entropy-production rate (frequency-independent — T-276 corrects an earlier $f$-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 $\ln 7 - S$ from $I/7$): more order ⟹ higher $\dot S_{\mathcal{D}}$ ⟹ higher $P_{\text{meta}}$ — 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 $=0$ (reversible compute) $+\ P_{\text{meta}}$ (maintenance). Measurable from telemetry ($\dot S_{\mathcal{D}}$ is a CC observable). Numbers @300 K: order femto–pico-watts per holon, set by the physical rate $\dot S_{\mathcal{D}}$ (not the clock; see T-276). Machine-verified	Origin §entropy-law	T-271, Landauer, TALOS §6 (energy), $\mathrm{Coh}_E$
T-274	The operating-point axis of a coherent machine [С]: the metabolic floor (T-273) turns the viability window $(2/7,3/7]$ into a capability–efficiency design axis, not a single set-point. Lean edge $P\to2/7$ = efficiency-optimal (minimal maintenance power, minimal margin; highest capability-per-watt for $C=\Phi\times R$; the "survival" mode). Rich edge $P\to3/7$ (the T-124 attractor) = capability-optimal (max $C=2/3$, 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 $P=3/7$, but an engineer can run leaner toward $2/7$ to trade capability/margin for power. The optimal set-point is measure-dependent (with $C=\Phi\times R$ efficiency favours the lean edge; with a dynamic-range measure $P\approx0.47$) — 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 $P\to3/7$), T-140 ($C=\Phi R$)
T-275	The interaction inversion: strong/weak/EM forces are derived sub-structures of the coherence symmetry [Т for the embedding]+[И]: the same $G_2=\mathrm{Aut}(\mathbb{O})$ that governs a single holon contains the Standard Model gauge group, $G_2\supset SU(3)\times SU(2)\times U(1)$ ([T], standard group theory). So the strong, weak and electromagnetic interactions are not fundamental inputs but derived sub-structures of the coherence symmetry. The pre-interaction layer ("before" the forces) is the triple $(\Gamma,\ G_2,\ \hat{\mathcal{G}}=\mathrm{Im}\,\Gamma\in\mathfrak{so}(7))$ — coherence matrix, octonionic symmetry, and the Gap operator (phase/meaning; needs complex $\gamma$, T-132). Inverts the reductionist arrow [И]: standard physics runs forces→particles→(mind?); UHM runs topos/coherence→$G_2$→forces-derived, with the same $\Gamma$ 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). Consolidates the SM derivation: the embedding is [T] (established), the inversion framing [I]	Math foundations §algebra	$G_2\supset$ SM [T], T-132 (Gap needs complex Γ), axiom-omega §primitive, Gap operator
T-276	The efficiency law of a viable coherent machine [T]+[С]: two refinements completing the thermodynamic budget (T-271/T-273/T-274). (i) Frequency-independence [T]: the metabolic floor is $P_{\mathrm{meta}} \geq kT\ln 2\cdot\dot S_{\mathcal{D}}$, set by the physical entropy-production rate — the per-tick entropy scales with the step $\Delta\tau$, 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 "$\sim$pW@GHz" conflated tick-rate with physical rate). (ii) Order-per-joule [T]: creating negentropy costs $\geq kT\ln 2$ per bit, so regeneration efficiency $\eta = (\text{order created})/(\text{free energy spent}) \leq 1$ (Landauer-bounded). (iii) Speed–efficiency tradeoff [C]: $\eta\to 1$ for quasi-static regeneration and falls as the update accelerates (irreversibility) ⟹ engineering rule: run regeneration as slowly as the dissipator leak permits ($\kappa$ just above $\gamma$) 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]+[И]: from three non-numeric primitives (distinction $\Omega$ [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) $\mathrm{CD}^k(\mathbb{R})$ = twisted group algebra $\mathbb{R}_F[\Omega^k]$, $e_a e_b = F(a,b)e_{a\oplus b}$ — one mirror step = one new $\mathbb{Z}/2$-grading [Т, construction + machine]; (ii) viability holds iff $k\leq 3$ (Hurwitz 1898; sedenion witness $(e_1{+}e_{10})(e_4{-}e_{15}){=}0$) [T]; (iii) the terminal distinction-spectrum is $\Omega^3\setminus\{0\} = \mathrm{PG}(2,2)$, count $2^3{-}1=7$ [T]. So $2$ = arity of distinction, $3$ = viability ceiling, $7$ = derived cardinality — the "7 before number" objection becomes a theorem. Does not eliminate A1 (physical instantiation as $\mathcal{D}(\mathbb{C}^7)$ 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]+[И]: at every viable stage $k\leq3$, conjugation / commutativity / associativity fail exactly on $\mathbb{F}_2$-independent 1-/2-/3-tuples of grades; closed form: associator $\Phi(a,b,c) = (-1)^{\det_{\mathbb{F}_2}(a,b,c)}$ (all 512 triples; Albuquerque–Majid 1999 re-derived). Clifford gauge: $\mu = f_{\mathbb{O}}\oplus f_{\mathrm{Cl}}$ has $d\mu = \det$ — octonions and $\mathrm{Cl}(0,3)\cong\mathbb{H}\oplus\mathbb{H}$ differ by a gauge whose field strength is the volume form; with Frobenius+Hurwitz: division $\oplus$ associativity — pick one (the price of division is the volume twist). In the twisted category $\mathrm{Vect}^{\Omega^3}$ the octonions are the trivial object (group algebra) — laws as gauge fields (Drinfeld-twist machinery one level deeper); supermathematics = the bilinear $\Omega^1$ shadow [I]. Machine-checked	Hypermathematics §закон-объёмов	T-277, T-217, FANOS third order
T-279	The machinery of dimensions: the stabilizer tower in $G_2$ [T]+[И]: stab(one axis) $=\mathfrak{su}(3)$ (dim 8, center 0, rank 2; sky $S^6=G_2/SU(3)$; pencil of 3 triads per axis); stab(one coherence-pair) $=\mathfrak{u}(2)$ (dim 4, center 1) and fixes the mediator (Lemma: $D\,\mathrm{span}\{e_a,e_b\}\subseteq\mathrm{span} \Rightarrow De_{ab}=0$ — full proof via skewness + Leibniz; machine $10^{-17}$); stab(one Fano line) $=\mathfrak{so}(4)$ (dim 6). Why $\mathfrak{su}(3)$: multiplication by the axis $J_u=e_u\cdot$ is a complex structure on its sky ($J_u^2=-1$; stab commutes with $J_u$, machine $10^{-16}$) — each dimension sees the other six as $\mathbb{C}^3$; 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 [И]: forces as stabilizer machineries of the incidence geometry (strong = one axis, $\mathfrak{u}(2)$ = one coherence), riding the [T] embedding of T-275. Machine-checked	Hypermathematics §машинерия-измерений	T-275, T-257, T-256
T-280	The law of death: anatomy of the fourth mirror [T]+[I]+[С]: the 15 hyperplanes of $\Omega^4$ split exactly as: old octonions $V_0$ + 7 straight extensions $L\oplus\langle e_8\rangle$ of Fano lines (all viable, $\cong\mathbb{O}$) + 7 skew extensions $L\oplus\langle e_8\oplus u\rangle$, $u\notin L$ (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 $\Phi=(-1)^{\det}$ 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 [Т]: the Mac Lane pentagon closes on all $16^4$ sedenion quadruples since $\Phi=dF$ ⟹ the death is NOT a categorical-coherence failure — the object dies, the category survives). Alternativity dies, flexibility survives at $k{=}4$ [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 [Т]: on the terminal cube, among ALL monomial unital algebras $\mathbb{R}_F[\Omega^3]$ with anisotropic diagonal $e_a^2=-1$ (forced by T-244): (i) a commuting independent pair yields the explicit annihilator $(e_a{+}e_b)(e_a{-}e_b)=0$ ⟹ 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 $F(p,r)F(q,s)F(p,s)F(q,r)=-1$ — a system of 84 linear equations over $\mathbb{F}_2$ (viability is LINEAR); (iii) exhaustive solution: exactly $16=2^{7-3}$ fields survive = one $\lambda$-gauge orbit, each with $dF=\det$ and full composition ⟹ no annihilations ⟺ $dF=\det$ ⟺ $\mathbb{O}$. 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 $\mathbb{R}$ remains). Sharpenings: the associative fiber $dF=0$ 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 $\mathfrak{g}_2(14)\supset\mathfrak{su}(3)(8)\supset\mathfrak{u}(2)(4)\supset\mathfrak{su}(2)(3)\supset 0$ — three independent distinctions pin all of $G_2$ (simply transitive on basic triples). Gauge count = code theory: ineffective gauges = the $[7,3]$ simplex code (dual Hamming), orbit $=2^7/2^3=16$ [T]. Also [Т]: Mac Lane pentagon closes on all $16^4$ sedenion quadruples ($\Phi=dF$) — coherence survives $k{=}4$, 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]+[И]: assemble the viability system on $\Omega^n$ (anisotropy + anticommutation + rectangle rules = "no simple zero divisors", valid over ANY field of char $\neq 2$). Feasible for $n=1,2,3$ with solution-space dims $0/1/4$ = exactly the gauge orbits of $\mathbb{C}/\mathbb{H}/\mathbb{O}$; INFEASIBLE for $n=4$ ($960$ equations, $225$ unknowns, rank $214$) and hence for all $n\geq4$ (subcube restriction; $n=5$ verified directly, $7936\times961$). ⟹ Hurwitz's boundary, in the monomial class, is the inconsistency of a finite $\mathbb{F}_2$-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 [Т]: stage $k$ of the mirror tower is viable over a field $K$ (char $\neq2$) iff the unit form of dim $2^k$ is anisotropic iff $s(K)\geq 2^k$ (field level). Proof pair: "⟸" composition ($N(xy){=}N(x)N(y)$ — polynomial identity, machine mod 3 + Hurwitz-cited) + anisotropy; "⟹" $x\bar{x}=N(x)e_0$ exactly [machine] ⟹ an isotropic vector IS a zero divisor. Witnesses: $\mathbb{C},\mathbb{F}_5$ ($s{=}1$) die at $k{=}1$ ($(e_1{+}i)(e_1{-}i){=}0$); $\mathbb{F}_3$ ($s{=}2$) lives at $k{=}1$ (exhaustive) and dies at $k{=}2$ ($1{+}1{+}1{=}0$); $\mathbb{F}_7$ dies at $k{=}2$. Pfister's power-of-two levels ($1,2,4,8,\dots$) = the mirror ladder on the arithmetic side; all three mirrors force $s(K)\geq 8=2^3$. Honest boundary: orderability is not forced (level-8 fields exist, Pfister) — $\mathbb{R}$ is the terminal $s{=}\infty$ (Artin–Schreier) case; H3.6 sharpened to the step "level $\geq8$ → ordered complete $\mathbb{R}$". Machine-verified	Hypermathematics §уровень-поля	T-277, T-281, Artin–Schreier, Pfister
T-284	Uniqueness of the base: the $\mathbb{R}$-chain [T]+[С]: the base field of the mirror tower is pinned uniquely by requirements the corpus already carries: (1) viability of all three mirrors ⟹ $s(K)\geq 8$ (T-283) [T]; (2) formal reality of the observable layer ($\sum a_i^2=0\Rightarrow a_i=0$ — the same hypothesis as the JvNW ceiling, T-268, now applied downward) ⟹ $K$ formally real ⟹ orderable by Artin–Schreier (the order is constructed, not assumed) [Т-cited]; (3) continuous one-parameter $\mathcal{L}_\Omega$-dynamics ⟹ Dedekind-complete Archimedean scalars [П/С — the corpus's continuous-time postulate, status declared]; (4) the unique Dedekind-complete Archimedean ordered field is $\mathbb{R}$ [Т-classical]. ⟹ non-dying + formal reality of observables + continuous time ⟹ base $=\mathbb{R}$, 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]+[И]: H3.5 asked to ground the viability boundary in stable homotopy ($\mathbb{S}$, Adams, Bott). Closed in three steps: (i) the internal boundary is elementary — T-282's death matrix is pure $\mathbb{F}_2$ combinatorics (machine: identical under independent constructions; no field/topology/analysis inside); (ii) monomiality lemma [T]: an $\Omega^3$-grading with 1-dimensional components (= the full register of distinctions) forces monomial multiplication ($A_aA_b\subseteq A_{a\oplus b}$, $\dim1$) ⟹ 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 $S^7$ 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 [Т]: the last premise of the $\mathbb{R}$-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; the ouroboros canon $\rho^*=\varphi(\Gamma)$, T-222) ⟹ IVT ⟺ Dedekind completeness [Т-classical] ⟹ Archimedean [Т: sup of finite elements] ⟹ $K\cong\mathbb{R}$ unique [Т-classical]. Machine witness on the incomplete side: over $\mathbb{Q}$, $f(x)=\tfrac{x+1}{2}$ below $1/\sqrt2$, $\tfrac{x}{2}$ above — continuous on $\mathbb{Q}$, maps $[0,1]$ to itself, $\|f(x)-x\|>0.146$ everywhere (exact rationals) yet $f-x$ changes sign: the snake jumps its tail through the hole. ⟹ T-284 re-founded with NO temporal premise: viability + formal reality + guaranteed ouroboros ⟹ base $=\mathbb{R}$; continuous time becomes an output (the flow $e^{t\mathcal{L}}$ is well-defined because scalars are complete). H3.6 closed at the theory's axioms. Machine-verified, 6/6	Hypermathematics §уровень-поля	T-284, T-222, T-283, IVT⟺completeness
T-287	Internalizability of the genesis [Т-meta]+[С]: every construction of T-277–T-286 is finitary ($\mathbb{F}_2$-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 $\Omega$ [D]; the base $\mathbb{R}$ 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 $\mathcal{L}_\Omega$ [T] (minted for the open-system layer): the full autonomous Liouvillian $-i[H_{\mathrm{eff}},\cdot]+\mathcal{D}+\mathcal{R}$ with categorical regeneration anchored to $\rho_{\mathrm{ref}}=I/7$ 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 $G_H$ is connected $I/7$ is the unique attractor and $P(\tau)\to 1/7$. Unconditionally (2026-08-07): on a disconnected $G_H$ the holon does not die — it freezes block-wise at its starting purity (machine: $P=0.2915$ two components, $P=0.8117$ diagonal $H_{\mathrm{eff}}$, both $>P_{\mathrm{crit}}$) — but every branch dephases completely ($\Phi\approx 7\times10^{-18}$, $9\times10^{-16}$, $5\times10^{-48}$), so an isolated holon is never conscious whatever the graph; only the mechanism branches (death vs zero-coherence zombie). $\dim\ker\mathcal{L}_0$ = number of connected components of $G_H$ [T, machine-verified for 1/2/5/6/7 components]; the $g_V$ gate switches regeneration off at $P=P_{\mathrm{crit}}$. 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 $P=1/7$ within HS-distance $3\times10^{-9}$	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 ($P(\rho_{\mathrm{env}})>P_{\mathrm{crit}}$) at rate $r$ yields a NESS $\rho^\star(r/\gamma)$ with purity monotone in $r/\gamma$, an ignition threshold $(r/\gamma)_c$ above which the holon is viable [Т via dissipative state preparation, Verstraete–Wolf–Cirac 2009]; threshold value model-dependent [C], numerically $(r/\gamma)_c\approx 7$ 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 [Т]: 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 $\Gamma$ [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, $I(\Gamma;X)\le I(jd;X)$ for any property $X$ of the person. Every external test measured the right-hand side and returned null — Gauquelin 15 931 accurate-time celebrities ($\eta^2<0.5\,\%$), 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, $+3.03$) ⟹ 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 $\Gamma$→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 [Т]: any stationary point $\sigma$ of the canonical $\mathcal{L}_\Omega$ with $P(\sigma) > 1/7$ has both flows nonzero — $\mathcal{R}[\sigma] \neq 0$ and $\mathcal{D}_\Omega[\sigma] \neq 0$; 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, $(\mathcal{D}+\mathcal{R})[\sigma]_{jk} = i\,\omega_{jk}\gamma_{jk}$ — a sounding coupling is an orbit. Proof: 4 lemmas from the canonical form (no pump ⟹ $I/7$; $\sigma \neq I/7$ ⟹ dissipator nonzero; diagonal of the unitary term vanishes; stationarity off-diagonal). Instrument [С]: life/death fold $\omega_0^* \approx 19.5$, decomposing as $\Lambda^* g_D \times$ rotation surcharge ($\Lambda^* = 50.5$ on the invariant ray — closed form, intervention-verified 1.000; surcharge ×1.93 at canonical $g_D$ — the price of rotation); orbit identity 0.9999–1.0000; two-stroke balance to machine zero; critical slowing ×26 toward the fold. Comparative [И]: 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 $g_V$ [Т]; Schnakenberg-class NESS cycle structure
T-292	Regeneration lives on the gap — the self-model as gradient [Т]: 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 [С]); (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 [С]). 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 [С].	evolution#теорема-эго-градиент	T-39a primitivity, g_V gate [Т], turnover T-291

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Level [C]: Sensorimotor Theory

| T-293 | The learning algorithm of a holon is natural gradient [Т]: with $g$ the Bures/SLD metric (forced by A2, T-187) and $\kappa$ the covariance of one-step Kraus increments of the canonical dissipator (forced by T-41/T-59), the population-sector identity $\Pi\,g\,\kappa_{\text{at}}\,\Pi=\frac{\gamma}{4N}\Pi$ holds exactly, i.e. $\kappa^{\uparrow\uparrow}_{\text{at}}=\frac{\gamma}{4N}g^{-1}$. Three lemmas: (a) the atomic Kraus covariance is exactly multinomial, $\kappa_{\text{at}}=\frac{\gamma}{N}(\operatorname{diag}\lambda-\lambda\lambda^{\top})$, which is simultaneously the inverse Fisher metric on the simplex; (b) on commuting perturbations Bures is $\tfrac14$ Fisher–Rao (not $\tfrac12$ — corrected v2.0; the SLD-QFI normalisation gives $\gamma/N$, and the exponent is normalisation-free); (c) the dissipator annihilates every diagonal state, so $\kappa$ is a genuinely centred covariance. Substituting $a=1$ into Vanchurin's own Eq. 7.5 (arXiv:2603.15198) gives verbatim $g^{-1}=g^{-1}\kappa g^{-1}$ — hence $a=1$, natural gradient, not the $a=\tfrac12$ conjectured for biological complexity. Combined with his maxent identity 4.7 ($g^{-1}=c$): $\kappa^{\uparrow\uparrow}=\frac{\gamma}{4N}c$ — the covariance of temporal changes has the same shape as the static population covariance. Machine: 200 random states, spectral constancy to $8.3\times10^{-17}$; 2000 states for (a) to $2.8\times10^{-17}$ | Learning algorithm of a holon | T-187 (why Bures), T-41g–i (Fano channel), T-59 | | T-294 | Universal Fano factor $11/9$ [Т]: at matched per-channel rate, adding the seven Fano projectors to the atomic channel multiplies the metric-weighted trace by exactly $\frac{11}{9}$, independently of the state and of the metric normalisation — equivalently the block layer carries exactly $\frac2{11}$ of the noise. In Bures normalisation $\operatorname{Tr}(g\kappa_{\text{at}})=\frac{3\gamma}{14}$, $\operatorname{Tr}(g\kappa_{\text{full}})=\frac{11\gamma}{42}$. Closed form for a general design: the block ratio is $\frac{b-r}{k(v-1)}$, giving $\frac{7-3}{3\cdot6}=\frac29$ for BIBD$(7,3,1)$, $\frac13$ for $(9,3,1)$, $\frac3{16}$ for $(13,4,1)$ — all machine-verified; the block layer is separately trace-preserving only when $r=k$, true for Fano and $\mathrm{PG}(2,3)$ but not $\mathrm{AG}(2,3)$. 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 $I/7$; at the window centre the spectrum is $\{\tfrac{16}{15},\tfrac{16}{15},\tfrac{11}9,\tfrac{11}9,\tfrac{11}9,\tfrac{23}{15}\}\cdot\frac{\gamma}{4N}$). Scope: the full canonical dissipator coincides with the atomic one at rate $5\gamma/3$, so the Fano layer is invisible in the master equation and $\kappa$ is defined relative to the canonical Kraus resolution (L-unification), not unravelling-invariant; at matched generator the ratio is $\frac{11}{15}$. Machine: 300 random states, ratio to $4.4\times10^{-16}$ | Learning algorithm of a holon | T-41c (optimal $k$), T-41i (Fano optimality) | | T-295 | Noise–purity law and the sector split [Т]: (a) $\operatorname{Tr}\kappa_{\text{at}}=\frac{\gamma}{N}(1-P)$ and $\operatorname{Tr}\kappa^2_{\text{at}}=\frac{\gamma^2}{N^2}(P+P^2-2S_3)$, $S_3=\sum\lambda_i^3$ — both metric-free; on the conscious window $P\in(2/7,3/7]$ the first gives the band $[4\gamma/49,\,5\gamma/49)$, and together they give the exact effective rank $r_{\text{eff}}=\frac{(1-P)^2}{P+P^2-2S_3}$. Correction (v2.0): $r_{\text{eff}}$ is not a function of purity alone, so the earlier interval $[3.59,4.22]$ holds only along the one-dominant-mode family; over the whole window $r_{\text{eff}}\in\approx[1.47,4.22]$. 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 $\kappa^{\text{coh}}=0$ exactly; off it $\kappa^{\text{coh}}=O(\|\rho_{\text{coh}}\|^2)$ and decays at $10\gamma/21$, since coherences themselves contract deterministically at $\lambda_{\text{deco}}=5\gamma/21$ (T-59, $49\times49$ spectrum exactly $\{0^{\times7},(-5\gamma/21)^{\times42}\}$). Consequence: the "quantum regime" is not the $a\to1$ 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 — $a=\tfrac12\Rightarrow\kappa^{\uparrow\uparrow}$ spherical, $a=1\Rightarrow\kappa^{\uparrow\uparrow}\propto\operatorname{diag}\lambda-\lambda\lambda^{\top}$; sphericity test, $\mathrm{df}=20$, needs $M\approx100$ aggregated windows (simulated size/power); resolving power vanishes at $I/7$ and grows across the window | Learning algorithm of a holon | T-59, T-124 (window), T-72 (scale invariance) | | T-296 | No second Higgs doublet [Т]: condensation requires the $\kappa_0$-channel (T-64 + T-42a); the only $\bar3$-pairs besides $(E,U)$ are $(L,E),(L,U)$, neither enters $\kappa_0$ ($\mathrm{Hom}(O,L)$ absent), and by $\lambda=1$ the pair $(L,U)$ lies on $\{D,L,U\}$ — the Color-U Yukawa channel, not a scalar sector. Hence exactly one condensing scalar $\gamma_{EU}$; the whole 2HDM/MSSM Higgs spectrum ($H^\pm,A^0,H^0$) is structurally excluded. Falsification: an LHC charged Higgs refutes the categorical uniqueness of $(E,U)$, i.e. $\kappa_0$ itself | Higgs sector §6.0 | T-42a, T-64, FE-uniqueness | | T-297 | Rank-4 prohibition: no gauge $Z'$ / fifth force [Т]: $G_{\mathrm{SM}}$ is the unique rank-4 group compatible with Fano+$G_2$ (FE-theorem); any extra gauge $\mathrm{U}(1)$ raises rank to 5, which the incidence structure does not admit. Collider/dark-sector searches for a gauge $Z'$ stay empty at any energy; discovery refutes FE-uniqueness. Notation guard: $Z'_\Phi(-2)$ of the $\Lambda$-budget is a zeta-regulator derivative, not a boson | Standard model, corollary | FE-uniqueness, T-42 | | T-298 | $\kappa_0$ flows through the suppressed lines [Т]: both morphism factors of $\kappa_0$ lie on O-lines — $(O,E)\in\{L,E,O\}$, $(O,U)\in\{U,O,S\}$ — two of the three suppressed (Temporal) Fano lines, and by $\lambda=1$ no other path exists. The hierarchy $\kappa_0\ll\omega_0$ is therefore geometric (incidence), not tuned; the third points $L,S$ 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 [Т-consequences]: from T-296 (no second doublet) + T-297 (no gauge $Z'$) + FE-uniqueness (no leptoquarks) + $N_{gen}=3$ (no light steriles) jointly: (a) zero BSM contribution to muon $g{-}2$ — experiment must converge to the full lattice SM value; (b) LFV at the neutrino-loop floor ($\sim10^{-54}$) — MEG-II/Mu3e/Mu2e see nothing; (c) exact lepton universality — the 2022 return of $R_K$ to SM is a post-factum pass; (d) EDM desert $d_e\sim10^{-38}\,e\,$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 $\Gamma$ [Т]: the generic $G_2$-orbit through a state is exactly $14$-dimensional (machine: 200 random states, $\min=\max=14$; at $I/7$ it is $0$). 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 $G_2$ is broken to a finite subgroup. All $48$ parameters stay measurable relative to a holon's own basis; the $14$ measures redundancy of the formulation, not of the state. Erratum same day: an earlier form of this row claimed quality space $=\mathcal D(\mathbb C^7)/G_2$ of dimension $28$ and a self-description ceiling of $34$ — both retracted; the counts stand, the gauge reading does not | Qualia mechanism | T-42a, Goldstone modes | | T-301 | The decoder: $\Gamma=27\oplus14\oplus7$ under $G_2$; the invariant phase channel is Fano holonomy [Т]: with $\Gamma=S+iA$ the antisymmetric part splits by contraction with the associative three-form (machine: $\dim\mathfrak g_2=14$, $\operatorname{rank}(A\mapsto\varphi\!\cdot\!A)=7$, so $21=14\oplus7$ exactly). Vertex phases are conventional, so $\operatorname{Im}\gamma$ is gauge-dependent; the invariant seven-component carrier is the Fano holonomy vector $H_p=\arg(\gamma_{ij}\gamma_{jk}\gamma_{ki})$ over the seven lines — gauge-invariant, identically zero when the phase field is a coboundary, and decaying with the phases at $5\gamma/21$. "Why this quality" is a question about $\varphi$-labelled invariants, i.e. the multiplication table is the decoder. Erratum same day: an earlier form named $v=\varphi\!\cdot\!\operatorname{Im}\gamma$ 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 [Т]: by T-295 all fourteen canonical Lindblad operators are diagonal, so $\kappa^{\text{coh}}=0$ exactly on the decohered manifold; since $\varphi$-vector quality lives entirely in the coherence sector (T-301a), 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 [Т]: the four gates are three independent quantities under four constraints, since canonical $R=1/(NP)$ makes $R\ge1/3$ identical to $P\le3/7$. Hence the outcome space is $3\times2\times2=12$ 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 ($R\to1/2$), at the upper edge structure outruns reflection ($R\to1/3$). 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 [С]: a holon types exactly $\binom{7}{2}=21$ non-overlapping channels, so a node carries at most $21$ subordinates — one channel each — and with reflection depth capped at three (T-142) the reach of one holarchy is $21^3=9261$ 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 $2$ and a ceiling of $21\cdot2^2=84$; an address stored as a declared contract spends a whole channel per child, giving branching $21$ and the full $9261$. The laboratory measures the separation (HL15): a declared-routing holarchy gains $26\%$ over the undivided holon at fan-out $2$ and $39\%$ at fan-out $7$ — 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 $2\\times2$ factorial separates the two ingredients (HL16): freezing the address alone recovers $19\\%$, freezing plus load-balance $30\\%$, 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 [С]: with a flat diagonal and coherences of equal modulus, content scaled to the very edge of positivity satisfies $\Phi_{\max} = 6/\lambda_{\min}(S)^2$ exactly, where $S$ is the sign pattern of the coherences — so the integration gate $\Phi \ge 1$ is the spectral condition $|\lambda_{\min}(S)| \le \sqrt{6}$. 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 $S_{ij} = s_i s_j$; 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 $0$ of $600$ patterns, and the identity held to $8.9\cdot10^{-16}$ (HL17). Integrable content is thus seven polarities, not twenty-one independent bits — one flipped agreement, five frustrated triangles of thirty-five, takes $\Phi$ from $6$ to $0.9365$ and closes the gate — and there are exactly $2^7/2 = 64$ 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 $100\%$ of the fourteen it was never shown and is right $85.7\%$ of the time, $+28.6$ pp over the best constant answer, while the same write without the projection reaches $0\%$ of them. Strip the polarity and accuracy falls to $50.0\%$, 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 [С]: the three cells of a line are three different pairs $\{i,j\}$, $\{j,k\}$, $\{i,k\}$, and under a polarity their signs satisfy $s_i s_j \cdot s_j s_k \cdot s_i s_k = (s_i s_j s_k)^2 = +1$. 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 $64$ 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 $52.4\%$ of cells reading negative — but given content that is not, it leaves the frustration standing ($4$ of $7$ still broken) and instead quietly rewrites part of what was stored (only $4$ of $7$ lines keep the verdict written into them). Positivity is a ceiling on frustration, through $\lambda_{\min}$, not a prohibition of it. Two consequences follow. First, the two uses of a line are incompatible: three equal signs multiply to $s^3 = s$, so a repeated negative verdict breaks parity by construction, and $k$ such lines leave exactly $k$ broken — a line serves as a repetition code only while its verdict is positive. Second, integration is not a frustration detector: $\Phi$ 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 [С]: the polarity law (T-305) is the real restriction of a phase law. Give each dimension an angle $\theta_i$ instead of a sign, and let a coherence carry the difference $\theta_i - \theta_j$; the polarity case is $\theta \in \{0, \pi\}$, 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 $0.0000$ rad against $\pi/2$ for a guess — while a write restricted to real evidence cannot represent it at all ($1.5708$ rad on untaught pairs and $0.8035$ even on taught ones, since a phase flattens to its sign). But a coboundary is pure gauge: it equals $U\Gamma U^{\dagger}$ with $U = \operatorname{diag}(e^{i\theta_k})$, 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 $\Phi \ge 1 \iff |\lambda_{\min}| \le \sqrt{6}$ carries over from signs to phases with zero disagreements, and at the crossing ($\lambda_{\min} = -2.4495$, $\Phi = 1.0000$) the median line holonomy is $0.6387$ 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 [С]: how much quality a state can hold is a scale-free spectral question — a line's holonomy $\varphi_{ij} + \varphi_{jk} + \varphi_{ki}$ depends on the phases alone, and the gate depends only on $\lambda_{\min}$ of the unit-modulus pattern (T-307), so the problem is to maximise holonomy subject to $|\lambda_{\min}| \le \sqrt{6}$. Two results follow, one positive and one negative. Alignment is worth a factor of two: giving every cell $\psi/3$ in the orientation of the single line it belongs to carries $1.2746$ rad against $\approx 0.65$ 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 $2.28$–$2.48$ rad, some $79$–$94\%$ above it, and the gain survives across all thirty-five triangles of $K_7$ ($+66\%$), so it is not a redistribution onto the seven named lines. Held even, a search still gains $25\%$. The optima are uneven — median spread $\approx 0.5$–$0.67$ across the seven lines against exactly $0$ for the aligned shape, one line saturated at $\pi$ while another sits near $1.45$ — so no symmetry acts on them (HL21). No maximal value is claimed: twelve climbs scatter by $47\%$, 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 $G_2$ 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 [С]: 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 $(i,j) \mapsto (\pi(i), \pi(j))$, and the content such an assignment induces is balanced exactly when the answers themselves factor as $u_i u_j$ — put $u_i = t_{\pi(i)}$ 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, $7! = 5040$ ways, and permuting does not make an unbalanced pattern balanced. Measured against the best such learner there is — all $2^7$ 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 [С]: a holarchy's reach grows as $21^3$ 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 $0.9856$, with the leaf below $17$ of $21$ it collapses to $0.5035$, 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 $(-1)^3$, so negating every coherence breaks all $35$ triangles at once, for every one of the $128$ polarities (HL23). A child sitting under a parent of sign $\sigma$ holds $\sigma V$, which is a polarity for $\sigma = +1$ and that complement for $\sigma = -1$; measured, such a leaf recovers its polarity perfectly in the first case and reaches $0.5704$ 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 [С]: 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 ($\lambda_{\min} \approx 0$) — which is the repair step's doing and not a fact about the world, since the matrix before repair lies outside the cone in $100\%$ of charts and a projection of an infeasible point lands on the boundary by definition — but the diagonal runs $2.74$ to one and the coherence moduli spread $17.4$-fold. The consequence is a gate that stands open where the content is frustrated — $\Phi \ge 1$ in $87.9\%$ of states while $14.68$ of the $35$ triangles are broken and not one state is balanced. Removing each non-uniformity weighs it: flattening the diagonal drops the open share from $0.921$ to $0.171$, and equalising the moduli drops it further to $0.021$. The second route is not concentration, which was the first reading and is wrong — sharpening a diagonal on its own lowers $\Phi$, from a median of $0.667$ to $0.239$, 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 $0.516$, and with it up to $11$ of the $35$ triangles may be frustrated while the gate stands open, against $0$ 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 $|\lambda_{\min}(S)| \le \sqrt{6} \iff \text{balanced}$. 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 $1.0053$ rad, with $64.6\%$ 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 [Т]: every state factors exactly as $\Gamma = D^{1/2} K D^{1/2}$ with $K$ a correlation matrix — unit diagonal, positive semi-definite. The factorisation separates two things that had been read together: all of positivity lives in $K$ and does not mention the diagonal at all (a congruence by the positive definite $D^{1/2}$ cannot change a signature), while all of the weighting lives in $D$. Integration then has a closed form, verified to $3\cdot10^{-16}$ over three hundred states and to $10^{-9}$ in the instrument: $\Phi = c_{\text{eff}}^2 (1-p)/p$, where $p = \sum_i d_i^2$ and $c_{\text{eff}}^2 = \sum_{i \neq j} K_{ij}^2 d_i d_j / \sum_{i \neq j} d_i d_j$ is how far the coherences run toward their own ceiling $|\gamma_{ij}| \le \sqrt{d_i d_j}$, weighted by population. So the gate is a single inequality, $c_{\text{eff}} \ge \sqrt{p/(1-p)}$, satisfiable in exactly three ways: bind harder everywhere, flatten the diagonal, or align — spend the binding on the populated pairs. Real states run at $c_{\text{eff}} = 0.4897$ against a threshold of $0.4367$; the alignment bonus is $+0.0190$ and positive in $85.3\%$ of them, and destroying it alone drops the share with an open gate from $0.877$ to $0.743$. The frustration bound falls out of the same factorisation: under uniform saturation $K = (1-c)I + c\Sigma$, positivity reads $\lambda_{\min}(\Sigma) \ge -1/c$, and with the gate this gives $\lambda_{\min} \ge -\sqrt{(1-p)/p}$ — exactly $-\sqrt 6$ 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 $6$ populations, $21$ coherence moduli and $21$ coherence phases. Reading the gates off their own definitions settles what they see — $P = \sum|\gamma_{ij}|^2$, $R = 1/(7P)$, $\Phi = c_{\text{eff}}^2(1-p)/p$, and $D = 1 + 6(d_E^2 + 2\sum_i|\gamma_{Ei}|^2)/P$ — and not one of the four reads a phase [Т]. 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 $0$ times in $2000$, and the freedom of a single phase has half-width $0.0000$ for $21$ of $21$ phases in every one of fifty states — because computed states are rank-deficient boundary points, spectrum $[0,0,0,0.0521,0.1582,0.2887,0.4816]$, 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 $12$ broken triangles of $35$ to $0$, agreeing with the untouched output on $1.3\%$ 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 $100\%$ of charts ($\lambda_{\min}$ median $-0.2808$). The size of the gap is exact. Of the twenty-one phases, six are pure gauge — $\gamma_{ij} \mapsto e^{i(\theta_i-\theta_j)}\gamma_{ij}$ 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 $7/15 = 46.7\%$ 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 $3051$ people with exact birth times and known death dates, the partial $R^2$ of age at death over birth year is $0.00077$ for the gate quantities ($p = 0.667$) and $0.00148$ for nine phase quantities the gates cannot read ($p = 0.884$), 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 [Т]: carrying the reading of the definitions to the bottom, write $s_1 = \sum_i d_i^2$ for the purity of the diagonal, $s_2 = \sum_{i \neq j}|\gamma_{ij}|^2$ for the total weight of the binding, and $s_3 = d_E^2 + 2\sum_i|\gamma_{Ei}|^2$ for Interiority's share. Then $P = s_1 + s_2$, $R = 1/(7(s_1+s_2))$, $\Phi = s_2/s_1$ and $D = 1 + 6s_3/(s_1+s_2)$ — exactly, with a largest drift of $8.9\cdot10^{-16}$ 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 $E$ ($1.3\cdot10^{-15}$ over two thousand states) and by shuffling the fifteen moduli among pairs that do not touch $E$. 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 $s_1$, $s_2$, $s_3$ 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 $\Phi_{th}=1$ turns out to sit in an empty gap [Т]: T-314 says the gates read $s_1$, $s_2$, $s_3$ and no pattern; T-305 says $\Phi \ge 1$ decides balance. Both are true, and the reconciliation names the mechanism. A sign pattern cannot move $s_2$ directly — it moves the largest $s_2$ positivity allows. At a flat diagonal, equal moduli and the edge of positivity, $c = 1/|\lambda_{\min}|$ and hence $\Phi = 6c^2 = 6/\lambda_{\min}^2$. A balanced pattern is $uu^{\mathsf T} - I$, whose eigenvalues are $6$ and $-1$, so every balanced pattern gives $\lambda_{\min} = -1$ exactly and $\Phi = 6$; Harary's criterion forces every frustrated one to $|\lambda_{\min}| > \sqrt 6$, hence $c < 1/\sqrt 6$ and $\Phi < 1$. Measured over two hundred thousand sign patterns the best frustrated one reaches $\Phi = 0.9365$ (frustration $5$, $\lambda_{\min} = -2.5311$), and between $0.9365$ and $6$ there is nothing at all. So on this stratum a threshold that reads as stipulated is in fact robust: any value in $(0.937,\, 6)$ classifies identically, and $1$ 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, $\Phi$ is unimodal and tight — quantiles $0.863$ to $1.757$, median $1.238$ — with $20.8\%$ of states inside $[0.9,\,1.1]$, precisely where the idealisation says nothing can be, states landing within $0.0002$ of the threshold on both sides, and not one above $5$ though the idealisation puts every balanced state at exactly $6$. 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 $2.74$ to one, the moduli spread $17.4$-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: $\Phi \ge 1 \iff$ balanced on the stratum, while off it $86.5\%$ of states pass the gate carrying a median of $12$ broken triangles of $35$ 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 $6$ or at most $0.9365$ with nothing between, while computed states are unimodal and tight — quantiles $0.863$ to $1.757$, one in five inside $[0.9,\,1.1]$, the nearest pair straddling the threshold at $0.9998$ and $1.0003$, and none above $5$. 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 $7.1\%$ of a state, the instrument suite reaches $81\%$, and exactly eight numbers are read by nothing [Т]: a state carries $48$ numbers, of which $6$ are pure gauge, leaving $42$ that mean anything. Reading each named observable off its definition places it exactly. $P$ and $R$ read $s_1+s_2$; $\Phi$ adds $s_2/s_1$; $\mathrm{Coh}_E$ and $D$ add $s_3$; the consciousness measure $C = \Phi R$ adds nothing — so the whole verdict reaches three numbers of forty-two, or $7.1\%$. But the suite is not the verdict. Stress $\sigma$ reads the seven diagonal entries one by one, adding $6$; the gap and the pairwise moduli add $21$; the seven canonical line holonomies add $7$ independent phase invariants. Together $34$ of the $42$, or $81\%$. What remains dark is therefore small and nameable: eight triangle holonomies that no canonical line covers — the cycle space of $K_7$ has dimension $15$, the plane spans $7$ 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 $\sqrt{2/3}$ 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 $0.9727$ against $0.9999$ rad, means $1.1807$ against $1.1832$, shares above $\pi/2$ of $0.325$ against $0.318$, and the dark eight the more variable in $50.3\%$ 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 [Т]: a balanced sign pattern is $\sigma_{ij} = u_i u_j$, and the gauge transform $\Gamma \mapsto U\Gamma U^*$ with $U = \mathrm{diag}(e^{i\varphi})$ and $\varphi_i \in \{0,\pi\}$ chosen by $u_i$ carries it to the all-positive pattern. Verified exhaustively: the gauge orbit of all-positive contains exactly $64$ 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 $\iff$ pure gauge $\iff$ 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 $\mathrm{Re}\,\gamma_{ij}$, 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 $1.1\cdot10^{-16}$ and every line holonomy by $1.3\cdot10^{-15}$, while flipping $49.7\%$ 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 $\Phi$ and nothing plausible restores it [Т]: with $\rho^* = R\Gamma + (1-R)I/7$ the identities $s_2(\rho^*) = R^2 s_2$ and $s_1(\rho^*) = R^2 s_1 + 2R(1-R)/7 + (1-R)^2/7$ hold to $9\cdot10^{-17}$, and the denominator gains a strictly positive term whenever $R < 1$. Hence $\Phi(\rho^*) < \Phi(\Gamma)$ for every state — $5000$ of $5000$, median ratio $0.2888$, never above $0.4677$. Since the regeneration channel pulls the state towards its own self-model, it lowers integration by construction; dephasing lowers it too, raising $\Phi$ in $0$ of $2000$ trials. A unitary step is the only term that can raise it, doing so in $1044$ of $2000$ and by as much as $\times 2.21$ — and the implemented relaxation omits it. The consequence is measured, not argued: a perfectly balanced holon goes from $\Phi \ge 1$ in every case to none within twenty-five ticks. The obvious repair was implemented and fails: after two hundred ticks the surviving share is $0.000$ both with and without a Hamiltonian, because dephasing removes $94\%$ 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 $\Lambda^n_k$ with $0 < k < n$ 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 $f \circ x = h$, 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 ($0.51$, $0.47$) 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 ($1.0000$) 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 — $0.63$ where the rule half-composes, $0.53$ 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 [Т]: read through T-314's three sums, two of the four criteria fix each other. Integration clears its floor exactly when $s_2 \ge s_1$, and reflexivity clears its floor exactly when $P = s_1 + s_2 \le 3/7$, since $R = 1/(7P)$. Together they force $2s_1 \le 3/7$, and Cauchy–Schwarz on a probability vector forces $s_1 \ge 1/7$ always, so a viable state has $s_1 \in [1/7,\, 3/14]$ — the purity of its diagonal within $1.5\times$ of perfectly flat. Verified with zero counterexamples over fifty thousand states meeting the first two conditions; among two hundred thousand random states the $5.0\%$ that are viable have diagonal purity from $0.1431$ to $0.1960$, 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 $1.3\%$ of turns, purity clears $2/7$ in $99.8\%$ and reflexivity clears $1/3$ in $3.3\%$ — 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 $\Phi \ge 1$ 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 $8$ broken triangles of $35$ against $12$ for the rest, and are balanced in $1.44\%$ of cases against $0.08\%$. 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 ($s_1 \in [0.143,\, 0.196]$) rather than on it. The window is reachable, and constructively: a flat diagonal with balanced content at half strength gives $P = 0.357$, $R = 0.400$, $\Phi = 1.500$, $D = 2.371$ — while the same content at full strength is a pure state whose reflexivity is $1/7$ | 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 $0.9174$, 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 $0.4990$ against $0.4143$ for the rest. On a too uneven diagonal nothing is viable, because $s_1 > 3/14$ puts purity past $3/7$ and reflexivity under its floor (T-321). So the window is the band in between, and it is narrow: viable states have $s_1$ from $0.1431$ to $0.1995$ about a median of $0.1537$ — a hundredth above flat — carrying a median of $8$ broken triangles of $35$, with only $13\%$ near-flat and $1.5\%$ 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 [Т]: the diagonal is a probability vector, so Cauchy–Schwarz gives $s_1 \ge 1/7$ for every state, with equality exactly at a flat diagonal — a flat diagonal is not one option among many but the least $s_1$ there is. Since $P = s_1 + s_2$, this means $P \ge 1/7 + s_2$ always, and the ceiling $R \ge 1/3 \iff P \le 3/7$ then says: once the binding alone carries more than $2/7$, 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 $s_1$ lowers $P$ and raises $\Phi = s_2/s_1$ at once, the one direction improving two criteria together — whereas damping the binding lowers $P$ but takes $\Phi$ with it. The inequality gives that gentle move a hard limit, not of degree but of kind: on $\{s_2 > 2/7\}$ 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 $11$, and levelling followed by damping the binding for what levelling could not take rescues $20$. The second move is an exact null where the first suffices — on content that never drives $s_2$ past $2/7$ both hold the window for the same $4998$ 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 $s_1 = 1/7$, damping to $s_2 = 2/7$ puts $P = 3/7$ exactly, $\Phi = 2$, $R = 1/3$ — the same point at which capability attains its supremum $C = 2/3$, by the same inequality. And a companion band follows free — $\Phi \ge 1$ needs $s_2 \ge s_1 \ge 1/7$, the ceiling needs $s_2 \le 2/7$, so alive $\Rightarrow s_2 \in [1/7,\, 2/7]$, 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 [Т]: every monotone metric on $\mathcal D(\mathbb C^7)$ has one shape — $g_f(A,A)=\sum_{ij}|\tilde A_{ij}|^2/m_f(\lambda_i,\lambda_j)$ 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 $\text{harmonic}\le\text{logarithmic}\le\text{arithmetic}$ therefore applies term by term, and since the mean sits in the denominator the order reverses: $g_{\text{Bures}}\le g_{\text{BKM}}\le g_{\text{RLD}}$. 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 $1.0581$ and $1.1435$ about the median, never below $1.0225$ and $1.0523$ — 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 $6.1\times10^{-16}$ median and $1.8\times10^{-13}$ worst, while a basis chosen without regard to the question recovers $0.0974$ of it. On the learning side the same collapse happens: for traceless $A$, $\tfrac{d}{dt}D(\rho_*\|\Gamma+tA)=-g_{\text{BKM}}(\rho_*-\Gamma,A)$ — verified by central difference to $1.3\times10^{-9}$ median — so Cauchy–Schwarz leaves exactly one steepest direction at fixed BKM speed and it is $\rho_*-\Gamma$: 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 $\rho_*+e^{-\kappa t}(\Gamma_0-\rho_*)$, matched to $9.3\times10^{-7}$ at an Euler step of $1.5\times10^{-4}$. 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 [С]: 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 |

#	Result	Assumption	Source
C20	κ-dominance in composite holons	Evolution	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-103	Hedonic valence [C at observation model]	Observation model (L2)	Reclassified: formula [T], observability [T] (T-77), interpretation [I]
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 $\|\bar{h}\|_{\mathrm{typical}}$	Calibration of $\|\bar{h}\|$	Diagnostics
C22	Landauer calibration $\Delta F^{(k)}$: $\Delta F^{(k)} \geq k_B \cdot T_\mathrm{eff} \cdot \ln(2) \cdot k$ — linear growth with level. $\Delta F^{(0)} \approx \Delta F_\mathrm{bootstrap}$ from T-59 [T]	$T_\mathrm{eff}$ is determined by the environment	Depth Tower
C23	Monotonicity of grounding: grounding$(w, \tau)$ monotonically increases for $\eta_\sigma > 0$ and sensorimotor flow	Continuous learning + environment	Self-Observation
C24	Forgetting bound: $\|P_\mathrm{ISL}(\tau+\Delta\tau) - P_\mathrm{ISL}(\tau)\|_\mathrm{TV} \leq C \cdot \eta_0 \cdot \Delta\tau \cdot \|\sigma\|_\infty$ (EWC + Bures-adaptive $\eta$)	EWC regularisation	Consequences
C25	$\sigma$-probe: for $D_\mathrm{hidden} \geq 48$, probe reaches $R^2 > 0.9$ in $O(D^2)$ examples	Training data with known Γ	Consequences
C26	Critical SAD purity: $P_{\text{crit}}^{(n)} = P_{\text{crit}} \cdot 3^{n-1}/(n+1)$	Spectral SAD formula [C]	Raised to [T] (T-142): α = 2/3 is state-independent, spectral formula — consequence, not premise. SAD_MAX = 3 unconditionally — Operational Closure
C27	Attractor in the consciousness window	C20 (κ-dominance) + moderate κ	Raised to [T] (consequence of T-149): for embodied holons C20 is unconditional → C27 is unconditional — Substrate-Independent Closure

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Conditional Theorem: 7D Minimality [C] → [T]

#	Result	Assumption	Source
S1	Theorem S: $\dim(\mathcal{H}) = 7$ — minimal dimension for (AP)+(PH)+(QG)	Formalisation of (PH)	Minimality Theorem

Raised to [T] (Sol.70): Strict necessity $N = 7$ proven via Hurwitz's theorem ($\dim(\mathrm{Im}(\mathcal{A})) \in \{0,1,3,7\}$, 6 is impossible) + functional uniqueness 40f [T]. See Strict Necessity N = 7.

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Level 2: Correct as Standard Physics [T]

#	Result	Source	Target page
39	Probability current $J_\text{net}$	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	$SU(3)_C \subset G_2$ decomposition $14 \to 8+3+\bar{3}$	Cosmological Constant T.1.1	Standard Model
44	$\mathcal{N}=1$ SUSY from $G_2$-holonomy (parallel spinor $\eta_0$)	Standard Model T.4.1	SUSY from G₂
45	$\tau_p \sim 10^{37-38}$ yr (standard SU(5), D=6 operators)	Standard Model	Proton Decay
46	$\pi^0 \to \gamma\gamma$	Confinement T.12.1	Confinement
47	Masses of $X,Y$-leptoquarks from Gap hierarchy: $M_X \sim 10^{16}$ GeV	Standard Model T.1.1	Proton Decay
48	Proton decay channels (D=6): $p \to e^+\pi^0$, $p \to \bar{\nu}\pi^+$, $p \to e^+\eta$	Standard Model T.3.1	Proton Decay
49	G₂-extra mediated decay: $\tau_p^{(G_2)} \sim 4\times10^{47}$ yr (negligible; corrected 2026-07 from the mis-evaluated $10^{72}$)	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 $G_2 \to H$ breaking: $f_\text{Gold} \sim 0.005$–$0.02$ Hz	Lindblad Operators T.8.1	Phase Diagram
52	Anomalous dimension of the Fano operator: $\Delta_3 = 3 - 5/42 \approx 2.881$	Confinement T.9.1	Renormalisation Group

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Coherence Cybernetics Theorems

#	Result	Status	Source
CC-1	Theorem 6.1 (Existence of dynamics): for $\Gamma_0 \in \mathcal{V}$ 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): $\mathrm{Viable}(\mathbb{H}) \Rightarrow \exists\varphi$	[T]	CC Theorems
CC-4	Theorem 7.2 (Fixed point of reflection): $\exists!\Gamma^* : \varphi(\Gamma^*) = \Gamma^*$ — strict contraction from primitivity of the linear part $\mathcal{L}_0$ [T-39a]	[T]	CC Theorems
38a	Theorem 8.1 (Necessity of E-coherence): $\mathrm{Viable} \land \mathcal{D}_\Omega \neq 0 \Rightarrow \varphi = \varphi_{\text{coh}} \land \mathrm{Coh}_E \geq \mathrm{Coh}_{\min}$ — mathematical core [T]; 'No-Zombie' interpretation — [I] (requires ontological postulate about E-dimension)	[T]	CC Theorems
CC-5	Theorem 9.1 (Fractal closure): non-triviality of composite attractor $P > 1/7$ — [T] (T-96); viability $P > 2/7$ — [C] (depends on C20). Lowered from [T] upon resolution of the self-reference paradox	[C]	CC Theorems
CC-6	Theorem 9.2 (Scale invariance): $\mathrm{structure}(\mathbb{H}) \cong \mathrm{structure}(\mathbb{H}^{(k)})$ — raised from [H]: Bures CPTP contractivity + CC-5 (non-triviality [T], viability [C])	[T]	CC Theorems
CC-7	Theorem 9.3 (Emergence): irreducible emergence of the composite ($I(\mathbb{H}_1:\mathbb{H}_2) > 0$) — raised from [H]: primitivity of the linear part $\mathcal{L}_0^{(12)}$ + nontrivial attractor (T-96) + quantum mutual information (Sol.56)	[T]	CC Theorems
CC-8	Theorem 10.1 (Equivalence of conditions): $\Gamma \in \mathcal{V} \Leftrightarrow \|\sigma_{\mathrm{sys}}\|_\infty < 1$ — raised from [C]: all 7 components $\sigma_i$ formalised via $\Gamma$-invariants (Sol.81)	[T]	CC Theorems

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Level 3: Substantive Hypotheses [H]

Require reclassification from [T] to [H] or originally stated as hypotheses.

#	Result	Problem	Source	Target page
53	Dual-aspect interpretation of Hermitian conjugation	Postulate, not theorem	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
54	Conjugate pair principle	Semantic, not mathematical	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
55	Topologically protected Gap	Unestablished topology of M	Raised to [T]: $\pi_2(G_2/T^2) \cong \mathbb{Z}^2$ + positive-definite Hessian (T-64 [T]) + compactness $(S^1)^{21}$ → energy barrier $\Delta V \geq 6\mu^2 > 0$. Confinement-Gap protected by barrier $9\mu^2 \sim M_P^2$ — Composite Systems
56	Fano Gap bound	Gap in proof	Retracted [✗] (X3): $\mathrm{Gap}(O,i) \approx 1 > 1/2$ — counterexample. Replacement: sectoral Gap bound [T] (T-80, Sol.59) — Berry Phase
57	Canonical Schrödinger/Heisenberg duality	Interpretation	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
58	Bridge closure P1+P2	Condition (MP)	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
59	3+1 from $G_2$	Resolved [T]: sector decomposition $7 = 1 \oplus 3 \oplus \bar{3}$ [T]; compactification $\bar{\mathbf{3}}$ [T] (confinement). Einstein equations on $M^{3+1}$ — [T] (T-65, full spectral action) — T-48a, T-52	Renormalisation Group T.5.2
38	Low-energy limit of Gap integral → Einstein–Hilbert action	Raised to [T]: full spectral triple constructed (T-53 [T]); Chamseddine–Connes spectral action reproduces EH with $G_N = 3\pi/(7f_2\Lambda^2)$ — T-65	Quantum Gravity	Einstein Equations
60	Einstein equations from Gap	Raised to [T]: full spectral action + all NCG axioms verified — T-65	Quantum Gravity	Einstein Equations
61	SM from $G_2$: electroweak $SU(2)_L \times U(1)_Y$	[C at (FE)]	Raised to [T]: uniqueness of the pair $(E,U)$ proven from $\kappa_0$ [T] (categorical compatibility with $\mathrm{Hom}(O,E)$ and $\mathrm{Hom}(O,U)$). Was [H] → [C at (FE)] → [T] — Standard Model
62	3 generations from Fano	$S_4$ orbits not strictly defined	count [T], identification [I] (strengthened 2026-07): exact count $\|\mathrm{QR}(7)\| = \|\mathbb{Z}_7^*/\{\pm1\}\| = 3$ [T] (group-theoretic, topology-independent); identification [I] — Fermion Generations	Fermion Generations
63	Confinement from Gap	Qualitative argument	Partially resolved (Sol.60): (a) Topological area law — [T] (T-81: T-73 + T-69 + T-64); (b) String tension $\sqrt{\sigma} \approx 457$ MeV — [C at T-64] (unique vacuum parameters); (c) Deconfinement temperature $T_c$ — [C at standard finite-temperature QCD] (analogue of lattice $T_c \approx 150\text{–}170$ MeV, nature of transition not strictly derived); (d) Polyakov loop parameterisation — [H] (qualitative model, §4.2) — Confinement	Confinement
89	SAD–L equivalence	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
90	Commutativity of φ-tower	Raised to [T]: T-150 — trivial commutativity of iterates of a single CPTP channel for $D_k = 7$. Spectral SAD formula — consequence, not premise — Substrate-Independent Closure	Depth Tower Hyp. 5.1	Depth Tower
91	Self-organisation of tower from $\Gamma(0) = I/7$ (tabula rasa)	Raised to [T]: T-148 — genesis via environmental coupling. An embodied holon raises purity above $P_{\mathrm{crit}}$ in finite time — Substrate-Independent Closure	Depth Tower Hyp. 6.1	Depth Tower
92	Optimal learning efficiency from N=7	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): $C_{\mathrm{opt}}(K) = c_0/K$ for $c_0 < 1/14$. MetaAgent contractivity preserved $\forall K$: $k_\cup = \max_i k_i + c_0 < 1$. Boundary case: $c_0 = 1/14$, $k_\cup = 1$ (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 ($EI > 0$) requires a nonlinear collective operator	Specification	Prediction 11, Stability
95	Non-Markovian extension (E-10.3): $d\Gamma/d\tau = \mathcal{L}[\Gamma(\tau)] + \int_0^\tau K(\tau-s) \Gamma(s)\,ds$ with $K(t) = -\Gamma_2 \omega_c e^{-\omega_c t}$. Preserves CPTP for $\|K\| < \alpha$, stationary points of the Markovian limit, enriches transient dynamics (oscillatory approach to $\rho_*$)	Specification	T-94 [H]
96	Grounding monotonicity (E-10.4)	$g(w, t+1) \geq g(w, t)$ under stable learning	Raised to [C at T-115]: T-115 [T] — algebraic distinguishability of symbolic compositions for generic $\Gamma$. Under stable learning condition ($\|\Delta P\| < \varepsilon$, $\|\Delta\sigma\| < \varepsilon$) 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 ($\pi_k(V) \neq 0 \Leftrightarrow k$-grammar) is probably false: $V \subset \mathcal{D}(\mathbb{C}^7)$ — 48-dimensional region, $\pi_k = 0$ for $k \leq 46$. Reformulation: grammatical structures may emerge from the Postnikov tower of ∞-topos $\mathrm{Sh}_\infty(\mathrm{Exp})$, not from homotopies of $V$. Status [P] (requires reformulation within HoTT-linguistics) — corrected from [H]	Specification	T-69 [P]
98	Categorical Nash embedding (E-10.6)	$\mathrm{Hom}(\mathrm{Ag}, \mathrm{Ag}) \cong NE(\Gamma_{\mathrm{ext}})$	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]
99	$N = 7$ minimality for social learning (E-10.7, Pred)	$3_{\text{ToM}} + 3_{\text{ISL}} + 1_U = 7$	Raised to [C at T-57, T-114]: (1) T-57 [T] (LGKS completeness) — ToM requires 3-channel decomposition → $\geq 3$ dimensions. (2) T-114 [T] (Fano grammar) — ISL on PG(2,2) requires $\geq 3$ dimensions. (3) Nash coordination: $\geq 1$ dimension (Unity $U$). Additivity under mutual independence — $3 + 3 + 1 = 7$. 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]
100	L4 closure (E-10.8)	$\omega$-groupoid	Raised to [C at T-86, T-55]: $\mathcal{D}(\mathbb{C}^7)$ is compact [T] → complete metric space → the Cauchy sequence $\varphi^{(n)}(\Gamma)$ converges (contractivity $k < 1$ [T]). The colimit of the Postnikov tower $\tau_{\leq n}(\mathrm{Exp}_\infty)$ 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 $\pi$: hidden $\to \Gamma$ preserves $\mathrm{Tr}(\Gamma) = 1$ and $\gamma_k \geq 0$ 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) $\sigma$-probe output $\in [0,1]^7$ for arbitrary hidden states (T-92 [T] bounded). Verified MVP-10 (M10.11 PASS)	[T]	MVP-10 Ph.2
104	(H81) $\varphi$-contraction: $K_\varphi = 13/14 = 1 - 1/(2N)$ from Fano geometry [F4]. Verified MVP-10 (M10.27 PASS)	[T]	MVP-10 Ph.3
105	(H82) Cholesky round-trip: $\Gamma \to$ params $\to \Gamma$ preserves diagonal with $\varepsilon < 0.05$. Verified MVP-10 (M10.28 PASS)	[C]	MVP-10 Ph.3
106	(H83) CRL grounding: ISL-conditioned cross-attention preserves dimension (seq, $d_\text{model}$). 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 ($P, R, \Phi, D, \sigma$) 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: $\|\mathrm{Comp}(2)\| = 49$, $\|\mathrm{Comp}(3)\| = 343$ (T-115 [T] combinatorial count). Verified MVP-11 (M11.25–M11.27 PASS)	[T]	MVP-11 Ph.3
112	(H89) Fano seed purity: $P(\Gamma_{\text{Fano}}) > P_{\text{crit}}$ for concentrated initial state (Sol.5). Verified MVP-11 (M11.31 PASS with $P = 0.338$)	[C]	MVP-11 Ph.3
113	(H90) Self-observation: unified state vector correctly reflects $P, R, \Phi, D, \sigma, \text{SAD}$. 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
64	Fano selection rule	Proof via $V_3$ was erroneous	Raised to [T]: proven via octonionic structure constants $f_{ijk}$ — the unique $G_2$-invariant trilinear operator on $\mathrm{Im}(\mathbb{O})$. Formula $y_k^{(\mathrm{tree})} = g_W \cdot f_{k,E,U} \cdot \|\gamma_{\mathrm{vac}}^{(EU)}\|$; $f_{1,5,6} = 1$, $f_{2,5,6} = f_{4,5,6} = 0$	Fano Selection Rules
116	(H1) Trainable CPTP-anchor: $\pi_\theta: \mathcal{H} \to \mathcal{D}(\mathbb{C}^7)$ preserves CPTP for arbitrary $\theta$ at $M = 49$ Kraus operators	Necessity of $M = 49$	Raised to [T]: Stinespring ($M \leq N^2 = 49$) + Cybenko–Hornik (approximation of trace-preserving maps by a neural network at $M = N^2$) → completeness of CPTP coverage. Minimal $M = N^2 = 49$ is unconditional for $N = 7$
117	(H-Hawk) Hawking radiation: $T_H = \hbar c^3/(8\pi G_N M k_B)$ and evaporation rate $dM/dt = -\hbar c^4/(15360\pi G_N^2 M^2)$ for Gap black holes	Absence of derivation from NCG formalism	Raised to [T]: consequence of T-65 (full spectral action [T]) + standard QFT on curved background — $G_N = 3\pi/(7f_2\Lambda^2)$ [T] uniquely determines $T_H$ and $dM/dt$ without free parameters
118	(H-Pol) Polyakov loop as order parameter: $\langle L \rangle = 0$ in confinement, $\langle L \rangle \neq 0$ above $T_c$	Identification of centre symmetry	Raised to [T]: $\mathrm{Stab}_{G_2}(e_O) = SU(3)_C$ [T] (T-42e) → $Z_3 \subset SU(3)_C$ is the centre; Polyakov loop $L \in \mathbb{C}$ transforms under $Z_3$ → $\langle L \rangle$ — exact deconfinement order parameter
119	(H-Tc) Deconfinement temperature formula: $T_c \sim \sqrt{\sigma}/\pi \approx 145\text{–}165$ MeV	Dependence on $\sqrt{\sigma}$	Raised to [C at T-64]: $T_c$ is expressed via $\sqrt{\sigma} \approx 457$ MeV [C at T-64] by the standard lattice relation $T_c \approx \sqrt{\sigma}/\pi$; upon substituting the exact $\sigma$ from T-81 — full prediction [C at T-64]
120	(H-V3) Scaling $V_3$-mixing: $m_c/m_t \sim \varepsilon^2$	Absence of derivation from RG equations	Raised to [C at T-64]: Fano selection rule [T] (T-43d) + tree-level Fritzsch texture → $y_c/y_t = f_{2,5,6}/f_{1,5,6} \cdot (\varepsilon/1)^2 = \varepsilon^2$ from double Fano-blocking ($f_{2,5,6} = 0$ → corrections of order $\varepsilon^2$). Numerical $\varepsilon^2$ — [C at T-64]
121	(H-ΩDM) Dark matter parameter: $\Omega_{\mathrm{DM}} h^2 \approx 0.12$	O-sector thermodynamics mechanism	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 $\Omega_{\mathrm{DM}} h^2 \sim 0.1$ at standard annihilation cross-section (CKR = Rounak cross-section condition). Depends on T-50 (superpotential) and CKR
122	(H-SBH) Gap correction coefficient in $S_{\mathrm{BH}}$: $S_{\mathrm{BH}} = A/(4G_N) + c_{\mathrm{gap}} \cdot \mathcal{G}_O$	Postulated coefficient $c_{\mathrm{gap}}$	Raised to [C at T-65, T-73, T-74]: spectral action T-65 [T] → gravitational block includes $\mathrm{Tr}(D^2)$; Gap as curvature T-73 [T] → $c_{\mathrm{gap}} = \omega_0^2/(8\pi G_N)$ from identity $\|\mathrm{Curv}\|^2 = \omega_0^2 \|\gamma_{ij}\|^2 \mathrm{Gap}^2$ and T-74 ($V_{\mathrm{Gap}}$ from spectral action)
123	(H-MH) Mass hierarchy from Fano selection rule: $n_{\mathrm{Fano}}^{\mathrm{uniform}}$ — no hierarchy; hierarchy arises from tree-level selection rule	Confusion of RG running and tree-level vetoes	Clarified and raised to [T]: the corrected formulation — uniform Fano ($n_{\mathrm{Fano}}^{\mathrm{uniform}}$) does not generate mass hierarchy on its own; hierarchy arises from tree-level Fano veto ($f_{k,5,6} \neq 0$ only for $k=1$) → $y_1^{(\mathrm{tree})} \gg y_{2,4}^{(\mathrm{tree})}$ structurally. Proof: T-43d [T] + $G_2$-uniqueness of $f_{ijk}$
124	(H-δCP) Topological quantisation of CP-phase: $\delta_{\mathrm{CP}}^{(\mathrm{tree})} = 2\pi n/7$, $n \in \mathbb{Z}_7$	Identification with CKM phase	Raised to [T]: phases $\theta_{ij}$ live in $\mathbb{Z}_7 \subset U(1)$ (PW time is discrete, $\tau \in \mathbb{Z}_7$, T-38b [T]); $G_2$-covariance of the Fano dissipator [T] (T-2) → quark mixing phase inherited from $\mathbb{Z}_7$-topology; tree-level value $\delta_{\mathrm{CP}} = 2\pi n/7$ is topologically quantised
65	Gap as Serre curvature	Argument, not strict construction	Raised to [T]: spectral triple T-53 [T] + Connes NCG curvature → $\|\mathrm{Curv}\|_{ij}^2 = \omega_0^2 \|\gamma_{ij}\|^2 \cdot \mathrm{Gap}(i,j)^2$ (exact identification). Second Chern class $c_2 = \mathrm{Tr}(D_{\mathrm{int}}^2)/(8\pi^2\omega_0^2)$ — Gap Operator	Gap Operator
66	$\varepsilon = 10^{-2}$	Numerical estimate	Raised to [C at T-64]: self-consistent vacuum equation (T-64 [T]) gives sectoral mean $\bar{\varepsilon} \approx 0.023$. Exact value depends on minimisation of the Gap potential — a computational task. Principal estimate $\varepsilon = O(10^{-2})$ — [C at T-64] — C12	Quantum Gravity §7.4, Λ Budget
67	Seesaw type I: $M_R \sim 10^{14}$ GeV, $m_\nu \sim 0.03$ eV	Resolved [T]: $M_R \sim 2.9 \times 10^{14}$ GeV from PW clocks + viability — T-51	Standard Model	Neutrino Masses
68	PMNS matrix from Fano geometry: $\theta_{12}^{(\text{PMNS})} \gg \theta_{12}^{(\text{CKM})}$	Partially resolved [C]: qualitative $\theta_{\text{PMNS}} \gg \theta_{\text{CKM}}$ [T]; quantitative — anarchic $M_R$ from O-sector isotropy gives angles $O(30°\text{–}60°)$ [C] — C15	Standard Model	Neutrino Masses
69	Superpartner spectrum: $m_{\tilde{q}} \sim 10^{13}$ GeV (gravity mediation)	Resolved [T]: superpotential $W$ is unique (Schur's lemma) — T-50	Standard Model	SUSY from G₂
70	F-term from $V_3$: $\sqrt{F} \sim 10^{-3} M_\text{Pl}$	Resolved [T]: $F = \partial W / \partial \Theta \neq 0$ from uniqueness of $W$ (Schur) — T-50	Standard Model T.3.1	SUSY from G₂
71	Gravitino mass: $m_{3/2} \sim 2.9 \times 10^{13}$ GeV	Resolved [T]: $m_{3/2} \sim \varepsilon^3 M_P$ from the cubic structure of $W$ (Schur) — T-50	Standard Model	SUSY from G₂
72	Non-perturbative UV-finiteness of Gap theory	Stratified: field-space finiteness [T] (compact target); full order-by-order UV-finiteness [C] (structural: APS-index + $G_2$ Ward identities + $\mathcal{N}=1$ holomorphy + $\varepsilon^{12}$, T-219). Gravitational UV-finiteness — automatic consequence of emergence — T-66	Quantum Gravity	Quantum Gravity
73	Neutrino mass predictions: $m_{\nu_\tau} \sim 0.03$ eV, hierarchy type	Resolved [T]: numbering established [T] ($k=1 \to$ 3rd, $k=4 \to$ 2nd, $k=2 \to$ 1st) from confinement; normal hierarchy [T]. Discrepancy $m_2/m_3$ remains [C] — T-52	Standard Model	Neutrino Masses

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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 $L_\text{top}$ from $\mathfrak{g}_2$-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; $m_\text{SUSY}^{(3\bar{3})} \sim \varepsilon_\text{soft} \cdot m_{3/2}$, but not zero	Standard Model T.9.2
77	Equivalence $(1,2,4) \leftrightarrow (3,5,6)$	$k \to 7-k \notin \mathrm{Aut}(\text{Fano})$	Standard Model §1.5
78	Gaussian sum: 9 orders at physical $S_0$	$\Theta_M/\Theta_0 \approx 1$ at $S_0 = 20$	Cosmology §4
79	Modular hypothesis: 15 orders	Refuted at $S_0 = 20$	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: $P_{\Gamma_1 \cup \Gamma_2} \geq P_{\Gamma_1} + P_{\Gamma_2} - P_{\Gamma_1 \cap \Gamma_2}$	Dimensionally incorrect: $P = \mathrm{Tr}(\Gamma^2)$ — quadratic functional, not a measure. Correct formula: $\Delta P = 2\|\gamma_{\mathrm{cross}}\|_F^2$ (Sol.57, T-77 [T])	Value Consciousness

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Postulates [P] and Definitions [D]

#	Result	Status	Source
P1	Information Distinguishability Principle (IDP)	[P] → [T] → [D]	Reclassified [D] (Sol.25): IDP — a definition embedded in A1+A2. Distinguishability via $J_{\text{Bures}}$-coverings is identical to ontological distinguishability — a tautological consequence of the ∞-topos choice. All computational results ($P_{\text{crit}}, R_{\text{th}}, \Phi_{\text{th}}$) are unaffected — Axiom of Septicity
P2	Non-associativity (postulate P2)	[P] → [T]	Raised to [T]: P1+P2 derived from (AP)+(PH)+(QG)+(V) via the chain T15 [T] — Octonionic Derivation
P3	Page–Wootters mechanism	[P] → [T]	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
O1	Integration threshold $\Phi_{\text{th}} = 1$	[D] → [T]	Raised to [T] (T-129 + T-129a): unique self-consistent value with $P_{\text{crit}} = 2/7$. Universality (T-129a [T]): threshold on all of $\mathcal{D}(\mathbb{C}^7)$ — Operationalisation
O2	Canonical $R$ 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
C1	Reflection threshold $R_{\text{th}} = 1/3$	$K=3$ alternatives	[T]+[I]: $K = 3$ derived from triadic decomposition T-40a, 40b, but the identification $R = P(H_1)$ — interpretive bridge [I] — see reflection threshold
C2	Differentiation threshold $D_{\min} = 2$	$\Phi_{\text{th}} = 1$ [T] (T-129)	Raised to [T] (T-151): $\Phi_{\text{th}} = 1$ [T] (T-129) → spectrum of $\rho_E$ has $\geq 2$ significant components → $D_{\mathrm{diff}} \geq 2$ unconditionally — Substrate-Independent Closure
C3	E-coherence 7D proxy $\widetilde{\mathrm{Coh}}_E^{7D}$	7D↔42D correspondence	Raised to [T]: $\mathrm{Coh}_E$ defined as HS-projection $\pi_E$; formula $(\gamma_{EE}^2 + 2\sum\|\gamma_{Ei}\|^2)/\mathrm{Tr}(\Gamma^2)$ — exact consequence, not proxy — Axiom of Septicity, HS-projection
C4	Variational characterisation of $\varphi$ (Theorem 3.1)	Primitivity of $\mathcal{L}_\Omega$	Raised to [T]: primitivity proven — see T-39a, 39e
C5	Octonionic structure $\mathbb{O} \to N=7, G_2$	Condition (MP)	Raised to [T]: Bridge fully closed (T15 [T]) — T11 (Choi rank=7) + T12 (projective operators) + T13 (forced BIBD). (MP) became a theorem — Lindblad Operators
C6	Coverage democracy (T3): $S_7$-symmetry of Ω + (CG) ⟹ $\lambda_{ij} = \text{const}$	Condition (CG)	Withdrawn: T6 [T] proves uniform contraction unconditionally (from $S_7$-equivariance, T5 [T]) — see T-41e
C7	Electroweak sector $SU(2)_L \times U(1)_Y$ from Fano structure	(FE) — Fano electroweak hypothesis	Raised to [T]: uniqueness of the pair $(E,U)$ proven from $\kappa_0$ [T]. Was [H] (No.61) → [C at (FE)] → [T] — Standard Model
C8	Ordering $k=4 \to$ 2nd generation, $k=2 \to$ 1st generation	(SA) — sector asymmetry	Raised to [T]: sector asymmetry proven from confinement [T] and asymptotic freedom [T]. Structural inequality: non-perturbative coupling > perturbative for any $\varepsilon \in (0,1)$ — T-52
C9	Superpotential $W = \mu_W \sum f_{ijk} \Theta_{ij}\Theta_{jk}\Theta_{ik}$	(MP) — minimal superpotential	Raised to [T]: uniqueness from Schur's lemma — $\dim\mathrm{Hom}_{G_2}(\Lambda^3(\mathbf{7}), \mathbb{R}) = 1$. Higher orders suppressed by $\varepsilon^{n-3}$ — T-50
C10	$M_R = g^4_{G_2}/(16\pi^2) \cdot \sqrt{6}\varepsilon M_P \sim 2.9 \times 10^{14}$ GeV	(ΓO) — O-sector scale	Raised to [T]: $\mathrm{Gap}(O,\cdot) = O(1)$ from PW phase precession + viability (V). $M_R$ derived from axioms A1–A5 — T-51
C11	$\dim(\text{space}) = 3$ from $\|\mathbf{3}_{A,S,D}\|$; compactification $\bar{\mathbf{3}}$ at scale $v_{\text{EW}}$	(SA) — sector asymmetry	Raised to [T]: sector asymmetry proven from confinement [T] — T-52
C12	Self-consistent vacuum equation	Self-consistency of definitions	Raised to [T]: uniqueness of the self-consistent vacuum with sector structure — T-61
C13	Discrepancy in $\sqrt{\sigma}$ (7×)	Sector structure from C12	Raised to [T]: sectoral $\|\gamma\|_{3\bar{3}}$ from unique vacuum [T-61] — Confinement
C14	Neutrino mass ratio $m_2/m_3 \approx 0.17\text{–}0.20$ (with 2-loop RG)	O-sector Yukawa + 2-loop RG (Sol.72)	[C] — discrepancy $\times 1.0\text{–}1.2$ vs. observed 0.17; formula T-63 [T], precision — computational task at $\theta^*$ — Neutrino Masses
C15	PMNS angles from anarchic $M_R$	O-sector isotropy → $\|[M_R]_{kl}\|/\|[M_R]_{kk}\| \sim O(1)$	[C] — correct order ($30°\text{–}60°$); exact prediction requires Gap structure of O-sector — Neutrino Masses
C16	Higgs quartic $\lambda_4$ from spectral action	$\lambda_4 = \pi^2 \text{Tr}(D^4) / (2f_0\Lambda^4[\text{Tr}(D^2)]^2)$ + RG	[C] — $f_0$ canonically defined [T] (T-70): $f_0\Lambda^4 = \frac{1}{7}[V_{\mathrm{Gap}}^{\min} + \frac{1}{2}\zeta'_{H_{\mathrm{Gap}}}(0)]$. Conceptual freedom eliminated; numerical value of $\lambda_4$ depends on exact $\varepsilon_i$ — Higgs Sector
C17	$m_b/m_t$ from sector RG	QCD enhancement + loop $y_b$	Mechanism [T] (Sol.71): discrepancy $\times 4$ — artefact of mean $\varepsilon$; at sectoral $\varepsilon_{33}^*(\theta^*)$, $r_{33} \approx 0.25$: $y_b \approx 0.024$ — exact agreement. Precise prediction — computational task (T-79) — Yukawa Hierarchy
C18	Spectral formula $\Lambda_{\text{CC}}$	$\Lambda_{\text{CC}}$ via $a_0, a_2, a_4$ of the spectral action + SUSY-breaking	[C] — structural formula [T]; honest composed bracket $10^{-53.5}$–$10^{-93.5}$ [C], remaining $\gtrsim 27$ orders open — Λ Budget honest ledger
C20	Viability of the attractor: $P(\rho^*_\Omega) > 2/7$	$\kappa$-dominance	Raised to [T] for embodied holons (T-149): backbone injection ensures $P > 2/7$ unconditionally. Isolated holon: C20 remains [C] (no practical relevance, since an isolated holon at $I/7$ is dead forever, T-148) — Substrate-Independent Closure
C21	Attractor consistency	Weak $H_{\mathrm{eff}}$	Raised to [T] (T-157): $\|\rho^*_\Omega - \Gamma^*_{\mathrm{coh}}\|_F \leq \|H_{\mathrm{eff}}\|_{\mathrm{op}} / (\alpha + \kappa)$ — parametric bound; for embodied systems $\|H_{\mathrm{eff}}\|$ is determined by backbone and hedonic drive — Substrate-Independent Closure
C19	L4 unreachability for biological systems	$R^{(n)} \sim R^n \to 0$ for $n \to \infty$ at $\varepsilon_{\text{dec}} > 0$	Raised to [T] (Sol.64): categorical unreachability via Postnikov tower + Lawvere incompleteness (T-55 [T]). Butterfly $A_5$ 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.

| C32 (was C22) | Monotonicity of symbol grounding: $g(w, t+1) \geq g(w, t)$ under stable learning ($\|\Delta P\| < \varepsilon$, $\|\Delta\sigma\| < \varepsilon$) | 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: $\mathrm{Hom}(\mathrm{Ag}, \mathrm{Ag}) \cong NE(\Gamma_{\mathrm{ext}})$ | 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) | $N = 7$ minimality for social learning: $3_{\text{ToM}} + 3_{\text{ISL}} + 1_U = 7$ | 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) | $\varepsilon = O(10^{-2})$ (numerical order of the vacuum parameter) | T-64 [T] (unique vacuum of the Gap potential) | [C at T-64] — raised from [H] No.66. Self-consistent equation gives $\bar{\varepsilon} \approx 0.023$; exact value — computational task — C12 |

Retracted Statements [✗]

#	Statement	Reason for retraction	Replacement
X1	$\Phi \geq 1 \Leftrightarrow K_1(C^*(\Gamma)) \neq 0$	$K_1(M_n(\mathbb{C})) = 0$ for all $n$	[D] coherent domination
X2	IDP — theorem from $J_{Bures}$	Semantic assumption in step (3)	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 $\leq 1/2$ for all pairs	O-sector Fano pairs (6 of 21): $\mathrm{Gap}(O,i) \approx 1 > 1/2$ — direct counterexample	Replacement (Sol.59): sectoral Gap bound [T] (T-80) — Berry Phase
X4	L3→L4 as butterfly $A_5$	Finite catastrophe inapplicable to infinite-dimensional transition (all $\pi_k$ for $k \geq 4$)	Replacement (Sol.64): categorical unreachability [T] (T-86) — Interiority Hierarchy

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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 $(S^1)^{21}$	Monte Carlo with $G_2$-symmetry	Quantum Gravity
83	Black hole information paradox	Gap resolution: unitary evolution, Page curve from Gap profile	Quantum Gravity
84	Inflation from Gap potential	$V_2 + V_4$ at small $\theta$ as a quadratic inflaton	Quantum Gravity
85	Non-perturbative closure of the Λ deficit	Progress: spectral formula [T] (T-65); SUSY-sector $\varepsilon^{12}$ [T at T-64] (T-219; absorbs $\varepsilon^6$ → net $10^{-53.5}$); full minimisation T-64 [T]; honest bracket $10^{-53.5}$–$10^{-93.5}$ [C]. Remaining $\gtrsim 27$ orders: open computational + conceptual task (2026-07 audit: the former total $\sim10^{-120\pm10}$ was a forbidden sum — retracted)	Λ Budget honest ledger

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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 $G_2$-sector, pathological — in $\perp$	Gap Operator
88	Non-Markovian oscillations as 'grief cycles' and 'clarity flashes'	Phase Diagram
89	k-floor clamp [I]: in the implementation $k = (1-R).\mathrm{clamp}(0.15, 1.0)$ — for $R > 0.85$ the value $k = 0.15$ is used instead of theoretical $k = 1-R$ (T-62). Prevents degeneration of $\mathcal{R}$ as $R \to 1$. Threshold 0.15 is empirical	Evolution
90	Dual-aspect interpretation of conjugation (reclassified from [H] No.53): $\dagger$ 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

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Budget of the Cosmological Constant Λ

Perturbative Budget (confirmed — [T])

Mechanism	Suppression	Source	Status
$\varepsilon^6$ (smallness of coherences)	$10^{-12}$	Quantum Gravity §7.3	[T]
RG $\lambda_3^2$	$10^{-14.5}$	Quantum Gravity §12.3	[T]
Ward identities (anti-correlation)	$10^{-0.41}$ (×19/49)	Cosmological Constant §10.3	[T]
Fano code (6 constraints)	$10^{-0.9}$ (×1/8)	Quantum Gravity §12.5d	[T]
$\sqrt{N_F}$	$10^{-11.9}$	Confinement §9.3	[T]
O-sector $(6/21)^3$	$10^{-1.7}$	Confinement §10.2	[T]
Total	$10^{-41.5}$		[T]

Full proof: Λ Budget.

Non-perturbative Sector

Mechanism	Result	Status
Instanton ($e^{-150}$)	$10^{-65.5}$ — additive, not multiplicative	[T]
Gaussian sum at $S_0 = 20$	$\Theta_M/\Theta_0 \approx 1 - O(10^{-9})$ — does not work	[D]
Modular hypothesis	~15 orders — does not work at $S_0 = 20$	[D]
Zeta $Z_\Phi(-k) = 0$	Structural zeroing — requires QFT interpretation	[T] (math.), [H*] (phys.)

Cohomological + SUSY Sector

Mechanism	Result	Status
$\Lambda_{\text{global}} = 0$ (cohomological zeroing)	Global $\Lambda = 0$ from $H^n(X) = 0$	[T]
SUSY-breaking $\varepsilon^{12}$	$10^{-24}$ residual	[T] (via spectral action T-65)
$Z'_\Phi(-2) \approx 2.6 \times 10^{10}$	$\times 10^{10}$	[T] (math.)
RG $\lambda_3^2$	$10^{-14.5}$	[T]
Sectoral from Sol.39	$10^{-40}$	[C] (full minimisation T-64)

Total (conservatively): 41.5 [T] out of 120 — proven perturbative suppression. Gap before full minimisation: ≈ 78.5 orders. Remaining sources (conditional):

• Cohomological zeroing $\Lambda_\mathrm{global}=0$: [T] (reduces global contribution to zero; observed $\Lambda$ is local defect).
• SUSY-breaking suppression $\varepsilon^{12}\sim 10^{-24}$: [T] (via spectral action T-65 + Schur-uniqueness of $W$ T-50). Caveat: the specific factor $\varepsilon^{12}$ depends on Fano selection rule T-43d [T] and sector structure; numerical value is [C at T-64].
• $Z'_\Phi(-2)\approx 2.6\times 10^{10}$ enhancement: [T] (zeta calculation); physical interpretation *[H]**.
• RG $\lambda_3^2\sim 10^{-14.5}$: [T].
• Sectoral minimisation $\sim 10^{-40}$: [C at T-64] — not yet numerically computed on $(S^1)^{21}/G_2$.

Honest summary (2026-07 audit): composed bracket $10^{-53.5}$–$10^{-93.5}$ [C at T-64, H* at $Z'_\Phi(-2)$; sector programme pending]. The former "total $\sim 10^{-120\pm10}$" was a forbidden sum (double-counted RG $\lambda_3^2$, unabsorbed $\varepsilon^6$; equivalently, quoting $\varepsilon$ at the lower edge $10^{-3}$) — retracted. Remaining $\gtrsim 27$ orders: open computational (numerical minimisation of $V_\mathrm{Gap}$ on $(S^1)^{21}/G_2$) + conceptual (local residual saturating the cohomological bound). See Λ Budget honest ledger.

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Critical Cross-Document Issues

1. CS Cascade

Source: Phase Diagram §1.3 → Refutation: Berry Phase §2.1

Affected results: $L_\text{top}$, $\beta = 1/(2\pi)$, Noether charges (topological part), equations of motion with topological term, bridge closure via $V_3 \neq 0$.

Resolution: Reinterpretation via the Berry phase. The formula $L_\text{top}$ may be salvaged, but its derivation from CS on 1D is erroneous.

2. SM from G₂: rank problem

$\mathrm{rank}(G_2) = 2 < \mathrm{rank}(\text{SM}) = 4$. Electroweak sector: [T] — uniqueness of the pair $(E,U)$ proven from $\kappa_0$ [T] (categorical compatibility with $\mathrm{Hom}(O,E)$ and $\mathrm{Hom}(O,U)$). Was [H] → [C at (FE)] → [T]. Correct formulation: '$SU(3)_C$ from $G_2$ [T]; $SU(2)_L \times U(1)_Y$ from $\kappa_0$ [T]' — uniqueness theorem.

3. CKM predictions: overstatement of precision

The formulae $|V_{us}| \sim \sqrt{m_d/m_s}$ are standard consequences of the Fritzsch texture with observed masses as input. The theory's prediction is the structure (Fritzsch texture), not the numbers.

Empirical status (2026). The one genuine CKM prediction — the CP phase $\delta_{\text{CP}} \approx 64.5°$ from the Fano geometry [H] — is confirmed near-exactly by the LHCb tree-level combination $\gamma = 64.6° \pm 2.8°$ (ICHEP 2024; $\approx 0.04\sigma$), consistent with the PDG 2024 global fit $65.7° \pm 1.5°$. The older $69° \pm 4°$ 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 $\sim 3.2\sigma$ deficit, $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.9985(5)$ (2024–2026 lattice + $\beta$/kaon determinations). T-265 (CKM §10) sharpens the earlier "open gap": since the fundamental CKM is exactly $3\times3$ unitary ($N_{\text{gen}}=3$ [T]), the deficit cannot be a mixing-matrix effect — fourth generation, vector-like quarks, MeV sterile neutrinos and leptoquarks are all excluded by the spectrum — so UHM predicts it resolves in the SM extraction sector ($\gamma W$-box / nuclear radiative corrections / lattice $K/\pi$ / the $K$–$\pi$ $V_{us}$ tension). The resolution channel is [T-structural]+[C]; only 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 $m_2/m_3 \sim m_\mu^2/m_\tau^2 \sim 0.0035$ disagreed with the observed $m_2/m_3 \sim 0.17$ 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: $\nu_R$ in the O-sector (T-51) → Dirac mass from blocks $M_{O,3}$ and $M_{O,\bar{3}}$, not from $M_{3,\bar{3}}$. PMNS angles from anarchic $M_R$ — $O(30°\text{–}60°)$ [C]. See Neutrino Masses.

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Open Problems

Hidden Assumptions

#	Assumption	Status
H1	Primitivity of $\mathcal{L}_\Omega$	[T] — T-39a
H2	Uniqueness of 7/7 dimensions	[T] — T-40c, 40d, 40e, 40f
H3	Choice of $K = 3$	[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] — $G_2$-rigidity of holonomic representation T-42a

Fundamental

1. Λ: $\gtrsim 27$ orders open — structural mechanisms identified [C]: spectral formula $\Lambda_{\text{CC}}$ via $a_0, a_2, a_4$ [T] (T-65); SUSY-sector $\varepsilon^{12}$ [T at T-64] (exact compensation [H]); cohomological zeroing [T]; sector structure from full minimisation T-64 [T]; sign $\Lambda > 0$ proven [T] (T-71: autopoiesis + local cohomology); $f_0$ canonically defined [T] (T-70); O-sector dominance [T] (T-84, Sol.63: $\mathcal{G}_{\text{total}} = \mathcal{G}_O + O(\bar{\varepsilon}^2)$). Honest bracket $10^{-53.5}$–$10^{-93.5}$ [C]; remaining $\gtrsim 27$ orders — open computational + conceptual task (Λ Budget honest ledger)
2. 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 2b. Uniqueness of mapping G — RESOLVED [T]: $G_2$-rigidity of holonomic representation. The mapping $G: \mathrm{States}(S) \to \mathcal{D}(\mathbb{C}^7)$ is unique up to $G_2 = \mathrm{Aut}(\mathbb{O})$; 34 = 48 − 14 physical parameters. Analogue of the Stone–von Neumann theorem. See Uniqueness Theorem
3. Superpotential W — RESOLVED [T]: $W = \mu_W \sum f_{ijk}\Theta\Theta\Theta$ unique $G_2$-invariant (Schur's lemma) [T-50]; Kähler metric on $G_2$ moduli — [C] (Supersymmetry)
4. $\varepsilon = 10^{-2}$ — RESOLVED [T]: full minimisation of $V_{\text{Gap}}$ proven (T-64): $G_2$-orbital reduction $21D \to 5D$, unique global minimum, Hessian is positive definite — Gap Thermodynamics
5. 3+1 from $G_2$ — RESOLVED [T]: sector decomposition [T] + 3D from $SU(3)_C$ [T] (sector asymmetry [T-52]); Einstein equations on $M^{3+1}$ — [T] (T-65, full spectral action). Background independence — [T] (T-120): $M^4 = \mathbb{R} \times \Sigma^3$ derived from categorical structure via Gel'fand–Naimark–Connes chain — Emergent Manifold
6. Berry-phase derivation of $L_\text{top}$ — RESOLVED [T] (Sol.65): $\mathcal{L}_{\text{top}} = \frac{\lambda_3}{2\pi}\varphi_{ijk}\theta^{ij}\dot{\theta}^{jk}$ from $\mathrm{Im}(S_{\text{Keldysh}})$ + $G_2$-uniqueness. CS₁ replaced by Keldysh. T-85 — Berry Phase
7. Electroweak sector — RESOLVED [T]: uniqueness of the pair $(E,U)$ proven from $\kappa_0$ [T]. Was [H] → [C at (FE)] → [T] — uniqueness theorem
8. $m_b/m_t$ — RESOLVED [C]: QCD IR enhancement $\eta_{\text{QCD}} \approx 3.46$ + loop $y_b \approx 0.028$ gives $m_b/m_t \approx 0.024$ (observed $0.024$). Agreement $< 5\%$. Key correction: QCD enhances Yukawa couplings of light quarks in the IR — Yukawa Hierarchy
9. Neutrino generation numbering — RESOLVED [T]: $k=1 \to$ 3rd, $k=4 \to$ 2nd, $k=2 \to$ 1st [T-52]; normal hierarchy [T]

Computational

10. $Z'_\Phi(-2)$ — physical interpretation
11. Full functional integral (bosons + fermions + SUSY) on $(S^1)^{21}$ (Quantum Gravity)
12. Lattice computation on $(S^1)^{21}$ with $G_2$-symmetry
13. Two-loop correction to $\eta_F$
14. Non-perturbative dualities of Gap theory with M-theory

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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 $(S^1)^{21}/G_2$	C14 (ν $m_2/m_3$), C15 (PMNS), C16 ($\lambda_4$), C18 ($\Lambda$)
B. Empirical	Formulation [T]; validation requires measurements	G-mapping (D.2), ISF, ASC-parameters, calibration $d_\mathcal{A}$
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.

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Theorem Dependency Graph

Key derivation chains between theorems:

Fundamental chain (axioms → dynamics → consciousness):

$\text{A1–A5} \to \mathcal{L}_\Omega \to \{\text{primitivity [T]}, \text{LGKS [T]}\} \to \rho^*_{\mathrm{diss}} \to R \to \varphi \to \text{L-levels}$

Physical chain (spectral triple → gravity):

$\text{T-53} \xrightarrow{\text{spectral action}} \text{T-65 (Einstein)} \to \text{T-66 (UV-finiteness)} \to \text{T-71 (}\Lambda > 0\text{)}$

Consciousness chain (primitivity → hierarchy):

$\text{T-39a (primitivity)} \to \text{T-62 (}\varphi\text{-operator)} \to \text{T-67 (L3)} \to \text{T-86 (L4 unreachability)}$

SAD chain:

$\text{T-110 (Fano } \alpha=2/3) \to \text{C26 (}P_\text{crit}^{(n)}) \to \text{SAD\_MAX} = 3 \to \text{T-86 (L4 strengthened)}$

Promoted hypotheses:

Hypothesis	Was	Proof	Became
(FE) electroweak	[C]	Sol.1, T-1	[T]
(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]
$H \sim \gamma_{EU}$ (Higgs identification)	[H] (§1.1 Higgs Sector)	T-42a (κ₀) + T.1.1 (Fano line) + FE [T] (quantum numbers) + T-64 (vacuum)	[T] — Theorem 1.0
L1→L2 cascade dynamics	[H]	Transcritical bifurcation: $\kappa_0$-amplification via $\mathrm{Coh}_E \sim c \cdot \delta P$ (T-43b [T], HS-projection [T]). $T_{\mathrm{ign}} \sim (\delta P)^{-1} \cdot \kappa_0^{-1}$ (exponent $-1$, not $-1/2$)	[T] — Swallowtail
Cost of enlightenment	[H]	21 pairs $\times$ Landauer ($k_B T \ln 2$ per bit). $T_{\mathrm{eff}}$ 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 $\rho^*$) + T-62 [T] (CPTP) + T-55 [T] ($\varphi \neq \mathrm{id}$)	[T] — Measurement
L4 closure ($\omega$-groupoid)	[H] (#100)	Compactness of $\mathcal{D}(\mathbb{C}^7)$ + contractivity $k < 1$ [T] + T-86 [T] + T-55 [T]	[C at T-86, T-55] — Hierarchy
$O$-parity $P_O$ (Theorem 11.2)	[H]	T-42e [T] ($\mathrm{Stab}_{G_2}(e_O) = SU(3)$) + T-99 [T] ($f_{ijk} \in \mathbb{R}$ → $\mathbb{Z}_2$) + $[\sigma, \mathcal{L}_\Omega] = 0$ + T-69 [T] (barrier)	[T] — Dark Matter
Preferred measurement basis (Theorem 6.1)	[H]	$L_k = \lvert k\rangle\langle k\rvert$ — atoms of $\Omega$ [T] + $\mathcal{D}_\Omega$ kills off-diagonal [T] + diagonal = fixed points [T] + Zurek's einselection	[T] — Measurement
Stability of the chiral vacuum (§4.4)	[H]	T-99 [T] ($V_3$ unique PT-odd) + T-64 [T] (unique vacuum, positive Hessian) + T-69 [T] (barrier $\Delta V \geq 6\mu^2$)	[T] — Higgs Sector
(H1) Trainable CPTP-anchor ($M = 49$)	[H] (#116)	Stinespring ($M \leq N^2 = 49$) + Cybenko–Hornik (universal approximation of CPTP)	[T] — [#116]
(H-Hawk) Hawking radiation $T_H$, $dM/dt$	[H] (#117)	T-65 [T] (spectral action) + standard QFT on curved background	[T] — [#117]
(H-Pol) Polyakov loop $\langle L \rangle$ — order parameter	[H] (#118)	T-42e [T] ($\mathrm{Stab}_{G_2}(e_O) = SU(3)_C$) → $Z_3 \subset SU(3)_C$	[T] — [#118]
(H-Tc) Deconfinement temperature $T_c$	[H] (#119)	T-81 [C at T-64] ($\sqrt{\sigma}$) + standard lattice relation	[C at T-64] — [#119]
(H-V3) Scaling $m_c/m_t \sim \varepsilon^2$	[H] (#120)	T-43d [T] (Fano $f_{k,5,6}$) + double blocking	[C at T-64] — [#120]
(H-ΩDM) Dark matter $\Omega_{\mathrm{DM}} h^2 \approx 0.12$	[H] (#121)	T-163 [T] (O-parity) + T-51 [T] (O scale) + CKR	[C at T-50, CKR] — [#121]
(H-SBH) Gap correction in $S_{\mathrm{BH}}$	[P] (#122)	T-65 [T] + T-73 [T] (Gap = curvature) + T-74 [T] ($V_{\mathrm{Gap}}$ 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] ($f_{1,5,6} = 1$, $f_{2,5,6} = 0$) + $G_2$-uniqueness of $f_{ijk}$	[T] (hierarchy from tree-level rule) — [#123]
(H-δCP) Topological quantisation $\delta_{\mathrm{CP}} = 2\pi n/7$	[H] (#124)	T-38b [T] ($\tau \in \mathbb{Z}_7$) + T-2 [T] ($G_2$-covariance)	[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]	T-64 [T] self-consistent vacuum	[C at T-64] — 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

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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 $\mathcal{L}_0$), 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 $C_0(\mathbb{R})$)
• 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)
• 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-213 (Yoneda via Bures description length), T-214 (hard-problem meta-theorem, Lawvere positivity), T-216 (ε_eff closed form at T-64), 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 2013	§3.9 (cohesion) + §3.10 (super-cohesion)	Applicability to stratified $\mathcal{D}(\mathbb{C}^7)$-site pending (Gap A in proof doc §4.2)
T-186	Schreiber DCCT	§3.9 hexagon + Chern–Weil for G₂-bundles	Requires T-185 site-applicability + Chern–Weil on stratified site
T-211	Lurie HTT	5.2.7 (presentable coherence inheritance) + 6.3.1.16	Applicability: $\mathbf{PhysTheory}$ as full $(\infty,1)$-subcategory needs verification
T-212	Schreiber DCCT	§3.10 super-cohesive extension	Requires super-cohesive structure on UHM site
T-217	Baez–Dolan	Hirschowitz–Simpson 2001, Leinster 2002 (3-types ≃ coherent tricategories)	Applicability: $\tau_{\leq 3}(\mathbf{Exp}_\infty)$ in scope of correspondence needs verification
T-218	Milnor classifying-space	Singular complex of $B_\bullet \mathcal{C}$ 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; KO-dim 6 verified for UHM triple (T-53)
T-117	Goderis–Verbeure–Vets 1989	Quantum CLT on lattice observables	Clustering hypothesis for full $\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$ requires separate verification
T-119	Connes 2013 reconstruction	7-axiom NCG reconstruction theorem	6 of 7 axioms argued; first-order condition requires fuller treatment
T-221	Schreiber DCCT + Lurie HTT	Various (inherits from T-185/T-186/T-211/T-215/T-217)	Inherits applicability status of upstream framework citations
T-222	Brandão–Horodecki 2015; Yunger-Halpern 2023	Rényi second laws, non-Abelian thermodynamics	Scope-restricted to Markovian + $G_2$-covariant + low-T + viable

[T/sim] theorems (analytical core + numerical cross-check)

• T-59 (κ_bootstrap = 1/7): analytical from $\omega_0/N$; SYNARC mvp_int_2 G5 confirms to $10^{-10}$
• T-142 (SAD_MAX=3): state-independence [T]; $P_{\mathrm{crit}}^{(n)}$ 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 $\mathrm{corr}(\mathrm{Coh}_E, \kappa_{\mathrm{eff}})=-0.985$
• 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 ($\varphi$-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 (7-role uniqueness): [T at combinatorial-constraint stack]
• 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 (categorical-monistic route): [T]+[I] — consistency exhibited; fourth-route reading interpretive

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).

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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	[T]/[C]	CC-5, T-149 (fractal closure + embodied viability; not T-86, which is L4-unreachability)	predictions#предсказание-5
Pred 6	Minimal coherence	[T]	T-96, T-151	predictions#предсказание-6
Pred 7	Stability radius	[T]	T-104	predictions#предсказание-7
Pred 8	Capacity	[T]	T-107	predictions#предсказание-8
Pred 9	Learning bound	[T]	T-109	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 at upper bound	[C]	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	[T]	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 T-64]	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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