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• [✗] Retracted — proven erroneous or withdrawn

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)	[T at T-64]	Upgrades T-176 to closed form
T-217	UHM Fundamental Closures §11 (new 2026-04-17)	[T]	L3 tricategorical coherence via τ_{≤3}(Exp_∞) + Baez–Dolan; upgrades T-67 K=4 count to [T]
T-218	UHM Fundamental Closures §12 (new 2026-04-17)	[T]	SYNARC Cog = Sing(B·𝒞_FKraus) is Kan complex (Milnor); explicit horn-filler algorithm $O(\dim\mathcal D)$
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 + $G_2$-covariant). 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) All UHM observables $P,R,\Phi,\mathrm{Coh}_E,\Lambda,H,\pi_{\mathrm{bio}}$ are $G_2$-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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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	G₂-covariance of the Fano dissipator	Lindblad Operators T.11.2	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)	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	Exactly 3 fermion generations ($N_{\text{gen}} = 3$): upper bound $\leq 3$ from swallowtail $A_4$ + lower bound $\geq 3$ from uniqueness of the associative triplet $(1,2,4) \subset \mathbb{Z}_7^*$ + irreducibility of $\mathbb{Z}_3$ — raised from [H] (No.62)	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)
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: the finite spectral triple $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$ with KO-dimension 6 is constructed; $D_O$ and $D_3$ with opposite signs → $\eta_{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)$ — raised from [C]	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
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$\times 1.8$
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: compactness $(S^1)^{21}$ + $G_2$ Ward identities (14 constraints) + $\mathcal{N}=1$ SUSY (Seiberg) + APS-index = 0. All counterterms are forbidden	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)
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]
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	Equivalence of Freedom definitions (Sol.78): $\pi_0(\mathrm{Map}(\Gamma, T)) = \dim\ker(\mathcal{H}_\Gamma) + 1$ by Morse-Bott theory. $\mathcal{F}[\Gamma]$ — a Morse-Bott function on $\mathcal{D}(\mathbb{C}^7)$; gradient trajectories from $\Gamma$ to $\rho^*$ ↔ connected components $\pi_0$. Previously — upper bound	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 ($\sigma_A = 1 - \gamma_{AA}/P$, $\sigma_S = 1 - \mathrm{rank}(\Gamma_S)/3$, $\sigma_D = 1 - N\gamma_{DD}$, $\sigma_L = 7(1 - \gamma_{LL})/6$, $\sigma_E = 1 - D_{\mathrm{diff}}/N$, $\sigma_O = 1 - \kappa_0/\kappa_{\mathrm{bootstrap}}$, $\sigma_U = 1 - \Phi/\Phi_{\mathrm{th}}$). $\|\sigma_{\mathrm{sys}}\|_\infty < 1 \Leftrightarrow \mathcal{V}_{\mathrm{full}}$ (full viability, strictly stronger than $\mathcal{V}_P = \{P > 2/7\}$) — raised from [C] (CC-8)	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$)
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: $r_{\mathrm{stab}} = \sqrt{P(\rho^*_\Omega) - 2/7}$ — the maximum Bures perturbation preserving viability. From T-98 (balance) + CPTP contractivity + Fuchs–van de Graaf inequality	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-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$:$
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: $A_{\text{space}} \cong C(\Sigma^3)$ for the unique smooth 3-manifold $\Sigma^3$. From T-117 (commutativity) + Gel'fand–Naimark + Connes reconstruction (2008). Key new result — raised from [P]	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: $\Lambda_{\text{Gap}} > 0$ (T-71 [T]) $\Rightarrow$ $\Sigma^3 \cong S^3$ (closed), de Sitter metric. 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
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]$
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 at $\alpha=2/3$ state-independence] + [C at $P_{\mathrm{crit}}^{(n)}$ formula derivation]: $\alpha = 2/3$ state-independence from $\dim=7$ + PG(2,2) is rigorous [T]. The iterated critical purity formula $P_{\mathrm{crit}}^{(n)} = P_{\mathrm{crit}} \cdot 3^{n-1}/(n+1)$ is derived heuristically from the Fano contraction compounding; the $1/(n+1)$ denominator accounts for normalization by depth but its first-principles derivation is pending. Unconditional SAD_MAX = 3 rests on this heuristic formula; the specific inequality $R^{(3)} \leq 0.130 < 0.200$ is empirical. 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)}$
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$
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-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,
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
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 [С при 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 [С при 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 [С при 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]
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 (requires multi-holon clustering)	[С]	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
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 [О]	Dimensions
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
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
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
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
T-189	MaxEnt derivation of the Bures metric (Char-IV) (reframed 2026-04-17): set $g^{ij}=C^{ij}_{\mathrm{SLD}}$, where $C_{\mathrm{SLD}}$ is the SLD covariance — a Petz-free physical observable defined from $\partial\rho=\tfrac12(L_i\rho+\rho L_i)$ without reference to any metric. Then Bures is uniquely selected via $g_B^{-1}=C_{\mathrm{SLD}}$ (Braunstein–Caves 1994). Status [T]: the selector equation and uniqueness of Bures solving it are proven. Caveat: this is a physical recasting of Char-III (SLD Fisher), not a logically independent fourth witness. Adds physical-mechanism clarity: the metric is determined by the state's own fluctuation structure, not by interpretive choice. Inspired by Vanchurin (2026, arXiv:2603.15198)	Cohesive Closure §5.3 Char-IV
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
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
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
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
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 constant factor $C_2 \leq 4$: $d_\mathrm{free}/(14\varepsilon^2) \leq N_\mathrm{learn} \leq C_2\cdot d_\mathrm{free}/(14\varepsilon^2)$. Lower bound = QCR (T-109); upper bound via Polyak–Łojasiewicz on Bures manifold + $G_2$-equivariance of Fano channel (T-41g) + Lipschitz Bures Hessian $L \leq 4/(14\omega_0)$. Closes learning-efficiency gap in AGI-sufficiency (A4). Derived in SYNARC paper Appendix G (Theorem G.3)	SYNARC paper App. G.3
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
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
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
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
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
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
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
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
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)
T-204	Pareto-optimal bounded rationality [T]: для resource budget $\mathcal{B} = (C, M, \varepsilon)$ (compute, memory, precision), effective dimension $d_\text{eff}(\mathcal{B}) = \min(49, \log_2 M, 14\varepsilon^2 C/c_\text{step})$. 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
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
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 $N_\text{samp} \geq 7/(14\varepsilon^2)$. 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
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
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
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}
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
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
T-213	Yoneda representability via Bures description length [T] : define $D_B(f):=\min	\mathrm{Kraus}(\rho_f)
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
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
T-216	Closed-form analytical εeff [T at T-64] : $\varepsilon_\mathrm{eff}=4 N_{33}^\mathrm{Fano}/(9	\bar\gamma
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
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. 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
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
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
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
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
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 reflection operator $R$ (T-96/T-98). 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
T-153a	Substrate-existence companion to T-153 — stratified [T at necessary conditions]+[T sketch at sufficiency]: 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 sketch]: the proof sketch gestures at Stinespring + Choi for $\dim > 7$ but the explicit construction of $G$ for general admissible substrates is not yet fully demonstrated — full proof is pending rigorous treatment. Removes the earlier ambiguity "any system might admit some faithful G" for the necessity direction.	Substrate-Independent Closure §T-153a
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

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Level [C]: Sensorimotor Theory

#	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: $\tau_p^{(G_2)} \sim 10^{72}$ yr (negligible)	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	Raised to [T]: $N_{\text{gen}} = 3$ exactly (upper bound $\leq 3$ from swallowtail $A_4$ [T] + lower bound $\geq 3$ from uniqueness of associative triplet $(1,2,4)$ [T] + irreducibility of $\mathbb{Z}_3$) — 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
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)
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)}
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}
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}
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
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	Raised to [T]: APS-index + $G_2$ Ward identities + $\mathcal{N}=1$ SUSY (Seiberg) — strict non-perturbative proof for the scalar-fermion sector. 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
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}}$~~
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 $
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}
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], numerical estimate $\sim 10^{-120 \pm 10}$ [C] — Λ Budget
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
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
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
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
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 $\varepsilon^{12}$ raised to [T]; full minimisation T-64 [T]; total ~$10^{-120 \pm 10}$ [C]. Remaining: computational task (numerical minimisation)	Λ Budget

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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 (2026-04-17 audit): total $\sim 10^{-120\pm10}$ [C at T-64, H* at $Z'_\Phi(-2)$, and computational task pending]. The $\pm 10$-order band reflects uncertainty in the not-yet-computed sectoral minimisation, not a robustly established prediction. Full closure requires numerical minimisation of $V_\mathrm{Gap}$ on $(S^1)^{21}/G_2$ — an explicit computational programme. See Λ Budget.

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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.

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. 79 orders of Λ — structurally closed [C]: spectral formula $\Lambda_{\text{CC}}$ via $a_0, a_2, a_4$ [T] (T-65); SUSY-breaking $\varepsilon^{12}$ [T]; 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: $\mathcal{G}_{\text{total}} = \mathcal{G}_O + O(\bar{\varepsilon}^2)$). Total $\sim 10^{-120 \pm 10}$ [C]. Full closure — computational task (Λ Budget)
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.
• [O] — 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]	T-86	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-113	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-125	predictions#предсказание-14
Pred 15	Attractor at upper bound	[C]	T-124	predictions#предсказание-15
Pred 16	L1→L2 avalanche	[T]	T-158	predictions#предсказание-16
Pred 17	Critical exponents	[T]	T-161	predictions#предсказание-17
Pred 18	Ward suppression	[T]	T-159	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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