Unitary Holonomic Monism

A Formal Theory of Reality and Consciousness

Unitary Holonomic Monism (UHM) is a formal theory describing the structure, dynamics, and phenomenology of reality through a single mathematical primitive — the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$.

Meta-theory status

UHM claims the role of a meta-theory (unifying physics, consciousness, and information within a single axiomatic framework). What is proven about this claim, as of 2026-09-26: The universal property [T] (T-174, restated) is carried by the kinematic object $u_0 = (A_{\text{int}}, \text{trivial dynamics})$, $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$, and goes from UHM to a theory: morphisms $u_0 \to (A, \sigma)$ in the $C^*$-typed part of $\mathbf{PhysTheory}$ are exactly the $A_{\text{int}}$-structures (a projection and two systems of $3 \times 3$ matrix units summing to 1) in the dynamically fixed algebra $A^\sigma$; on $\mathbb{C}^n$ a faithful one exists iff $n \geq 7$, and the multiplicity-free one exists only for $n = 7$, where it is unique up to $U(7)$; on states it acts through the unique trace-preserving conditional expectation — for $n = 7$ the sector pinching. The former claim — an essentially unique receiving morphism from every theory with $A_{\text{int}} \subset \mathcal{A}$, CPTP dynamics and $\leq 7$ observables into $\mathfrak{T}$, up to $G_2 \times \mathbb{R}_{>0}$ — is retracted [✗]: no $*$-homomorphism $M_7(\mathbb{C}) \to A_{\text{int}}$ exists, as monoid maps such morphisms are not unique, and the proof's "subtopos of modules", "$\mathrm{Mod}(A_{\text{int}}) \simeq \mathcal{D}(\mathbb{C}^7)$" and "Takesaki homomorphism" are false. $\mathbf{PhysTheory}$ is an $(\infty,1)$-category as a Grothendieck construction over $\mathbf{Topoi}_\infty$ (T-211 [T]; its former full embedding into $\mathbf{Topoi}_\infty$ is retracted). Rigidity of the primitive is stated [T] (T-173); T-174 no longer uses it. Alternative programmes (ToE embeddings): T-170 — [T] for the group coincidence $\mathrm{Stab}_{GL(7)}(\varphi_0) = \mathrm{Aut}(\mathbb{O}) = G_2$ (the holonomy group of M-theory compactifications), finiteness of the UHM integral at finite $M$ and existence of limit states; the equality of UHM and M-theory partition functions is a hypothesis [H] (its M-theory side is not defined). T-171 [T]: every finite spin network, with unbounded spin, is encoded injectively in a state of $\lvert V\rvert$ holons (T-171' is its corollary; the cluster construction is retracted). T-172 [T]: every finite causal set is encoded in a state, and finite posets embed fully faithfully as internal categories of $\mathbf{Sh}_\infty(\mathcal{D}(\mathbb{C}^7))$. These are encodings and a shared symmetry group, not derivations of LQG, causal-set or M-theory dynamics. The Standard Model in its NCG form ($\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$) contains no copy of $A_{\text{int}}$; it is reached from $A_{\text{int}}$ by construction (T-176). "Meta" therefore means: the $A_{\text{int}}$-structure of UHM is universal (corepresenting) and rigid exactly on $\mathbb{C}^7$, and the SM and gravity are derived from the primitive — not that the primitive receives every theory, nor that every known theory embeds into it.

The theory:

• Derives space, time, and metric from categorical structure
• Formalizes the connection between physics and consciousness
• Defines an interiority hierarchy (L0→L4): from minimal internal structure to reflective consciousness
• Derives the minimal structure of a self-sustaining system (7 dimensions)
• Establishes bounds of explanation — what the theory explains and what it takes as primitive

Etymology of the Name

• Unitary — from Lat. unus (one): reality is described by a single ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$; the underlying unitary evolution preserves information
• Holonomic — from Gr. holon (whole) + nomos (law): every part (Holonom) contains an image of the whole and obeys universal laws
• Monism — from Gr. monos (one): reality is one — there are no independent "layers" or "substances." In UHM this is a corollary of the terminal-object axiom ($H^*(X) = 0$ follows from Property 3), read ontologically through the PID definition — a theorem relative to the axioms, not an independent discovery

Theory Structure

Five Structural Properties of the Sole Primitive (Ω⁷)

Sole primitive

The ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ is the sole primitive of UHM theory. The notion of a "sheaf" in the ∞-topos is defined via a Grothendieck topology on the category $\mathcal{C}$.

#	Property	Formulation
1	Finite-dimensionality	$\text{Ob}(\mathcal{C}) \subset \mathcal{D}(\mathbb{C}^{42})$
2	Constraint	$\hat{C} \cdot \Gamma = 0$ (Page–Wootters)
3	Terminal object	$\forall \Gamma, \exists! f: \Gamma \to T$
4	Self-modeling	$\varphi \dashv i: \text{Sub}(\Gamma) \hookrightarrow \mathbf{Sh}_\infty$ (adjunction)*
5	Stratification	$X = \bigsqcup_\alpha S_\alpha$, $S_0 = \{T\}$

*The variational characterization $\varphi = \arg\min \mathbb{E}[S_{spec} + D_{KL}]$ was stated as a theorem about the categorically defined φ; it is retracted (2026-09-25): the functional equals the cross-entropy $-\mathrm{Tr}(\psi(\Gamma)\log\Gamma)$ and is minimised by a projection onto the top eigenvector of $\Gamma$, not by φ (FEP derivation, retraction box).

Connection to the Septicity Axiom

The Septicity Axiom (AP+PH+QG+V) is a set of consequences of Ω⁷ — operational requirements that any viable system must satisfy.

Theorem on degrees of freedom (consequence of Ω⁷)

The number of structurally distinct directions of development of a configuration $\Gamma$ — $\mathrm{Freedom}(\Gamma) = \dim\ker(\mathcal{H}_\Gamma) + 1$ — is a topological invariant. Systems with sufficient coherence possess a nontrivial choice space ($\mathrm{Freedom} > 1$).

Theorem S (justification of Axiom 3) [T]

N = 7 (Axiom 3) is the minimal dimension for satisfying (AP)+(PH)+(QG). All 7 dimensions are necessary and functionally unique [T]: A, S, D, L, U — algebraically; E, O — categorically (via the κ₀ formula). Proof →

Second justification (not independent of Theorem S: step T8 of the chain takes $N = 7$ from it; it read "second, independent" until 2026-09-25): theorems P1+P2 [T] (derived from (AP)+(PH)+(QG)+(V) via the T15 chain, whose step PG(2,2) → $\mathbb{O}$ takes the canonical orientation of the seven Fano lines — only 16 of the 128 orientations give a normed algebra, and they are the only orientation class invariant under the collineations of the design, T15-canon, registry row 41n; [C at (Alt)] earlier on 2026-09-25) yield $N = \dim(\mathrm{Im}(\mathbb{O})) = 7$ through the Hurwitz theorem. Structural derivation →

Key Results

Construction	Formula	Status
Base space	$X = \|N(\mathcal{C})\|$	[T] Derived
Cohomological monism	$H^n(X) = 0$ for $n > 0$ (locally constant coefficients)	[T] corollary of Property 3
Local physics	$H^*_{loc}(X, T) \neq 0$	[T] Theorem
Time	Clock register $\tau \in \mathbb{Z}_7$ (Page–Wootters)	Clock register [T]; the Page–Wootters link (constraint $\hat{C}\Gamma = 0$) [C] (T-87, step 4); the dynamics runs relative to the depth register — [T] (T-53b), with the time line $C_0(\mathbb{R})$ as its scaling limit (T-118 [T]); read "[T] Derived" until 2026-09-25
Arrow of time	$\dim(X_n) \geq \dim(X_{n+1})$ along the stratal depth $n \in \mathbb{N}$, monotone in the parameter $t$ of the Lindblad semigroup — the free energy $F$ is the Lyapunov functional of the full flow; $S_{vN}$ is monotone only for the unital part (the reset channel lowers it); the cyclic tick $\tau \in \mathbb{Z}_7$ carries no arrow; along the readings of the depth register the purity of the unital primitive part falls strictly (emergent time, Theorems 11.1–11.2)	[T] relative to the depth register (T-53b; "[T] Theorem" until 2026-09-25, then [C at an aperiodic time parameter] the same day)
Metric	$d_{strat}$ (Connes on strata)	[T] Derived
Evolution equation	All 3 terms ($H_{\text{eff}}$, $\mathcal{D}_\Omega$, $\mathcal{R}$) derived from axioms	[T] Fully
Conscious window (Goldilocks zone)	$P \in (2/7, 3/7]$: viability $\wedge$ reflexivity ($R \geq 1/3$ when $P \leq 3/7$)	[T] (T-124)
Octonionic structure	(AP)+(PH)+(QG) →[T1–T10]→ $\mathbb{O}$ → N=7, $G_2$, Fano, H(7,4)	[T]: steps 1–9 give the unoriented design PG(2,2) [T]; step 10 takes the canonical orientation, the unique collineation-invariant class (T15-canon, row 41n; [C at (Alt)] earlier on 2026-09-25); the strict necessity of $N = 7$ stays [C at (P1₆)]

7 Dimensions of the Holonom

Symbol	Dimension	Function	Mathematical operator
A	Articulation	Distinction, boundaries	Projector $P: P^2 = P$
S	Structure	Form retention	Hamiltonian $H: H^\dagger = H$
D	Dynamics	Change	Unitary operator $U(\tau) = e^{-iH_{eff}\tau}$
L	Logic	Coordination	Commutator $[A,B] = AB - BA$
E	Interiority	Experience	Density matrix $\rho_E$
O	Foundation	Vacuum coupling + internal clock	Page–Wootters, $H_O$, $V_O$
U	Unity	Integration	Trace $\mathrm{Tr}$

State space:

$$\mathcal{H}_{total} = \mathcal{H}_O \otimes \mathcal{H}_{6D} = \mathbb{C}^7 \otimes \mathbb{C}^6 = \mathbb{C}^{42}$$

Two formalisms: 7D and 42D

The theory uses two related formalisms:

Formalism	Dimension	Application
Minimal	$\mathbb{C}^7$	Conceptual basis, minimality theorems
Page–Wootters	$\mathbb{C}^{42} = \mathbb{C}^7 \otimes \mathbb{C}^6$	Operational calculations, emergent time

In the minimal formalism, $\mathcal{H}_O$ is one of the 7 dimensions (basis vector $|O\rangle$). In the extended formalism, $\mathcal{H}_O \cong \mathbb{C}^7$ is the internal clock space with 7 time states τ ∈ ℤ₇.

The formalisms are related by Morita equivalence [T]: $\mathrm{Sh}_\infty(\mathcal{C}|_7) \simeq \mathrm{Sh}_\infty(\mathcal{C}|_{42})$ (Lurie comparison theorem). All 7D formulas are exact, not approximations. See Coherence matrix.

Central Concepts

Coherence Matrix Γ (object of category $\mathcal{C}$)

$$\Gamma \in \text{Ob}(\mathcal{C}), \quad \Gamma^\dagger = \Gamma, \quad \Gamma \geq 0, \quad \mathrm{Tr}(\Gamma) = 1$$

• Diagonal elements $\gamma_{ii}$: probabilities of being in dimension $i$
• Off-diagonal elements $\gamma_{ij}$: coherences (quantum correlations) between dimensions

Purity

$$P = \mathrm{Tr}(\Gamma^2) \in \left[\frac{1}{7}, 1\right]$$

• $P = 1$: pure state (maximal coherence)
• $P = 1/7$: maximally mixed state (complete decoherence)
• $P > P_{\text{crit}} = 2/7 \approx 0.286$: viability condition (theorem)
• $P \in (2/7,\, 3/7]$: conscious window (Goldilocks zone) — viability $\wedge$ reflexivity $R \geq 1/3$; $P = 3/7$ is the upper bound (T-124)

Terminal Object T

$$T = \Gamma^* : \varphi(T) = T, \quad \forall \Gamma \in \mathcal{C}, \exists! f: \Gamma \to T$$

Interpretation: T is the global attractor toward which all trajectories converge. The arrow of time is the stratal collapse toward T.

Evolution Equation

With emergent internal time τ:

$$\frac{d\Gamma(\tau)}{d\tau} = \underbrace{-i[H_{eff}, \Gamma]}_{\text{unitary}} + \underbrace{\mathcal{D}[\Gamma]}_{\text{dissipation}} + \underbrace{\mathcal{R}[\Gamma, E]}_{\text{regeneration}}$$

where:

• τ — internal time (parameter of conditional states relative to O)
• $H_{eff} = H_{6D} + \langle\tau|H_{int}|\tau\rangle_O$ — effective Hamiltonian (from the Page–Wootters constraint)
• $\mathcal{D}[\Gamma]$ — Lindblad dissipator
• $\mathcal{R}[\Gamma, E] = \kappa(\Gamma) \cdot (\rho_* - \Gamma) \cdot g_V(P)$ — regenerative term [T] (full derivation from axioms)

Interiority Hierarchy

Level	Name	Condition	n-truncation
L0	Interiority	$\exists \rho_E$	$\tau_{\leq 0}$ (set)
L1	Phenomenal geometry	$\mathrm{rank}(\rho_E) > 1$	$\tau_{\leq 1}$ (groupoid)
L2	Cognitive qualia	$R \geq 1/3$, $\Phi \geq 1$, $D_{\text{diff}} \geq 2$	$\tau_{\leq 2}$ (bicategory)
L3	Network consciousness	$R^{(2)} \geq 1/4$ (metastable)	$\tau_{\leq 3}$ (tricategory)
L4	Unitary consciousness	$\lim_{n \to \infty} R^{(n)} > 0$, $P > 6/7$	$\tau_{\leq \infty}$ (∞-groupoid)

Threshold values (L2 thresholds):

• R (reflexivity) — measure of proximity to the dissipative attractor: $R = 1/(7P)$, where $P = \mathrm{Tr}(\Gamma^2)$
• Φ (integration) — connectivity measure: $\Phi = \sum_{i \neq j} |\gamma_{ij}|^2 / \sum_i \gamma_{ii}^2$
• $R^{(n)}$ (n-th order reflexivity) — meta-reflexivity measure: $R^{(n)} = \mathrm{Fid}(\varphi^{(n-1)}(\Gamma), \varphi^{(n)}(\Gamma))$

Threshold value statuses:

• $P_{\text{crit}} = 2/7$ [T] — lower bound of viability (Frobenius norm distinguishability)
• $P_{\text{max}} = 3/7$ [T] — upper bound of the conscious window: $R = 1/(7P) \geq 1/3$ holds if and only if $P \leq 3/7$; the Goldilocks zone $P \in (2/7, 3/7]$ is nonempty (T-124)
• $R_{\text{th}} = 1/3$ [T] — $K = 3$ from the triadic decomposition + Bayesian dominance
• $\Phi_{\text{th}} = 1$ [T] — the unique self-consistent value at $P_{\text{crit}} = 2/7$ (T-129)
• $D_{\min} = 2$ [D] — an independent L2 threshold, not a consequence of $\Phi_{\text{th}} = 1$ (T-151; it read "[T] — unconditional consequence of $\Phi_{\text{th}} = 1$" until 2026-09-25)

Level statuses

• L0–L2: stable states for biological systems
• L3: metastable state (finite lifetime $\tau_3$); threshold $K = 4$ [T] from quadratic decomposition (T-67)
• L4: theoretical limit; categorical unreachability [T] (T-86: $L4=\mathrm{colim}_{n}\tau_{\leq n}(\mathbf{Exp}_\infty)$ not reachable in finitely many steps, + Lawvere incompleteness T-55) — an attractor, not a physical state (the coherence-survival ratio $S^{(n)}\to 0$; the fidelity $R^{(n)}_{\mathrm{fid}}\to 1$ is a different quantity, see disambiguation)
• SAD metric [T], SAD_MAX = 3 [T] (T-142): generalization of L0–L4 to the continuous case via the representational tower; SAD = max{k : R^(k) > 1/(k+2)}, spectral formula [T], stress-dependent regime [T] — Depth tower

Formal Results

Theorem	Statement	Status	Reference
Cohomological monism	$H^n(X) = 0$ for $n > 0$ (locally constant coefficients; corollary of Property 3)	[T]	Consequences
Local nontriviality	$H^*_{loc}(X, T) \neq 0$	[T]	Consequences
7D minimality	$n < 7 \Rightarrow$ violation of (AP), (PH), or (QG)	[T]	Proof
Fixed point of φ	$\exists! \Gamma^* : \varphi(\Gamma^*) = \Gamma^*$ for a fixed anchor; for the canonical $\varphi_{\mathrm{coh}}$ it is $I/7$, $P = 1/7$ (not $2/7$, retracted 2026-09-25)	[T]	Proof
Attractors of an isolated holon	With a unital self-model (the canonical $\varphi_{\mathrm{coh}}$, every $G_2$- or $\Gamma_{\!\text{oct}}$-covariant linear one) the only stationary state is $I/7$ and $P$ never increases — dead isolation; with the self-registering $\varphi_s(\Gamma) = k\mathcal{P}_\alpha(\Gamma) + R\,\Gamma^2/\mathrm{Tr}\,\Gamma^2$ there are seven attractors with $P > 2/7$ for $\lVert H\rVert < h_0$ — self-sustaining attractors; under backbone dominance an embodied holon has exactly one (T-124c, restated). The former "at most one nontrivial attractor" and "the φ-tower converges for every holon" are retracted (T-124c, T-191 restated)	[T]	Theorem
Emergent time	Three constructions of the cyclic clock $\tau \in \mathbb{Z}_7$ (Page–Wootters, Bures, ∞-groupoid) are equivalent [T] (T-53a); the dynamics runs in the parameter of the Lindblad semigroup, whose finite carrier is the depth register: one state-independent Feynman–Kitaev constraint gives the conditional states exactly $e^{n\Delta t\,\mathcal{L}}\rho_0$ at every reading (T-53b), and the reading algebras converge to $C_0(\mathbb{R})$ (T-118)	[T]; dynamics [T] (it read [C at an assumed aperiodic time] until 2026-09-25)	Theorem
Arrow of time	Stratal collapse along the depth $n \in \mathbb{N}$: $\dim(X_n) \geq \dim(X_{n+1})$, monotone in the semigroup parameter $t$ (Lyapunov functional $F$; $S_{vN}$ only for the unital part), exact along the readings of the depth register; on a ring of readings the arrow breaks exactly once per period, which is optimal	[T] (it read [C at an aperiodic time parameter] until 2026-09-25)	Theorem
Critical purity	$P_{\text{crit}} = 2/N = 2/7$	[T]	Theorem
Necessity of interiority	$\text{Viable}(\mathbb{H}) \land \mathcal{D}_\Omega \neq 0 \Rightarrow \mathrm{Coh}_E \geq \mathrm{Coh}_{\min} > 1/7$	[T]	Theorem 8.1
$G_2$-rigidity	The holonomic representation is unique up to $G_2 = \mathrm{Aut}(\mathbb{O})$ kinematically and up to the finite frame group $\Gamma_{\!\text{oct}}$ dynamically; 34 kinematic $G_2$-invariants, 48 physical parameters (frame decision D-0910)	[T]	Theorem
Electroweak sector	$SU(2)_L \times U(1)_Y$ from the pair $(E,U)$ of $\kappa_0$ and the Higgs line $\{A,E,U\}$ (both [T]) — the group [C at (FE)]; its uniqueness [H]: no uniqueness theorem for $SU(2)\times U(1)$ exists, and the octonionic routes that derive Standard-Model structure end with an extra $U(1)$ (Furey and Hughes 2022: SM + $B-L$). Listed as "unique rank-4 construction [T]" until 2026-09-25. Through the Clifford system of $\mathbb{C}\otimes\mathbb{O}$ the whole group $(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6$ is the normaliser of colour in $\mathrm{Spin}(9)$ and the electroweak algebra is the centraliser of colour (T-326); in the doublet sector there is no extra $U(1)$, and $B-L$ returns only with the right-handed fields (T-329)	axis construction [C at (FE)], uniqueness there [H]; through T-326 [T] as mathematics, [C at (Cl)] in UHM	Theorem
Chirality of the doublets (T-327)	On $\mathcal{S} = \mathbb{C}\otimes\mathbb{O}$: $(\mathbf 3,\mathbf 2)_{1/6}\oplus(\mathbf 1,\mathbf 2)_{-1/2}$, not self-conjugate for every $G_{\mathrm{SM}}$-invariant complex structure — the Distler–Garibaldi test passed without a choice; right-handed singlets not derived	[T] as mathematics; [C at (Cl)]	Theorem
Three generations	$N_{\text{gen}} = 3$: exact count $\|\mathrm{QR}(7)\| = (7-1)/2 = 3$ [T]; physical identification [I]. Families must be horizontal (T-328): on one copy of $\mathcal S$ only $U(1)_B\times U(1)_L$ commutes with $G_{\mathrm{SM}}$, triality is not a family symmetry, and the clock register gives a horizontal three (three real harmonics of $\mathbb Z_7$) [T]; the identification with generations is [C at (GC)] — families	[T]+[I]; T-328 identification [C at (GC)]	Theorem
Fano Yukawa selection	$y_k = g_W \cdot f_{k,E,U} \cdot \|\gamma_{\text{vac}}^{(EU)}\|$ via octonionic $f_{ijk}$; the selection by $f_{k,E,U}$ is [T], and reading $\gamma^{(EU)}_{\text{vac}}$ as the Higgs vacuum value rests on $H \sim \gamma_{EU}$, a hypothesis [H]	[T] for the selection	Theorem
Source instability	$\Gamma_\odot = I/7$ is non-stationary: $F_0 \neq 0$, drift toward $\rho^*$, self-amplification	[T]	Proof
Free will	$\mathrm{Freedom}(\Gamma) = \dim\ker(H_\Gamma) + 1$; monotonicity under CPTP, $G_2$-invariance	[T]	Theorem
$A_4$-bifurcation	Swallowtail from 3 parameters $(\kappa, \alpha, \Delta F)$ + $\mathbb{Z}_2$-purity symmetry	[T]	Theorem
Gap-injection of L-levels	$L(\Gamma_1) \neq L(\Gamma_2) \Rightarrow [\mathrm{Gap}(\Gamma_1)] \neq [\mathrm{Gap}(\Gamma_2)]$	[T]	Theorem
Generation assignment	$k=1 \to$ 3rd [T] (unique nonzero tree-level Yukawa); $k=4 \to$ 2nd, $k=2 \to$ 1st [C at (SA)], with (SA) a hypothesis (it read [T] until 2026-09-25)	[T]; ordering [C at (SA)]	Theorem
Superpotential	$W = \mu_W \sum f_{ijk}\Theta\Theta\Theta$ — unique $G_2$-invariant (Schur's lemma)	[T]	Theorem
Right-handed neutrino mass	$M_R \sim 2.9 \times 10^{14}$ GeV from PW clock + viability	[T]	Theorem
3+1 from sector decomposition	Former statement: $7 = 1_O \oplus 3_{A,S,D} \oplus \bar{3}_{L,E,U}$; $\dim(\text{space}) = 3$. Retracted 2026-09-25: as an axis-labelled real decomposition it is false — $\mathrm{SU}(3)$ acts irreducibly on the six non-$O$ axes, and no three of them span an invariant subspace; the complexified $\mathbb{C}^7 = \mathbb{C} \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$ is Günaydın and Gürsey (1973) and gives colour, not space	[✗]	Retraction
3+1 that commutes with colour (48c)	$SU(3)_C$ fixes in the spin factor $\mathfrak h_2(\mathbb O) \cong \mathbb R^{1,9}$ exactly $\mathfrak h_2(\mathbb C_O)$: dimension 4, signature $(1,3)$ (the sign of $\det$, no reflection positivity needed); $SL(2,\mathbb C_O)$ acts on it commuting with colour, so Coleman–Mandula is respected; no rotation of the seven axes commutes with colour	[T] as mathematics; [C at (L)], (L) ⟺ (P) — 48e(f)–(i) as physical spacetime	Theorem
Sector hierarchy $\varepsilon$	The homogeneous vacuum is not stationary [T] (T-61, Theorem 14.1). Corrected 2026-09-25: the vacuum of the $G_2$-invariant $V_{\text{Gap}}$ is unique up to $G_2$ — for every $\kappa > 0$ off the transition curves [T]: $I/7$ (always for $\kappa \le \mu^2/48$) or one colour-invariant orbit $S^6$ (T-64); $\kappa$ itself is fixed by no derived source (T-331(e)), and the no-go is strengthened by T-331(f) [T]: the associator weight vanishes for every functional that does not tell triples of axes apart — everything the dynamics of an isolated holon determines — and a readout that does resolve the Fano lines can carry any weight, so $\kappa$ stays a free coupling — and has no sector structure, so the sector values are the hypothesis (SV) [H]. $\bar\varepsilon$ is the root mean square over the 15 non-O pairs, $\approx 0.027$ under (SV) — order $10^{-2}$ [C at (SV)] (registry C35); the earlier $0.023$ came from substituting $\varepsilon_O \approx 0.04$ against the table's $\varepsilon_O \sim 1$, with which the 21-pair mean is $0.53$	[T]; uniqueness [T]; value [C at (SV)]	Theorem
No topological $\Lambda$-term	$H^{n>0}(X) = 0$ forbids a $\Lambda$-contribution of the form $\int_X c$; the vacuum energy is not cancelled (a degree-0 quantity)	[T] narrow; the "global cancellation" reading retracted 2026-09-10 (Λ-budget §4.1)	Theorem
Einstein equations from spectral action	Full triple (T-53) → $S = \mathrm{Tr}(f(D_A/\Lambda))$ → EH, $G_N = 3\pi/(7f_2\Lambda^2)$: the heat-kernel coefficient $a_2$ sees the internal space only through $\mathrm{Tr}(1) = 7$. The "+ SM" part is imported from Connes' finite triple (no KO-dimension-6 structure exists on $\mathbb{C}^7$; T-178 is retracted as a derivation) and inherits its history: $m_H \approx 170$ GeV (Chamseddine–Connes–Marcolli 2007), excluded by CDF and D0 in 2008; the 2012 rescue adds a singlet $\sigma$ and a fitted parameter	EH [T] for the formula; the value needs the cut-off convention $f_2$ and the scale $\Lambda$ [D]; SM part imported	Theorem
UV-finiteness of Gap theory	Compactness of $(S^1)^{21}$ + $G_2$-Ward ($21 \to 7$) + $\mathcal{N}=1$ SUSY (Seiberg) + $\varepsilon^{12}$ suppression (T-219, a hypothesis [H] since 2026-09-25)	field-space [T], full order-by-order [C] (structural, with the [H] ingredient T-219)	Theorem
Lorentzian signature	$(1,3)$: one time direction from the Page–Wootters clock [T] as a count (the time line $\mathbb{R}$ itself is T-118, [T] through the depth register; [C at an aperiodic clock] earlier on 2026-09-25); three space directions from $S^3$ (T-119, [T] as mathematics since 2026-09-25, reading [I]); Lorentzian sign at reflection positivity (bounded-below PW generator / Osterwalder–Schrader). The row read "$(1,3)$-split [T]" until 2026-09-25. A second route to the same signature, without reflection positivity: Theorem 48c, [T] as mathematics, [C at (L)], (L) ⟺ (P) — 48e(f)–(i) as spacetime	[C] (registry row T-53)	Theorem
7D↔42D: section–retraction (T-58′)	$\pi\circ\iota = \mathrm{id}$; 7D formulas exact on their own. The Morita equivalence $\mathrm{Sh}_\infty(\mathcal{C}\|_7) \simeq \mathrm{Sh}_\infty(\mathcal{C}\|_{42})$ is retracted — it fails on dimension	[T] / [✗]	Theorem
Spectral gap of Fano dissipator	$\lambda_{\text{deco}} = 5\gamma/(3N)$ (BIBD symmetry); $\kappa_{\text{bootstrap}} = \omega_0/N \gg \lambda_{\text{gap}}/N$	[T]	Theorem
φ-operator (replacement channel)	$\varphi_k(\Gamma) = (1-k)\Gamma + k\rho_*$ — CPTP, monotonicity, fixed point $\rho_*$	[T]	Theorem
Global minimization of $V_{\text{Gap}}$ (T-64)	Corrected 2026-09-25: the page's cubic term is not $G_2$-invariant; every $G_2$-invariant cubic is PT-even, and the associator cubic $\mathcal A$ is the one that goes through the associator (T-331). For $V = \mu^2\mathcal G + \lambda_4\mathcal G^2 - \kappa\mathcal A$: vacuum $I/7$, unique, for $0 < \kappa \le \mu^2/48$; spontaneous Gap for $\kappa > \min(7\mu^2/48, \kappa_1)$ [T]; the vacuum is one orbit $S^6 = G_2/\mathrm{SU}(3)$ with colour unbroken [T] (the real twirl inequality (RT) proven 2026-09-25; two orbits coexist only on the transition curves). The $G_2$-orbit reduction $21D \to 5D$ and the Hessian $18/6/12\mu^2$ are retracted; the sector values are the hypothesis (SV)	[T]; (SV) [H]	Theorem
No-signalling of the full dynamics (Theorem 8.5)	The only state of a part that nothing done to the rest can change is its marginal, so the regeneration acts on unconditioned marginals, the selective (Lüders-in-the-dynamics) reading is not a UHM dynamics, and no operation at a distant partner changes a holon's statistics	[T] (the reading is forced; it was [C] until 2026-09-25)	Theorem
Computational power (Theorem 8.6)	The Abrams–Lloyd amplification runs on marginals: at the saddle of $\varphi_s$ the ideal regenerative dynamics decides satisfiability in time linear in $n$, so "UHM computes no more than BQP" would imply NP $\subseteq$ BQP; with noise open	[T] for the ideal dynamics; with noise [H]	Theorem
Higgs identification $H \sim \gamma_{EU}$ (Theorem 1.0)	A vacuum value of $\gamma_{EU}$ breaks $SU(3)_C$; no doublet on $\mathbb C^7$ nor among the operators on $\mathbb C\otimes\mathbb O$ or in the vector of $\mathrm{Spin}(9)$; the Higgs doublet is the colour-free Clifford plane of $\mathrm{Spin}(10)$ (standard model, Theorem 2.6(f); identification [H])	[H] (it read [T] until 2026-09-25)	Theorem
Fixing of semantic roles (T-177)	The collineation group of the Fano plane (168) acts regularly on ordered non-collinear triples; $O$ and the $\kappa_0$ pair $\{E,U\}$ fix $O$, $A$, $D$ and leave one binary choice $E\leftrightarrow U$ (with $L\leftrightarrow S$)	[T] (new form; the uniqueness from the axis sectors is retracted)	Registry
CC-7: Emergence (Theorem 9.3)	Weakly coupled embodied holons: a product stationary state exists iff the correlation part of $[H_{\mathrm{int}}, \rho_*^{(1)}\otimes\rho_*^{(2)}]$ vanishes; otherwise $I = \Theta(g^2) > 0$. The non-degeneracy (ND) holds off a closed set of anchors of measure zero (Theorem 9.4)	[T] for almost every anchor (it read [C at (ND)] earlier on 2026-09-25)	Theorem
Neutrino O-sector Yukawa	$m_D^{(k)} \propto \varepsilon_0 \sin(2\pi k/7)$; discrepancy $m_2/m_3$: $\times 50 \to \times 1.8$	[C]	Theorem
PMNS from anarchic $M_R$	O-isotropy → dense $M_R$ → angles $O(30°\text{–}60°)$	[C]	Theorem
Justification of $K=4$ for L3	Quadratic decomposition $3+1=4$; Bayesian dominance $R^{(2)} \geq 1/4$	[T]	Theorem
Unattainability of L4 for biosystems	Categorical: $L4=\mathrm{colim}_n\tau_{\leq n}(\mathbf{Exp}_\infty)$ not finitely reachable (T-86) + $S^{(n)}\sim 3^{-n}\to 0$; L4 = attractor	[T]	Theorem T-86
CC-5: Fractal closure	The canonical aggregate of a composite — the mean of the parts' marginals, the only permutation-invariant aggregation that returns a part on uncoupled copies — is viable whenever the parts are viable embodied holons and the coupling is weak, $\lvert g\rvert\,s(H_{\mathrm{int}}) < \varepsilon_V$ with $\varepsilon_V$ explicit (Theorem 9.5); at strong coupling it can be $I/7$ (Theorem 9.6). Earlier on 2026-09-25 conditional on (HOL), "the composite is itself a holon", which is still not derived in its full sense; before that an unconditional [T] from the retracted Morita equivalence T-58	[T at weak coupling]	Theorem
Topological protection of Gap vacuum	$\pi_2(G_2/T^2) \cong \mathbb{Z}^2$ [T]; the barrier values ($\geq 6\mu^2$) are Hessian eigenvalues of the retracted sector parametrisation of T-64, so the protection of the vacuum is [C at (SV)]	[T]+[C at (SV)] (it read [T] until 2026-09-25)	Theorem
Canonical definition of $f_0$	$f_0\Lambda^4 = \frac{1}{7}[V_{\text{Gap}}^{\min} + \frac{1}{2}\zeta'_{H_{\text{Gap}}}(0)]$; UV-finiteness + the unique vacuum, now the hypothesis (SV) (T-70)	[C at (SV)] (it read [T] until 2026-09-25)	Theorem
Structural necessity of $\Lambda > 0$	Autopoiesis + local cohomology → $\rho_{\text{vac}} > 0$; Lawvere incompleteness	[T]	Theorem
CC-6: Scale invariance (T-72)	Under (AGG) — the aggregation returns a part's state on uncoupled copies, and the coupling is weak — $P$, $R$, $\Phi$ and Gap of the aggregate stay within $O(\delta)$ of a part's; Theorem 9.5 proves (AGG) for weakly coupled embodied holons with $\delta = O(g)$, at the stationary state and along trajectories from the basin; preservation under any CPTP aggregation is retracted (the depolarising channel gives $I/7$), and at strong coupling the transfer fails (Theorem 9.6)	[T at weak coupling]	Theorem
Gap = curvature of Serre fibration	Spectral triple T-53 + NCG curvature → exact identification	[T]	Theorem
Internal theory (T-54)	$\mathrm{Th}_{\mathrm{UHM}} = \mathrm{Sub}_{\mathrm{closed}}(\Omega)$ — φ-invariant predicates	[T]	Theorem
Lawvere incompleteness (T-55)	$\mathrm{Th}_{\mathrm{UHM}} \subsetneq \Omega$ — from Cartesian closedness + nontriviality of φ	[T]	Theorem
Structural ToE (T-56)	φ-closed, finitely axiomatizable, principally incomplete, evolutionarily open	[T]	Theorem
Completeness of triadic decomposition (T-57)	LGKS theorem: unique decomposition $\mathcal{L} = \mathrm{Ham} + \mathrm{diss} + \mathrm{reg}$	[T]	Theorem
∞-groupoid $\mathbf{Exp}_\infty$ (T-91)	$\mathrm{Sing}(\mathcal{E})$ — Kan complex (Milnor's theorem); + T-76 → HoTT logic, Postnikov truncations	[T]	Theorem
Compression parameter $k = 1 - R$ (T-62)	$k$ is not free: $k = 1 - R$, $R = 1 - \|\Gamma - \rho^*\|_F^2/\|\Gamma\|_F^2$; adaptive self-modeling	[T]	Theorem
PW reconstruction algorithm (T-95)	4-step procedure $\Gamma \to \rho_E, D_{\text{diff}}, \sigma_L, C$; the 7D quantities survive the round trip $\pi\circ\iota = \mathrm{id}$ (T-58′). Registry: [T] → [C] (2026-09-10); the "zero error" step is retracted — it rested on the Morita equivalence T-58, and in 7D $\rho_E = \gamma_{EE}$ is a scalar compared with a 42D clock block	[C]	Theorem
Structural $\theta_{\mathrm{QCD}} = 0$ (T-99)	7-step derivation: reality of $f_{ijk} \in \mathbb{R}$ (A1) + unique vacuum (T-64) → $\theta_{\mathrm{QCD}} = 0$. Stratified 2026-09-25: step 2 ($V_3$ the only $PT$-odd term) [T] for the retracted cubic only — the $G_2$-invariant potential is PT-even (T-331); the conclusion uses the unique sector vacuum, the hypothesis (SV). Axion is purely DM. Retracted 2026-09-26: step 4 is false for $V_3$ itself; the Gap sector is CP-neutral but fixes no $\bar\theta$ (Theorem 3.1c), so $\bar\theta$ is free and strong CP is open [Pr]	[✗] (was [T]+[C at (SV)])	Theorem
Environment encoding (T-100)	CPTP functor Enc: ObsSpace → End(D(C⁷)), unique up to G₂. 3-channel decomposition from T-57	[T]	Theorem
Optimal action (T-101)	$a^* = \arg\min \|\sigma_{\mathrm{sys}}\|_\infty$ — from T-92 (equivalence of P and σ)	[T]	Theorem
Completeness of the 3-term equation (T-102)	$h^{\text{ext}} = h^{(H)} + h^{(D)} + h^{(R)}$, 4th type impossible (from T-57 LGKS)	[T]	Theorem
Hedonic valence (T-103)	$\mathcal{V}_{\text{hed}} = dP/d\tau\|_{\mathcal{R}}$: formula [T], observability at L2 [T] (T-77), phenomenal interpretation [I]	[T]+[I]	Theorem
Stability radius (T-104)	$r_{\text{stab}} \approx K\bigl(\sqrt{P-1/7}-\sqrt{1/7}\bigr)$, $K=\sqrt{35}\sqrt[4]{6}/10$ — Bures distance to $\{P=2/7\}$; the old $\sqrt{P-2/7}$ is refuted; most dangerous channel is $h^{(D)}$	[C]	Theorem
Landauer energy balance (T-105)	$\Delta F_{\min} = k_B T_{\text{eff}} \cdot \ln 2 \cdot \dot{S}_{\text{diss}}$; three metabolic regimes	[T]	Theorem
Information capacity of Enc (T-107)	$C_{\text{Enc}} \leq \log_2 7 \approx 2.81$ bits/observation (Holevo bound + T-102)	[T]	Theorem
Compositionality of Enc/Dec (T-108)	$\text{Enc}_{12} = \Phi_{\text{agg}} \circ (\text{Enc}_1 \otimes \text{Enc}_2)$ is CPTP for every CPTP aggregation (T-100 + closure of channels under $\otimes$ and $\circ$); that the diagnostics carry over across scales holds at weak coupling through the canonical aggregation (T-72, Theorem 9.5); the uniqueness claim is retracted	[T]; transfer [T at weak coupling]	Theorem
Information learning bound (T-109)	$n \geq \ln(1/(2\delta))/\xi_{\text{QCB}}$, $\xi_{\text{QCB}} \leq \ln 7$ (quantum Chernoff bound + T-107)	[T]	Theorem
Optimal learning bound (T-112)	$n_{\text{opt}} = \max(n_{\text{info}}, n_{\text{dyn}}, n_{\text{stab}})$ — three regimes	[T]	Theorem
N=7 minimality for learning (T-113)	Learning via regeneration is impossible for $N < 7$; $N = 7$ is Pareto-optimal	[T]	Theorem
Fano grammar (T-114)	Markov chain on PG(2,2) is ergodic, stationary distribution $\pi_i = 1/7$	[T]	Theorem
Composition distinguishability (T-115)	$\|\mathrm{Comp}(n)\| = 7^n$ for generic $\Gamma$ (algebraic distinguishability)	[T]	Theorem
PW Suzuki-Trotter (T-116)	$\varepsilon(T) \leq C_p \cdot T \cdot (\delta\tau)^{2p+1}$, for $p=2$: $\varepsilon \leq 10^{-5}$	[T]	Theorem
Landauer calibration (C22)	$\Delta F^{(k)} \geq k_B T_\mathrm{eff} \ln(2) \cdot k$ — linear growth	[C]	Theorem

Status legend

• [T] STRICT — mathematically proven without additional assumptions
• [C] CONDITIONAL — proven under explicit interpretational assumptions
• [P] PROGRAM — research direction

What the Theory Derives

From the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$:

1. Base space X = $|N(\mathcal{C})|$ — geometric realization of the nerve
2. Monism — H*(X) = 0 as a mathematical theorem
3. Local physics — H*_loc(X, T) ≠ 0 near the terminal object
4. Time — the cyclic clock τ ∈ ℤ₇ via the Page–Wootters mechanism (the clock register [T]; the constraint is assumed, [C])
5. Arrow of time — stratal collapse toward the terminal T, in the parameter of the dissipative semigroup, carried by the depth register ([T], T-53b)
6. Metric — d_strat (stratified Connes metric)
7. Dimensionality — dim(X) = 6 from N = 7
8. Octonionic structure — P1+P2 → $\mathbb{O}$ → N=7, $G_2$-symmetry, Fano plane, Hamming code, with the canonical orientation of the Fano lines (T15-canon; Track B)

Research program:

• Compactification 6D → 4D — connection to observed spacetime
• Einstein equations — [T] (T-53): full spectral action, see theorem
• Connection to the Standard Model — formalized program

Takes as primitive (categorical gap):

• Why the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ has an "inner side"
• Why this particular mathematical structure and not another

Premises in use (2026-09-26). The single list, with where each premise is used, its status, the independence models and the mergers, is Premises of UHM. In short:

• Axioms [P]: A1–A4 and the constraint $\hat C\Gamma = 0$ of A5 (the form of the timeless state; the clock register itself is [T], T-87); the metatheory, the CPTP formalism and the frame decision D-0910 are definitions [D].
• Bridge premises of physics [H]: (Cl₀) — fermions are vectors of the spinor module $\mathcal S = \mathbb C\otimes\mathbb O$; (P) — spacetime's tangent vectors are the Hermitian forms on the spinor factor of the fermion field, with a causal form preserved by every internal-structure-preserving transformation (⟺ (L) ∧ (W)). (P) as stated names $\mathcal S$, but the two are independent inputs; (W₀), a complex spinor factor, is not a separate premise.
• Principle of the self-model [Pr]: (MaxΦ) — the anchor is maximally integrated, $\Phi(\rho_a) = 6$ (⟺ (Eq-V)); it is independent of the axioms and of life in the window, and no variational principle of the corpus yields it.
• Strict necessity of $N = 7$ [H]: (Σ₆) — every decomposition is perfectly single-fault diagnosable (T-349); it replaced the stronger (P1₆) on 2026-09-28. $N \geq 7$ for the seven listed functions needs no premise.
• Free parameters: the associator coupling $\kappa$ of $V_{\text{Gap}}$ (free, T-331), $\mu^2$, $\lambda_4$, the regeneration rate and the Fano weight $\alpha$, $\omega_0$, $\bar\theta_{\mathrm{QCD}}$.
• Identification hypotheses [H]: (SV), (GC) in broken form, (UP) at leading order, (PQ) (strong CP open [Pr]), (FE) and (SA) in the axis frame, the Higgs identification, T-186(a), the reconstruction and aperiodic-clock conditions of T-119/T-120 as physics, (HOL) [I].

Results labelled "[T] as mathematics" use only the axioms; their physical readings carry the premises above. The former inputs (Alt), (MP), (MM), (Q), (RT), (Col), (Pure), (AGG), (ND) are discharged.

Minimality of the primitive

UHM's psychophysical primitive is one: the identity of being and experience (the inner side of the ∞-topos above), against two levels plus emergence for physicalism and two substances plus a causal link for dualism — a count of primitives about how experience relates to the physical, compared among these three types, [I] (justification). It is not a count of the theory's axioms. The independent axiomatic content is A1–A4 plus the constraint of A5 — five inputs, each independent (the independence of the constraint is [T]) — and the bridge premises and (MaxΦ) of the list above come on top; the identity of being and experience is not a sixth axiom but the reading of axiom Ω⁷ as a whole. (Until 2026-09-26 the box read "minimal among all possible axiomatic choices: one axiom instead of two or three"; the phrase is withdrawn as a count of axioms.) From it are derived: the form of experiential content (unique functor), identity of qualia (Yoneda lemma), immanence of description (closure via φ).

The relational identity of qualia via the Yoneda lemma was proposed before UHM: Tsuchiya and Saigo stated it for a category of experiences in an April 2020 preprint (doi:10.31219/osf.io/68nhy) and in Neurosci. Conscious. 2021, niab034; the category-theoretic approach to consciousness goes back to Tsuchiya, Taguchi and Saigo (Neurosci. Res. 107, 1–7, 2016), and a graded (enriched-category) version is in Tsuchiya, Phillips and Saigo (Conscious. Cogn. 101, 103319, 2022). UHM applies the lemma to its own category of experiences — rays of $\mathbb{P}(\mathcal{H}_E)$ with Fubini–Study distances.

Navigation

Section	Contents
Axiom Ω⁷	Five structural properties with the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ as primitive
Consequences	Cohomological monism, local-global dichotomy
Structure	Holonom and 7 dimensions
Dynamics	Evolution equations with terminal object T
Spacetime	Base space X, metric d_strat
Consciousness	Hierarchy L0→L1→L2→L3→L4
Emergent time	Page–Wootters, stratificational time
Categorical formalism	∞-topos, derived categories, IC cohomologies
Uniqueness theorem	G₂-rigidity: 34 kinematic invariants, 48 physical parameters (frame decision D-0910)
Standard Model	Colour $SU(3)$ from G₂ [T]; electroweak sector [C at (FE)], uniqueness [H]; 3 generations (count [T], identification [I])
Physics	Gauge symmetry, particles, gravity, cosmology
Neutrino masses	Seesaw from Gap, $M_R$ [T], O-sector Yukawa (formula [T] / numbers [C]), PMNS [C]
SUSY from $G_2$	Superpotential [T] (Schur), superpartner spectrum, gravitino
Gap thermodynamics	Potential $V_{\text{Gap}}$, global minimization [T], sector hierarchy $\varepsilon$
Quantum gravity	Spectral action [T], UV-finiteness (field-space [T], order-by-order [C]), Einstein equations [T]
Cosmological constant	$\Lambda > 0$ [T], spectral formula [T], honest bracket $10^{-53.5}$–$10^{-93.5}$ [C] ($\gtrsim 27$ orders open)
Composite systems	CC-5 (viability of the canonical aggregate at weak coupling, [T at weak coupling]), topological protection of Gap [T], emergent geometry
Interiority hierarchy	L0–L4, $K=4$ for L3 [T], categorical unattainability of L4 [T] (T-86)
Depth tower	SAD metric [T], depth dynamics (A₄-bifurcation, energy, stress, social), morphological agnosticism [H]
Glossary	Term definitions
Notation	Mathematical notation