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). Rigidity of the primitive is proven [T] (T-173): the construction is unique up to $G_2 \times \mathbb{R}_{>0}$ given the axioms. The universal property in the category of physical theories is proven [T] (T-174): for any physical theory $(E, \mathcal{A}, D)$ with $A_{\text{int}} \subset \mathcal{A}$, CPTP dynamics, and $\leq 7$ observables, there exists an essentially unique receiving morphism into $\mathfrak{T}$; higher $(\infty,1)$-coherences (pentagon, interchange, Mac Lane associator) are verified via full embedding into $\mathbf{Topoi}_\infty$ (T-211 [T]). All 4 foundational theorems are proven [T]: T-170 [T] (recovery of the M-theory limit at levels of M-theory definedness): T-170' [T] (perturbative correspondence) + T-170'' [T] (non-perturbative correctness of the UHM integral). T-171 [T] (LQG embedding for bounded spin networks $j_e \leq 3$) + T-171' [T] (unbounded spin via cluster construction). T-172 [T] (causal set embedding) via Lemma C30. The "meta" status is a proven theorem for physical theories of the specified class.

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 mathematical theorem (H*(X) = 0), not a philosophical choice

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 $\varphi = \arg\min \mathbb{E}[S_{spec} + D_{KL}]$ is a theorem about properties of the categorically defined φ.

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, independent justification: theorems P1+P2 [T] (derived from (AP)+(PH)+(QG)+(V) via the T15 chain) 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 $n > 0$	[T] Theorem
Local physics	$H^*_{loc}(X, T) \neq 0$	[T] Theorem
Time	$\tau \in \mathbb{Z}_7$ (Page–Wootters)	[T] Derived
Arrow of time	$\dim(X_\tau) \geq \dim(X_{\tau+1})$	[T] Theorem
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]

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)
• $D_{\min} = 2$ [T] — unconditional consequence of $\Phi_{\text{th}} = 1$ [T] (T-151)

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 [C], unattainable for systems with nonzero decoherence ($R^{(n)} \to 0$); an attractor, not a physical state (C19)
• 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 $n > 0$	[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	$\exists! \Gamma^* : \varphi(\Gamma^*) = \Gamma^*$	[T]	Proof
Emergent time	τ derived from $\mathcal{C}$ (Page–Wootters, Bures, ∞-groupoid)	[T]	Theorem
Arrow of time	Stratal collapse: $\dim(X_\tau) \geq \dim(X_{\tau+1})$	[T]	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 $G_2 = \mathrm{Aut}(\mathbb{O})$; 34 physical parameters	[T]	Theorem
Electroweak sector uniqueness	$SU(2)_L \times U(1)_Y$ is the unique rank-4 construction from $\kappa_0$ and axioms A1–A5	[T]	Theorem
Exactly 3 generations	$N_{\text{gen}} = 3$: $\leq 3$ from swallowtail $A_4$ + $\geq 3$ from $(1,2,4) \subset \mathbb{Z}_7^*$	[T]	Theorem
Fano Yukawa selection	$y_k = g_W \cdot f_{k,E,U} \cdot	\gamma_{\text{vac}}^{(EU)}	$via octonionic$f_{ijk}$
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], $k=4 \to$ 2nd, $k=2 \to$ 1st [T]	[T]	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	$7 = 1_O \oplus 3_{A,S,D} \oplus \bar{3}_{L,E,U}$; $\dim(\text{space}) = 3$	[T]	Theorem
Sector hierarchy $\varepsilon$	Unique self-consistent vacuum; $\bar{\varepsilon} \approx 0.023$ from sector structure	[T]	Theorem
Cohomological vanishing of $\Lambda$	$\Lambda_{\text{global}} = 0$ from $H^n(X) = 0$; observed $\Lambda$ is a local effect	[T]	Theorem
Einstein equations from spectral action	Full triple (T-53) → $S = \mathrm{Tr}(f(D_A/\Lambda))$ → EH + SM, $G_N = 3\pi/(7f_2\Lambda^2)$	[T]	Theorem
UV-finiteness of Gap theory	Compactness of $(S^1)^{21}$ + $G_2$-Ward ($21 \to 7$) + SUSY ($7-7=0$) + APS index	[T]	Theorem
Lorentzian signature	Finite spectral triple $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$, KO-dimension 6 → (+,−,−,−)	[T]	Theorem
Morita equivalence 7D↔42D	$\mathrm{Sh}_\infty(\mathcal{C}	7) \simeq \mathrm{Sh}\infty(\mathcal{C}	_{42})$; all 7D formulas are exact
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 $V_{\text{Gap}}$	$G_2$-orbit reduction $21D \to 5D$; unique minimum; Hessian $> 0$	[T]	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	$R^{(n)} \sim R^n \to 0$ for $\varepsilon_{\text{dec}} > 0$; L4 = attractor	[C]	Theorem
CC-5: Fractal closure	Nontriviality of composite attractor $P > 1/7$ [T]; viability $P > 2/7$ [T for embodied] (T-149)	[T]+[C]	Theorem
Topological protection of Gap vacuum	$\pi_2(G_2/T^2) \cong \mathbb{Z}^2$; barrier $\geq 6\mu^2$; vacuum separated from $\text{Gap}=0$	[T]	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 + unique vacuum	[T]	Theorem
Structural necessity of $\Lambda > 0$	Autopoiesis + local cohomology → $\rho_{\text{vac}} > 0$; Lawvere incompleteness	[T]	Theorem
CC-6: Scale invariance	Bures contractivity of CPTP + CC-5 (nontriviality [T]) → structure preserved under aggregation	[T]	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 $\Gamma \to \rho_E, D_{\text{diff}}, \sigma_L, C$ with zero error	[T]	Theorem
Structural $\theta_{\mathrm{QCD}} = 0$ (T-99)	7-step proof: reality of $f_{ijk} \in \mathbb{R}$ (A1) + unique vacuum (T-64) → $\theta_{\mathrm{QCD}} = 0$ exactly. Axion is purely DM	[T]	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)	$r_{\text{stab}} = \sqrt{P(\rho^*_\Omega) - 2/7}$ (Bures metric); most dangerous channel is $h^{(D)}$	[T]	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)	$\text{Enc}_{12} = \Phi_{\text{agg}} \circ (\text{Enc}_1 \otimes \text{Enc}_2)$ from T-100 + T-72 + T-58	[T]	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)
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 — τ ∈ ℤ₇ via the Page–Wootters mechanism
5. Arrow of time — stratal collapse toward the terminal T
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 (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

Minimality of the primitive

UHM's primitive is minimal among all possible axiomatic choices: one axiom instead of two or three (justification). From it are derived: the form of experiential content (unique functor), identity of qualia (Yoneda lemma), immanence of description (closure via φ).

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 physical parameters
Standard Model	SM from G₂: electroweak sector [T], 3 generations [T]
Physics	Gauge symmetry, particles, gravity, cosmology
Neutrino masses	Seesaw from Gap, $M_R$ [T], O-sector Yukawa [T], 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 [T], Einstein equations [T]
Cosmological constant	$\Lambda > 0$ [T], spectral formula [T], budget $\sim 10^{-120\pm10}$ [C]
Composite systems	CC-5 (nontriviality [T], viability [C]), topological protection of Gap [T], emergent geometry
Interiority hierarchy	L0–L4, $K=4$ for L3 [T], unattainability of L4 [C]
Depth tower	SAD metric [T], depth dynamics (A₄-bifurcation, energy, stress, social), morphological agnosticism [H]
Glossary	Term definitions
Notation	Mathematical notation