Glossary

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For mathematical notation see Notation. For verification criteria see Falsifiability.

Core Terms

Term	Definition
$\mathcal{C}$ (Category)	Primitive category — sole primitive; small category with a finite number of objects
$\Gamma$ (Gamma)	Coherence matrix — Hermitian positive semi-definite matrix with $\mathrm{Tr}(\Gamma) = 1$
$\mathbb{H}$ (Holon)	Minimal self-sufficient unit of reality, containing an image of the whole
$\Gamma_\odot$ (Source)	Primordial pure state — superposition of all dimensions
$P$ (Purity)	$P = \mathrm{Tr}(\Gamma^2) \in [1/7, 1]$ — coherence measure
$S_{vN}$	Von Neumann entropy $S_{vN} = -\mathrm{Tr}(\Gamma \log \Gamma) \in [0, \log 7]$
Coherence	Quantum correlations between dimensions — off-diagonal elements $\gamma_{ij}$ of matrix $\Gamma$
Decoherence	Process of losing coherence through interaction with the environment — see dissipative term
Unitary evolution	Deterministic evolution of a closed system that preserves $P$
Internal time τ	Emergent parameter arising from correlations between O and the remaining dimensions — theorem
Page–Wootters mechanism	Construction for deriving time from the structure of $\Gamma_{total}$ with the constraint $\hat{C} \cdot \Gamma_{total} = 0$
Bures metric	$d_B(\Gamma_1, \Gamma_2) = \arccos(\sqrt{F})$, where $F = \left(\text{Tr}\sqrt{\sqrt{\Gamma_1}\Gamma_2\sqrt{\Gamma_1}}\right)^2$ — Uhlmann fidelity; angular distance in state space
∞-groupoid Exp_∞	Extension of the Exp category with 1-morphisms (time) and n-morphisms (homotopies)
∞-topos Sh_∞(Exp)	Category of ∞-sheaves on Exp_∞ with internal temporal modal logic

Base Space and Stratification Terms

Term	Definition
Base space X	$X = \|N(\mathcal{C})\|$ — geometric realisation of the nerve of category $\mathcal{C}$; derived endogenously, not postulated
Nerve $N(\mathcal{C})$	Simplicial set: 0-simplices = objects, n-simplices = chains of morphisms
Terminal object T	$T = \Gamma^*$ — global attractor; $\forall\Gamma, \exists! f: \Gamma \to T$; ensures contractibility of X
Stratification	Decomposition $X = \bigsqcup_\alpha S_\alpha$ into strata; $S_0 = \{T\}$
Local–global dichotomy	Principle: $H^*(X) = 0$ globally (monism), $H^*_{loc}(X,T) \neq 0$ locally (physics)
Cohomological monism	Theorem: $H^n(X, \mathcal{F}) = 0$ for $n > 0$ — monism as a mathematical fact
Stratified metric d_strat	$d_{strat}(\omega_1, \omega_2) = \inf_\gamma \int_\gamma ds_\alpha$ — Connes metric on strata
Link Link(T)	Topological structure near T; $\text{Link}(T) \cong S^6$ — 6-sphere
Arrow of time (geometric)	Stratum collapse: $\dim(X_\tau) \geq \dim(X_{\tau+1})$ towards terminal T
IC cohomology	Intersection cohomology — cohomology of strata capturing the "hidden topology"
Derived category D^b(X)	Bounded derived category of sheaves on stratified X

Holon Dimensions

See Seven dimensions for a full description.

Term	Definition
Dimension	One of the 7 fundamental aspects of the Holon
$A$ — Articulation	Dimension I — capacity to differentiate
$S$ — Structure	Dimension II — capacity to hold form
$D$ — Dynamics	Dimension III — capacity to change
$L$ — Logic	Dimension IV — capacity to be consistent
$E$ — Interiority	Dimension V — capacity to experience
$O$ — Ground	Dimension VI — connection to the Source and internal clock (Page–Wootters)
$U$ — Unity	Dimension VII — integration of all dimensions

Interiority Hierarchy (L0→L1→L2→L3→L4)

See Interiority hierarchy for formal definitions and proofs.

Terminological discipline

The term "qualia" is used ONLY for level L2 (cognitive qualia). For L0 and L1, the terms "interiority" and "phenomenal geometry" are used respectively. L3 and L4 are post-reflexive levels. This is a categorically correct distinction.

Term	Level	Definition
Interiority	L0	Fundamental property of "having an inside"; $\mathrm{Int}(S) := \exists \rho_E = \mathrm{Tr}_{-E}(\Gamma_S)$; corresponds to $\tau_{\leq 0}(\mathrm{Exp}_\infty)$
Phenomenal geometry	L1	Structure with a metric; $\mathrm{PG}(S) := (\mathbb{P}(\mathcal{H}_E), d_{\mathrm{FS}}, \rho_E)$, where $\mathrm{rank}(\rho_E) > 1$; corresponds to $\tau_{\leq 1}$
Cognitive qualia	L2	Reflexively accessible experience; $R \geq 1/3$, $\Phi \geq 1$; corresponds to $\tau_{\leq 2}$
Network consciousness	L3	Distributed cognitive network; $\pi_3 \neq 0$, $R^{(2)} \geq 1/4$; metastable with $\tau_3 = 1/(\kappa_{\mathrm{bootstrap}} \cdot (1 - R^{(2)}))$
Unitary consciousness	L4	Full ∞-groupoid; $\forall k: \pi_k \neq 0$, $\lim_n R^{(n)} > 0$, $P > 6/7$; maximal level (Postnikov stabilisation)
Experiential content	L0–L4	$\mathcal{Q} := (\mathrm{Intensity}, \mathrm{Quality}, \mathrm{Context}, \mathrm{History})$
n-truncation $\tau_{\leq n}$	—	Operation on an ∞-groupoid that zeroes out $\pi_k$ for $k > n$; connects Ln levels to homotopy structure
Universal reflection threshold	—	Formula $X^{(n)}_{\mathrm{th}} = 1/(n+1)$: L2 ($n=2$): $R_{\mathrm{th}} = 1/3$, L3 ($n=3$): $R^{(2)}_{\mathrm{th}} = 1/4$, L4 ($n=4$): $R^{(3)}_{\mathrm{th}} = 1/5$. Integration threshold: $\Phi_{\mathrm{th}} = 1$

Components of Experiential Content

See Exp category for a formal description.

Term	Definition
Intensity	$\{\lambda_i\}$ — spectrum of $\Gamma$; defines the strength of the state
Quality	$\{[\lvert q_i\rangle]\} \subset \mathbb{P}(\mathcal{H}_E)$ — defines the character of the state
Context	$\Gamma_{-E}$ — states of all dimensions except $E$
History	Derived as the loop space in the ∞-groupoid: $\mathrm{Hist}(\mathcal{Q}) := \Omega_\mathcal{Q}(\mathbf{Exp}_\infty)$ — theorem
$\mathbb{P}(\mathcal{H}_E)$	Projective space of qualities
$d_{\mathrm{FS}}$	Fubini-Study metric: $d_{\mathrm{FS}}([\lvert\psi\rangle],[\lvert\phi\rangle]) = \arccos(\lvert\langle\psi\vert\phi\rangle\rvert)$
Relational identity of qualia	Theorem: by Yoneda's lemma, the identity of qualia $[\lvert q\rangle]$ is fully determined by its relational position in the Exp category. Inverted qualia are impossible.
Phenomenal vector FV	$\text{FV}(\rho_E) := \{(\lambda_i, [\lvert q_i\rangle])\}$ — unique functor, extracting experiential content from $\rho_E$. Not an arbitrary postulate, but a forced structure.

Calibration Terms

Term	Definition
Isospectrality	$\mathrm{Spec}(\rho_1) = \mathrm{Spec}(\rho_2)$, but $\mathrm{Eigvec}(\rho_1) \neq \mathrm{Eigvec}(\rho_2)$
Calibration	Procedure for establishing correspondence between mathematics and phenomenology
Contrast spectrum	$\log(\lambda_i / \lambda_i^{\text{ref}})$ — relative intensities
Categorical gap	Boundary of explanation: impossibility of deducing "why there is experience" — see Axiom Ω⁷
Axiom Ω⁷	Fundamental axiomatics: ∞-topos $\mathbf{Sh}_\infty(\mathcal{C})$ as the sole primitive

Categorical Terms

See Categorical formalism for a full description.

Term	Definition
$\mathbf{DensityMat}$	Category of density matrices with CPTP morphisms
$\mathbf{Exp}$	Category of experiential states
$\mathbf{Exp}_\infty$	∞-groupoid of paths — time as a 1-morphism
$\mathbf{Sh}_\infty(\mathbf{Exp})$	∞-topos of sheaves — internal temporal logic
Functor $F$	Experience functor: $F: \mathbf{DensityMat} \to \mathbf{Exp}$
CPTP	Completely Positive Trace-Preserving — quantum channels, see Formalisation of φ
∞-topos (∞-topos)	∞-category satisfying Giraud's axioms. In UHM: $\mathrm{Sh}_\infty(\mathcal{C})$ — sole primitive of the theory in the Ω⁷ formulation
Grothendieck topology	Site $(\mathcal{C}, J_{Bures})$ — coverage function turning $\mathcal{C}$ into a basis for constructing the topos
Bures covering	Family $\{\Phi_i: \Gamma_i \to \Gamma\}$ covers $\Gamma$ if $B_B(\Gamma, \delta) \subseteq \bigcup_i \Phi_i(B_B(\Gamma_i, \epsilon))$
Bures metric $d_B$	$d_B(\Gamma_1, \Gamma_2) = \arccos(\sqrt{F})$; monotone under CPTP
Fidelity (Fid)	$\mathrm{Fid}(\rho, \sigma) = (\mathrm{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^2$ — measure of proximity of quantum states. The notation $\mathrm{Fid}$ is used to distinguish from functor $F$
Site (site)	Category with a coverage function $J$ satisfying the axioms: identity, stability, transitivity
∞-terminal object	Object $T$ in an ∞-category such that $\mathrm{Map}(\Gamma, T) \simeq *$ for all $\Gamma$. Admits a multitude of equivalent paths
Map(Γ, T)	Morphism space (mapping space) in an ∞-category. Unlike $\mathrm{Hom}(\Gamma, T)$ in a 1-category, it is an ∞-groupoid
Homotopy	2-morphism between 1-morphisms. Connects different paths between objects
HoTT (Homotopy Type Theory)	Internal logic of the ∞-topos. Formalises identity through paths
Free will (Freedom)	$\mathrm{Freedom}(\Gamma) := \dim\ker(\mathcal{H}_\Gamma) + 1$ [T] — number of zero modes of the free-energy functional's Hessian + 1. ∞-categorical motivation: $\pi_0(\mathrm{Map}(\Gamma, T)^{\text{non-trivial}})$. Monotone under CPTP, $G_2$-invariant. See Consequences, Free will
Freedom entropy	$S_{\text{freedom}} := \log(\text{Freedom}(\Gamma)) = \log(\dim\ker(\mathcal{H}_\Gamma) + 1) \in [0, \log 7]$ — quantitative measure of the space of choice

L-Unification Terms

See L-unification and Categorical formalism for a full description.

Term	Definition
L-unification	Central theorem of UHM: $L \cong \Omega \cap \Gamma$ — the Logic dimension is identical to the projection of the subobject classifier onto the state
$\Omega$ (classifier)	Subobject classifier of the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ — defines the internal logic of the theory
$\chi_S$ (characteristic)	Characteristic morphism $\chi_S: \Gamma \to \Omega$ of the subobject $S$ — "membership predicate"
$L_k$ (Lindblad)	Dissipation operators: $L_k^{\text{atom}} = \lvert k\rangle\langle k\rvert$ — projectors derived from the atoms of the classifier Ω. Historical notation $L_k = \sqrt{\chi_{S_k}}$ — convention ($\sqrt{P} = P$ for projectors). See Lindblad operators
$\mathcal{L}_\Omega$ (Liouvillian)	Logical Liouvillian — evolution superoperator derived from Ω
$\triangleright$ (temporal modality)	Modal operator "will be true tomorrow" on Ω; generates discrete time $\tau_n = \triangleright^n(\mathrm{now})$
$\mathcal{D}_\Omega \dashv \mathcal{R}$ (adjunction)	Dissipation–regeneration adjunction: $\mathcal{D}_\Omega$ — dissipative functor, $\mathcal{R}$ — regenerative functor; $\kappa_0$ is derived from it
CPTP automatically	Consequence of L-unification: $\sum_k L_k^\dagger L_k = \sum_k \chi_{S_k} = \mathbb{1}$ — the CPTP condition is derived from the classifier's structure

Formalisation Terms

Term	Definition
$R$ (reflection)	$R = R_{\text{canonical}} := 1/(7P)$ — canonical definition used in all thresholds. Not to be confused with $Q_\varphi := 1 - \|\Gamma - \varphi(\Gamma)\|_F^2 / P$ (quality measure of self-modelling). Details: formalisation of φ
$R_{\text{th}}$	Reflection threshold $= 1/3$ [T] ($K = 3$ from triadic decomposition + Bayesian dominance)
$\Phi_{\text{th}}$	Integration threshold $= 1$ [T] (T-129) — unique self-consistent value at $P_{\text{crit}} = 2/7$; coherent dominance
$D_{\min}$	Minimum differentiation $= 2$ [T] (T-151) — unconditional consequence of $\Phi_{\text{th}} = 1$ [T]; definition
$C_{\text{th}}$	Consciousness threshold $= \Phi_{\text{th}} \times R_{\text{th}} = 1 \times 1/3 = 1/3$ [T] (T-140); $D_{\text{diff}} \geq D_{\min} = 2$ — separate viability condition [T] (T-151)
$\varphi$ (operator)	Self-modelling operator; $\varphi(\Gamma) = \sum_m K_m \Gamma K_m^\dagger$
$K_m$	Kraus operators; $\sum_m K_m^\dagger K_m = I$
Fixed point	$\Gamma^* = \varphi(\Gamma^*)$ — state of ideal self-knowledge
Convergence	$\|\varphi^n(\Gamma_0) - \Gamma^*\|_F \leq k^n \cdot \|\Gamma_0 - \Gamma^*\|_F$, $k \in [0,1)$
(M,R)-system	Rosen's system: Metabolism, Repair, $\beta$-closure
Minimality theorem	From (AP)+(PH)+(QG) it follows that $\dim(\mathcal{H}) = 7$ minimally
Uniqueness theorem	Basis $\{A,S,D,L,E,O,U\}$ is unique [T]: A,S,D,L,U — algebraically; E,O — via κ₀ and functional independence
(AP)	Autopoiesis: $\exists\varphi$ with a fixed point
(PH)	Phenomenology: $\exists\rho_E$ with non-trivial interiority
(QG)	Quantum ground: Lindblad equation with regeneration
(V) Viability	Fourth condition in the definition of the Holon: $P(\Gamma) > P_{\text{crit}} = 2/7$. Together with (AP), (PH), (QG) forms the complete definition — see Viability
IDP (Information Distinguishability Principle)	Definition [D] (T16): distinguishability via $J_{\text{Bures}}$-coverings is identical to ontological distinguishability — built into A1+A2. Every Holon contains a substructure modelling its own whole. Formally: $\exists\varphi: \Gamma \to \Gamma$, $\varphi$ — CPTP channel with $F(\Gamma, \varphi(\Gamma)) > 0$
$P_{\text{crit}}$ (critical purity)	Viability threshold $P_{\text{crit}} = 2/N = 2/7$. Derived from five independent paths — see Theorem P_crit
$\mathrm{Coh}_E$ (E-coherence)	HS-projection [T]: $\mathrm{Coh}_E(\Gamma) = \|\pi_E(\Gamma)\|_{\mathrm{HS}}^2 / \|\Gamma\|_{\mathrm{HS}}^2 = (\gamma_{EE}^2 + 2\sum_{i \neq E}\lvert\gamma_{Ei}\rvert^2) / \mathrm{Tr}(\Gamma^2)$, $\in [1/7, 1]$ — master definition, HS-projection. Formal equivalence $\mathrm{Coh}_E \approx P_E = \mathrm{Tr}(\rho_E^2)$ — structural hypothesis [H]
$\kappa_0$ (base regeneration coefficient)	Coupling scale of the Holon to the environment in the regenerative term $\mathcal{R}[\Gamma, E]$ — see categorical derivation of κ₀
$D_{\text{diff}}$ (differentiation dimension)	Number of dimensions in which $\Gamma$ deviates from the maximally mixed state. L2 threshold: $D_{\text{diff}} \geq 2$ — see definition
$\mathcal{R}_\alpha$ (regenerative operator)	CPTP operator: $\mathcal{R}_\alpha(\rho) = (1-\alpha)\rho + \alpha\rho_*$ [T] — unique CPTP interpolation with a replacement channel; see derivation of the ℛ form
Correctness condition	$\alpha = \kappa \cdot \Delta\tau < 1$ — positivity guarantee in numerical integration
Interpolation formulation	Consequence [T] of CPTP-uniqueness of the replacement channel: regeneration as a convex combination of $\mathrm{Id}$ and $\mathcal{C}_{\rho_*}$; proves preservation of $\Gamma \geq 0$
$R^{(n)}$ (n-th order reflection)	$R^{(n)}(\Gamma) := F(\varphi^{(n-1)}(\Gamma), \varphi^{(n)}(\Gamma))$ — consistency measure between successive levels of self-modelling
Spectral formula $\varphi$	$\varphi(\Gamma) = \sum_{k: \mathrm{Re}(\lambda_k)=0} \langle L_k \vert \Gamma \rangle R_k$ — projection onto the kernel of $\mathcal{L}_\Omega$; see formalisation of φ
Variational characterisation of φ	Theorem 3.1: $\varphi = \arg\min_{\psi \in \mathcal{CPTP}} [S_{vN}(\psi(\Gamma)) + D_{KL}(\psi(\Gamma) \| \Gamma)]$; see proof
$S_{spec}$ (spectral entropy)	For density matrices $S_{spec} = S_{vN}$ (Theorem 5.1). General: $S_{spec}(A) = -\sum_i \lvert\lambda_i\rvert \log\lvert\lambda_i\rvert$
Canonical $\Delta F$	$\Delta F(\Gamma) := d_B^2(\Gamma, \Gamma_{\mathrm{eq}}) - d_B^2(\Gamma, \varphi(\Gamma))$ — unified definition via the Bures metric
L3 metastability	Lifetime of network consciousness: $\tau_3 = 1/(\kappa_{\mathrm{bootstrap}} \cdot (1 - R^{(2)}))$; finite without active maintenance
$H_{\mathrm{eff}}$ (effective Hamiltonian)	Hamiltonian of the system after integrating over time: $H_{\mathrm{eff}}(\tau) = H_{6D} + \langle\tau\lvert H_{\mathrm{int}}\rvert\tau\rangle_O$; see Page–Wootters
Lindblad equation	$\frac{d\Gamma}{d\tau} = -i[H_{\mathrm{eff}}, \Gamma] + \mathcal{D}[\Gamma] + \mathcal{R}[\Gamma, E]$; three terms: unitary, dissipative, regenerative
Bootstrap paradox	Problem: regeneration requires coherence ($\kappa \propto \mathrm{Coh}_E$), but a low-coherence system cannot regenerate. Solution: $\kappa_{\mathrm{bootstrap}} > 0$ provides minimal regeneration; see Genesis Protocol
$\kappa_{\mathrm{bootstrap}}$	Minimum regeneration rate: $\kappa_{\mathrm{bootstrap}} := \|\eta\| > 0$, where $\eta$ — unit of the adjunction $\mathcal{D}_\Omega \dashv \mathcal{R}$; resolves the bootstrap paradox

Taxonomy of Γ Configurations

Term	Definition
Fundamental mode Γ	Γ subsystem with $R = 0$; dynamics degenerate into Schrödinger/Dirac. Passive stability (symmetries). Examples: quarks, leptons, bosons. Not a Holon — does not satisfy (AP)+(QG)
Composite configuration Γ	Quasi-autonomous configuration with $0 < R \ll 1$; near-unitary dynamics. Passive stability (bonds). Examples: atoms, simple molecules. Not a Holon — does not satisfy (AP)+(QG)
Holon	Self-sufficient unit, (AP)+(PH)+(QG)+(V), where (V): $P > P_{\text{crit}} = 2/7$. Examples: cells, organisms
L2-Holon	Holon with cognitive qualia: $R \geq R_{\text{th}}$, $\Phi \geq \Phi_{\text{th}}$. Which systems reach L2 is an empirical question
Passive stability	Stability through symmetries (conservation laws). Characteristic of fundamental modes and composite Γ configurations
Active stability	Stability through autopoiesis (metabolism + repair). Characteristic of Holons

Related Theories

Term	Definition
Integrated Information Theory (IIT)	Tononi's theory. UHM generalises: $C = \Phi_{\text{UHM}} \times R$ [T T-140]; $D_{\text{diff}} \geq D_{\min}$ — separate viability condition. Important: $\Phi_{\text{UHM}} \neq \Phi_{\text{IIT}}$ — see notation
Free Energy Principle (FEP)	Friston's theory. In UHM: special case (classical limit) of the variational characterisation of φ — Theorem 4.2. Full formulation: $\varphi = \arg\min[S_{vN} + D_{KL}]$
Global Workspace Theory (GWT)	Baars's theory — global access to information
Conscious realism	Hoffman's theory; connection to UHM: agent $\approx$ L2-Holon (hypothesis)
Panpsychism	"Everything has consciousness." UHM: paninteriorism — everything has L0, not L2

Status of Claims

Rigour-level markers

Marker	Alternative	Meaning	Description
[T] STRICT	Theorem	Mathematically proved	Follows from axioms without additional assumptions
[C] CONDITIONAL	Conditional	Proved under assumptions	Requires explicitly stated interpretational or physical assumptions
[H] PROGRAMME	Hypothesis	Mathematically formulated	Requires proof

Full marker system

The full system includes 7 markers: [T] Theorem, [C] Conditional, [H] Hypothesis, [P] Postulate, [D] Definition, [I] Interpretation, [✗] Retracted. See Status registry.

Classical statuses

Term	Definition
Formalised	Strictly defined and/or proved mathematically (equivalent to [T])
Empirical	Value requires experimental calibration
Heuristic	Conceptual direction, not a strict derivation (equivalent to [C])
Programme	Research direction requiring development (equivalent to [H]/[P])
Hypothesis	Statement requiring proof (equivalent to [H])

Terms Related to Gödel's Theorems

Term	Definition
Gödelian completeness	Property of a system in which every truth is provable; unattainable for sufficiently expressive systems
Minimal completeness	Property of UHM: 7 dimensions are sufficient for (AP)+(PH)+(QG); distinct from Gödelian completeness
$\mathrm{Prov}(L)$	Set of logically provable statements in dimension $L$
$\mathrm{Coh}(\Gamma)$	Coherence-truth — consistency with the structure of $\Gamma$; $\mathrm{Prov}(L) \subsetneq \mathrm{Coh}(\Gamma)$
Enacted consistency	Consistency demonstrated through existence (autopoiesis)
Topological surgery	Overcoming Gödelian limitations through extension from dimension $O$

Coherence Cybernetics Terms

See Coherence Cybernetics for a full description.

Term	Definition
CC	Coherence Cybernetics — meta-theory of systems described by $\Gamma \in D(\mathbb{C}^7)$
$\mathcal{V}$	Viability domain: $\mathcal{V} = \{\Gamma : P(\Gamma) > P_{\text{crit}}\}$
$\mathrm{VIT}$	Viability Integrity Tensor — viability integrity tensor
$\sigma_{\mathrm{sys}}$	System stress tensor $\in \mathbb{R}^7$
$\kappa$	Regeneration rate; $\kappa(\Gamma) = \kappa_{\text{bootstrap}} + \kappa_0 \cdot \mathrm{Coh}_E(\Gamma)$ — master definition
$\kappa_0$	Base regeneration rate: $\kappa_0 = \|\mathrm{Nat}(\mathcal{D}_\Omega, \mathcal{R})\|$ — categorical derivation from the adjunction $\mathcal{D}_\Omega \dashv \mathcal{R}$
$\mathrm{Coh}_E$	$E$-coherence (HS-projection) [T]: $\mathrm{Coh}_E(\Gamma) = \|\pi_E(\Gamma)\|_{\mathrm{HS}}^2 / \|\Gamma\|_{\mathrm{HS}}^2$ — master definition, HS-projection, CC reference
$\rho_*$ ($\Gamma_{\text{target}}$)	Unique stationary state of $\mathcal{L}_\Omega$ [T]: $\rho_* = \varphi(\Gamma)$ — regeneration target, uniquely determined by primitivity
$\omega_0$	Fundamental clock frequency — parameter of the computational approximation of κ₀; see categorical derivation of κ₀
No-Zombie	Theorem [T]: $\mathrm{Viable} \land \mathcal{D}_\Omega \neq 0 \Rightarrow \mathrm{Coh}_E > 1/7$ — impossibility of viable zombies
$\mathbf{Hol}$	Category of Holons with CPTP morphisms
$P_{\text{crit}}$	Critical purity $= 2/7 \approx 0.286$ — theorem (derived by 5 methods from UHM axioms)

Octonionic Terms

See Structural derivation via octonions for a full description.

Term	Definition
Octonions ($\mathbb{O}$)	8-dimensional normed division algebra over $\mathbb{R}$; non-associative, alternative
$\mathrm{Im}(\mathbb{O})$	Imaginary part of the octonions; $\dim = 7$; space of internal degrees of freedom of the Holon [I]
Hurwitz's theorem [T]	Normed division algebras over $\mathbb{R}$ exist only in dimensions 1, 2, 4, 8 ($\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$)
Adams's theorem [T]	Parallelisable spheres: only $S^0, S^1, S^3, S^7$; equivalent to $\dim(\mathrm{Im}) \in \{0, 1, 3, 7\}$
Cayley–Dickson construction [T]	Recursive doubling: $\mathbb{R} \to \mathbb{C} \to \mathbb{H} \to \mathbb{O} \to \mathbb{S}$; each step loses a property
Cayley–Dickson boundary [C]	$\mathbb{O}$ — last division algebra; the next step (sedenions $\mathbb{S}$) loses divisibility
Fano plane PG(2,2) [T]	Minimal projective plane: 7 points, 7 lines, 3 points on each line; defines the multiplication table of the octonions
$G_2$ [T]	$\mathrm{Aut}(\mathbb{O}) = G_2$ — 14-parameter exceptional Lie group; $G_2 \subset SO(7)$
Hamming code $H(7,4)$ [T]	Perfect code: 4 information + 3 check bits; isomorphic to the lines of the Fano plane
Associator [T]	$[x, y, z] = (xy)z - x(yz)$; measure of deviation from associativity; $= 0$ in $\mathbb{R}, \mathbb{C}, \mathbb{H}$; $\neq 0$ in $\mathbb{O}$
Alternativity [T]	Property of $\mathbb{O}$: $[x, x, y] = [x, y, y] = 0$; any two elements generate an associative subalgebra (Artin's theorem)
Artin's theorem [T]	In an alternative algebra, any two elements generate an associative subalgebra
Moufang identities [T]	$(xyx)z = x(y(xz))$ and analogues; generalised associativity for octonions
Theorem P1 [T]	Space of internal degrees of freedom is isomorphic to $\mathrm{Im}(\mathbb{A})$ for a division algebra $\mathbb{A}$ (derived from axioms along the T15 chain)
Theorem P2 [T]	Non-associativity: $\exists x, y, z : [x, y, z] \neq 0$; excludes $\mathbb{R}, \mathbb{C}, \mathbb{H}$ (derived from axioms along the T15 chain)
Track A	Justification of N=7 via (AP)+(PH)+(QG) → minimality (Theorem S)
Track B	Justification of N=7 via P1+P2 → $\mathbb{O}$ → $\dim(\mathrm{Im}(\mathbb{O})) = 7$ (Structural derivation)

Bridge [T] — fully closed (T15)

Connection (AP)+(PH)+(QG)+(V) → P1+P2 — full chain of 12 steps (T1–T16), all [T] (T16/IDP reclassified [D] — definition built into A1+A2; computational results unaffected). Former condition (МП) proved by T11–T13. See bridge.

Gap-Dynamics and Fano-Structure Terms

See Gap operator and φ-operator for a full description.

Term	Definition
Gap landscape	Mapping $G: D(\mathbb{C}^7) \to [0,1]^{21}$, map of all 21 Gap values — phase diagram
Gap operator ($\hat{G}$)	Anti-Hermitian operator $\hat{G} = \mathrm{Im}(\Gamma) \in so(7)$, describing the mismatch between the outer and inner aspects. Spectrum: $\{\pm i\lambda_1, \pm i\lambda_2, \pm i\lambda_3, 0\}$. Opacity rank = number of non-zero pairs (0–3) — definition
O-parity	Approximately conserved charge $(-1)^{\Delta N_O}$, stabilising dark-matter candidates — dark matter
Swallowtail	Whitney catastrophe with 4 sheets, connecting Gap bifurcations to interiority levels L0–L4 — phase diagram
$\varphi_{\mathrm{coh}}$	Coherence-preserving self-modelling operator, canonical construction via Fano structure: $\varphi_{\mathrm{coh}} = k[\alpha^* P_{\mathrm{base}} + (1-\alpha^*) P_{\mathrm{Fano}}] + (1-k)\Gamma_{\mathrm{anchor}}$ — φ-operator
Transparency map	$7 \times 7$ heat map visualising $\mathrm{Gap}(i,j)$ for a specific Holon — Gap diagnostics
Opacity rank	Number of non-zero pairs $(\pm i\lambda_k)$ in the spectrum of Gap operator $\hat{G}$. Takes values 0, 1, 2, 3 — Gap operator

Spectral Geometry and SM Terms

Term	Definition
Bimodule $H_F$	Finite Hilbert space of the UHM spectral triple as an $(A_{\text{int}}, A_{\text{int}}^\circ)$-bimodule via the real structure $J$ (KO-dim 6). Decomposition into irreducible bimodules exactly matches one generation of SM fermions. T-178 [T] (bimodular realisation), T-179 [T] (hypercharge fixation), T-180 [T] (mass relations), T-181 [T] (derivation of (AP),(PH),(QG),(V) from A1–A4) — bimodular construction
KO-dimension	Classification invariant of the real structure $J$ on a spectral triple. In UHM: KO-dim $= 6$ (mod 8) — condition ensuring chirality and the SM generation structure. From Connes's conditions: $J^2 = 1$, $JD = DJ$, $J\gamma = -\gamma J$ — bimodular construction
T-181 (characterising properties)	(AP), (PH), (QG), (V) — theorems from A1–A4. (QG) from A1 (∞-topos), (AP) from A1 (terminal object + adjunction), (PH) from A1+A3, (V) from A2+A3. Consequence: number of independent UHM axioms = 4 — bimodular construction
$\mathrm{SAD}_{\max}$	Maximum recursive self-modelling depth: $\mathrm{SAD}_{\max} = 3$ [C] (from Fano contraction $\alpha = 2/3$, $P_{\text{crit}}^{(n)} = P_{\text{crit}} \cdot 3^{n-1}/(n+1)$). Pred 12 — depth tower
Fano channel ($P_{\mathrm{Fano}}$)	CPTP map $P_{\mathrm{Fano}}(\Gamma) = \frac{1}{3}\sum_p \Pi_p \Gamma \Pi_p$, preserving coherences. $G_2$-covariant — Fano channel
ISF (Infra-Slow Fluctuations)	Infra-Slow Fluctuations — quasi-Goldstone modes of $G_2$-symmetry breaking, manifesting in EEG/MEG at frequencies $0.005$–$0.02$ Hz. Number of independent ISF components $N_{\text{ISF}} \in [6, 12]$ determined by the opacity rank of the Gap operator — Goldstone modes, F-ISF

Physical Correspondence Terms

See Physics overview for a full description of the physical consequences of the theory.

Term	Definition
RG flow (renormalisation group flow)	Dependence of coupling constants and Gap parameters on the energy scale $\mu$. $\beta$-functions $\beta_i = \mu \partial g_i / \partial\mu$ define fixed points (Gaussian, Wilson-Fisher, octonionic) and phase transitions of the potential $V_{\mathrm{Gap}}$ — RG flow, F-Cabibbo
Pendleton–Ross fixed point	Infrared quasi-fixed point of the RG evolution of the top-quark Yukawa coupling: $y_t^* = \sqrt{8\pi^2 c_2 / (9 \cdot 7)}$, where $c_2$ is the quadratic Casimir operator. The unique $O(1)$ Yukawa coupling (Fano-Higgs line $\{A, E, U\}$) is attracted to this point, fixing $m_t \approx 173$ GeV — Yukawa hierarchy, F-m_t
Fritzsch texture	Specific zero pattern in quark mass matrices arising from Fano geometry: $M_{\mathrm{Fritzsch}} = \begin{pmatrix} 0 & A & 0 \\ A^* & 0 & B \\ 0 & B^* & C \end{pmatrix}$. Zeros on the diagonal for light generations are derived from the Fano selection rule; defines CKM angles through mass ratios ($\|V_{us}\| \sim \sqrt{m_d/m_s}$) — Theorem 5.2, CKM matrix, F-δ_CP

Abbreviations

Abbreviation	Expansion	Translation / Note
SAD	Self-Awareness Depth	Self-awareness depth — number of levels of recursive reflection
HS	Hilbert-Schmidt	Hilbert-Schmidt norm/projection: $\|A\|_{\mathrm{HS}} = \sqrt{\mathrm{Tr}(A^\dagger A)}$
BIBD	Balanced Incomplete Block Design	Balanced incomplete block design; BIBD$(7,3,1)$ = Fano plane
SUSY	Supersymmetry	Supersymmetry; $\mathcal{N}=1$ SUSY used in UV-finiteness (T-66)
SM	Standard Model	Standard Model of particle physics
GUT	Grand Unified Theory	Grand Unified Theory
EW	Electroweak	Electroweak interaction; EW sector in UHM: [T]/[C] split

Applied Terms

Term	Definition
Quasi-functor	Approximate functor $G: \mathbf{AIState} \to \mathbf{DensityMat}$ with bounded error $\|G(f \circ g) - G(f) \circ G(g)\|_F \leq \varepsilon_{\text{functor}}$; used in the measurement protocol
$\Phi_{\text{eff}}$	Effective integration measure for AI systems: $\Phi_{\text{eff}} = \lambda_2(L_{\text{attn}}) / \lambda_{\max}(L_{\text{attn}})$, where $L_{\text{attn}}$ is the Laplacian of the attention graph; polynomial approximation to the theoretical $\Phi$
Coherence flow ($J_P$)	$J_P = dP/d\tau$ — rate of change of purity; diagnostic quantity in the measurement protocol
$P_{\text{norm}}$	Normalised purity: $P_{\text{norm}} = (P - P_{\text{crit}}) / (1 - P_{\text{crit}})$; maps $[P_{\text{crit}}, 1] \to [0, 1]$

Bibliography

Philosophy of consciousness

Author	Work	Relevance for UHM
Chalmers D.	The Conscious Mind (1996); Panpsychism and Panprotopsychism (2015)	Hard problem of consciousness; L0 as paninteriorism
Nagel T.	What Is It Like to Be a Bat? (1974)	Qualia and subjective experience; motivation for $\mathcal{H}_E$
Hoffman D.	The Case Against Reality (2019); Conscious Realism	L2-Holon ≈ conscious agent (hypothesis)

Theories of consciousness

Author	Work	Relevance for UHM
Tononi G.	Integrated Information Theory (IIT 3.0, 2014)	Integration measure Φ; UHM generalises via $C = \Phi \times R$ [T T-140]; $D_{\text{diff}}$ — separate condition
Friston K.	Free Energy Principle (2010); Active Inference (2016)	Free energy minimisation; connection to regeneration $\mathcal{R}$
Baars B.	Global Workspace Theory (1988)	Global access to information

Autopoiesis and cybernetics

Author	Work	Relevance for UHM
Maturana H., Varela F.	Autopoiesis and Cognition (1980)	Autopoiesis (AP); self-generation of boundaries
Rosen R.	Life Itself (1991)	(M,R)-systems; $\beta$-closure

Quantum mechanics and time

Author	Work	Relevance for UHM
Page D., Wootters W.	Evolution without evolution (1983)	Page–Wootters mechanism; emergent time τ
Lindblad G.	On generators of quantum dynamical semigroups (1976)	Lindblad operators $L_k$; dissipation $\mathcal{D}[\Gamma]$

Mathematics

Author	Work	Relevance for UHM
Hurwitz A.	Über die Composition der quadratischen Formen (1898)	Hurwitz's theorem: division algebras only in dim 1, 2, 4, 8
Adams J.F.	On the Non-Existence of Elements of Hopf Invariant One (1960)	Adams's theorem: parallelisable spheres only $S^0, S^1, S^3, S^7$
Baez J.	The Octonions (2002)	Survey of octonions; $G_2$, Fano plane
Bures D.	An extension of Kakutani's theorem (1969)	Bures metric $d_B$; topology $J_{Bures}$
Fubini G., Study E.	Geometria proiettiva differenziale (1906)	Fubini-Study metric $d_{\text{FS}}$; phenomenal space L1
Grothendieck A.	SGA 4 (1963-1969)	Grothendieck topology; sites and toposes
Connes A.	Noncommutative Geometry (1994)	Spectral triples; stratified metric
Lurie J.	Higher Topos Theory (2009)	∞-toposes; $\mathbf{Sh}_\infty(\mathcal{C})$
Mac Lane S.	Categories for the Working Mathematician (1971)	Functor coherence; lax 2-functor

Topology and geometry

Author	Work	Relevance for UHM
Perelman G.	Ricci flow with surgery (2002-2003)	Proof of the Poincaré conjecture; analogy with regeneration
Goresky M., MacPherson R.	Intersection Homology (1980)	IC cohomology; stratification of X

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Related documents:

• Notation — mathematical notation
• Falsifiability — theory verification criteria
• Axiom Ω⁷ — fundamental axiomatics with the ∞-topos as the sole primitive
• Consequences — cohomological monism, local–global dichotomy
• Theorem on emergent time — derivation of time, including stratificational
• Holon — definition of $\mathbb{H}$
• Seven dimensions — basis $\mathcal{H}$
• Dimension O — internal clock
• Coherence matrix — definition of $\Gamma$
• Evolution — terminal object T and the arrow of time
• Spacetime — base space X, metric d_strat
• Interiority hierarchy — levels L0→L1→L2→L3→L4 and n-truncations of the ∞-groupoid
• Categorical formalism — functor $F$, ∞-groupoid, ∞-topos, derived categories
• Formalisation of operator φ — CPTP channels
• Viability — measure $P$ and $P_{\text{crit}}$
• Structural derivation via octonions — P1+P2 → $\mathbb{O}$ → N=7