Potential notation conflicts
In UHM theory, some symbols have multiple meanings depending on context:
D D D Dynamics dimension vs D diff D_{\text{diff}} D diff differentiation measure H \mathcal{H} H vs H H H vs H Γ \mathcal{H}_\Gamma H Γ Freedom )Φ \Phi Φ integration measure . For denoting arbitrary CPTP channels, Ψ \Psi Ψ R R R reflection measure vs R \mathcal{R} R C \mathcal{C} C vs C C C consciousness measure . The context space in the Exp category is denoted Γ − E \Gamma_{-E} Γ − E γ i j \gamma_{ij} γ ij vs γ k \gamma_k γ k Recommendation: use Γ 2 \Gamma_2 Γ 2 Theorem 8.1 ) Context usually makes the meaning unambiguous.
Connection to IIT (Integrated Information Theory)
The integration measure Φ \Phi Φ differs from Φ \Phi Φ
Parameter UHM IIT Definition Φ UHM = ∑ i ≠ j ∣ γ i j ∣ 2 / ∑ i γ i i 2 \Phi_{\text{UHM}} = \sum_{i \neq j} \lvert\gamma_{ij}\rvert^2 / \sum_i \gamma_{ii}^2 Φ UHM = ∑ i = j ∣ γ ij ∣ 2 / ∑ i γ ii 2 Φ IIT \Phi_{\text{IIT}} Φ IIT Interpretation Coherence between dimensions Integrated information Computational complexity O ( n 2 ) O(n^2) O ( n 2 ) NP-hard
UHM generalises IIT: the consciousness measure C = Φ × R C = \Phi \times R C = Φ × R [T T-140] includes integration Φ \Phi Φ R R R D diff ≥ D min D_{\text{diff}} \geq D_{\min} D diff ≥ D m i n
Core Symbols
Symbol Meaning Definition C \mathcal{C} C Primitive category Small category with a finite number of objects — sole primitive
| Γ \Gamma Γ Coherence matrix | Γ ∈ L ( H ) \Gamma \in \mathcal{L}(\mathcal{H}) Γ ∈ L ( H ) Γ † = Γ \Gamma^\dagger = \Gamma Γ † = Γ Γ ≥ 0 \Gamma \geq 0 Γ ≥ 0 T r ( Γ ) = 1 \mathrm{Tr}(\Gamma) = 1 Tr ( Γ ) = 1 H \mathbb{H} H Holon | Minimal self-sufficient unit of reality |
| H \mathcal{H} H H = C 7 \mathcal{H} = \mathbb{C}^7 H = C 7 Seven dimensions |
| P P P Purity | P = T r ( Γ 2 ) ∈ [ 1 / 7 , 1 ] P = \mathrm{Tr}(\Gamma^2) \in [1/7, 1] P = Tr ( Γ 2 ) ∈ [ 1/7 , 1 ] S v N S_{vN} S v N S v N = − T r ( Γ log Γ ) ∈ [ 0 , log 7 ] S_{vN} = -\mathrm{Tr}(\Gamma \log \Gamma) \in [0, \log 7] S v N = − Tr ( Γ log Γ ) ∈ [ 0 , log 7 ] τ \tau τ Internal time | Evolution parameter derived from the structure of C \mathcal{C} C τ ∈ Z 7 \tau \in \mathbb{Z}_7 τ ∈ Z 7 t , t ′ t, t' t , t ′ τ \tau τ t = n ⋅ δ τ t = n \cdot \delta\tau t = n ⋅ δ τ H eff H_{\text{eff}} H eff Effective Hamiltonian | H eff ( τ ) = H 6 D + ⟨ τ ∣ H int ∣ τ ⟩ O H_{\text{eff}}(\tau) = H_{6D} + \langle\tau\vert H_{\text{int}}\vert\tau\rangle_O H eff ( τ ) = H 6 D + ⟨ τ ∣ H int ∣ τ ⟩ O d B d_B d B Bures metric | Angular: d B a n g l e = arccos ( F ) d_B^{\mathrm{angle}} = \arccos(\sqrt{F}) d B angle = arccos ( F ) d B c h o r d = 2 ( 1 − F ) d_B^{\mathrm{chord}} = \sqrt{2(1-\sqrt{F})} d B chord = 2 ( 1 − F ) convention below |
Base Space and Stratification
Symbol Meaning Definition X X X Base space X = ∥ N ( C ) ∥ X = \|N(\mathcal{C})\| X = ∥ N ( C ) ∥ N ( C ) N(\mathcal{C}) N ( C ) Nerve of the category Simplicial set: n-simplices = composable chains of morphisms T T T Terminal object T = Γ ∗ T = \Gamma^* T = Γ ∗ ∀ Γ , ∃ ! f : Γ → T \forall\Gamma, \exists! f: \Gamma \to T ∀Γ , ∃ ! f : Γ → T S α S_\alpha S α Stratum Component of the stratification X = ⨆ α S α X = \bigsqcup_\alpha S_\alpha X = ⨆ α S α S 0 = { T } S_0 = \{T\} S 0 = { T } d s t r a t d_{strat} d s t r a t Stratified metric d s t r a t ( ω 1 , ω 2 ) = inf γ ∫ γ d s α d_{strat}(\omega_1, \omega_2) = \inf_\gamma \int_\gamma ds_\alpha d s t r a t ( ω 1 , ω 2 ) = inf γ ∫ γ d s α Link ( T ) \text{Link}(T) Link ( T ) Link of the terminal object Link ( T ) ≅ S 6 \text{Link}(T) \cong S^6 Link ( T ) ≅ S 6 H ∗ ( X ) H^*(X) H ∗ ( X ) Cohomology H n ( X , F ) = 0 H^n(X, \mathcal{F}) = 0 H n ( X , F ) = 0 n > 0 n > 0 n > 0 H l o c ∗ ( X , T ) H^*_{loc}(X,T) H l oc ∗ ( X , T ) Local cohomology H l o c ∗ ( X , T ) ≅ H ~ ∗ − 1 ( S 6 ) ≠ 0 H^*_{loc}(X,T) \cong \tilde{H}^{*-1}(S^6) \neq 0 H l oc ∗ ( X , T ) ≅ H ~ ∗− 1 ( S 6 ) = 0 D b ( X ) D^b(X) D b ( X ) Derived category Bounded derived category of sheaves on X I C ( S α ) IC(S_\alpha) I C ( S α ) IC sheaf Intersection cohomology sheaf of stratum S α S_\alpha S α
Dimensions
Seven basis states of space H \mathcal{H} H :
Symbol Dimension Associated structure More A A A Articulation Projectors, measurements → S S S Structure Hamiltonian H H H → D D D Dynamics Unitary evolution U ( τ ) U(\tau) U ( τ ) → L L L Logic Operator algebra → E E E Interiority Density matrix ρ E \rho_E ρ E → O O O Ground Vacuum state ∣ 0 ⟩ \vert 0\rangle ∣0 ⟩ Page–Wootters ) → U U U Unity Trace operation T r \mathrm{Tr} Tr →
State Space Basis
H = s p a n { ∣ A ⟩ , ∣ S ⟩ , ∣ D ⟩ , ∣ L ⟩ , ∣ E ⟩ , ∣ O ⟩ , ∣ U ⟩ } = C 7 \mathcal{H} = \mathrm{span}\{|A\rangle, |S\rangle, |D\rangle, |L\rangle, |E\rangle, |O\rangle, |U\rangle\} = \mathbb{C}^7 H = span { ∣ A ⟩ , ∣ S ⟩ , ∣ D ⟩ , ∣ L ⟩ , ∣ E ⟩ , ∣ O ⟩ , ∣ U ⟩} = C 7 Orthonormality: ⟨ i ∣ j ⟩ = δ i j \langle i|j\rangle = \delta_{ij} ⟨ i ∣ j ⟩ = δ ij i , j ∈ { A , S , D , L , E , O , U } i, j \in \{A, S, D, L, E, O, U\} i , j ∈ { A , S , D , L , E , O , U }
Clock Algebra (Page–Wootters)
Symbol Meaning Definition H O H_O H O Clock Hamiltonian H O = ω 0 ∑ k = 0 N − 1 k ∣ k ⟩ ⟨ k ∣ O H_O = \omega_0 \sum_{k=0}^{N-1} k \vert k\rangle\langle k\vert_O H O = ω 0 ∑ k = 0 N − 1 k ∣ k ⟩ ⟨ k ∣ O V O V_O V O Time shift operator V O N = 1 V_O^N = \mathbb{1} V O N = 1 V O H O V O † = H O + ω 0 1 V_O H_O V_O^\dagger = H_O + \omega_0 \mathbb{1} V O H O V O † = H O + ω 0 1 A O \mathcal{A}_O A O Clock C*-algebra A O = C ∗ ( H O , V O ) ≅ M N ( C ) \mathcal{A}_O = C^*(H_O, V_O) \cong M_N(\mathbb{C}) A O = C ∗ ( H O , V O ) ≅ M N ( C ) H int H_{\text{int}} H int Interaction Hamiltonian Coupling of O with E and U C ^ \hat{C} C ^ Page–Wootters constraint C ^ = H O ⊗ 1 6 D + 1 O ⊗ H 6 D + H int \hat{C} = H_O \otimes \mathbb{1}_{6D} + \mathbb{1}_O \otimes H_{6D} + H_{\text{int}} C ^ = H O ⊗ 1 6 D + 1 O ⊗ H 6 D + H int H t o t a l \mathcal{H}_{total} H t o t a l Global space H t o t a l = H O ⊗ H 6 D \mathcal{H}_{total} = \mathcal{H}_O \otimes \mathcal{H}_{6D} H t o t a l = H O ⊗ H 6 D dim = 42 \dim = 42 dim = 42 ω 0 \omega_0 ω 0 Fundamental frequency Base frequency of clock O ∣ τ n ⟩ \vert\tau_n\rangle ∣ τ n ⟩ Clock basis Eigenstates of V O V_O V O
Evolution Equation
Full evolution equation with emergent internal time τ:
d Γ ( τ ) d τ = − i [ H eff , Γ ] + D [ Γ ] + R [ Γ , E ] \frac{d\Gamma(\tau)}{d\tau} = -i[H_{\text{eff}}, \Gamma] + \mathcal{D}[\Gamma] + \mathcal{R}[\Gamma, E] d τ d Γ ( τ ) = − i [ H eff , Γ ] + D [ Γ ] + R [ Γ , E ] where:
Unitary term :
− i [ H eff , Γ ] = − i ( H eff Γ − Γ H eff ) -i[H_{\text{eff}}, \Gamma] = -i(H_{\text{eff}}\Gamma - \Gamma H_{\text{eff}}) − i [ H eff , Γ ] = − i ( H eff Γ − Γ H eff ) Here H eff H_{\text{eff}} H eff
Dissipative term :
D [ Γ ] = ∑ k γ k ( L k Γ L k † − 1 2 { L k † L k , Γ } ) \mathcal{D}[\Gamma] = \sum_k \gamma_k \left( L_k \Gamma L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \Gamma\} \right) D [ Γ ] = k ∑ γ k ( L k Γ L k † − 2 1 { L k † L k , Γ } ) Regenerative term [T]:
R [ Γ , E ] = κ ( Γ ) ⋅ ( ρ ∗ − Γ ) ⋅ g V ( P ) \mathcal{R}[\Gamma, E] = \kappa(\Gamma) \cdot (\rho_* - \Gamma) \cdot g_V(P) R [ Γ , E ] = κ ( Γ ) ⋅ ( ρ ∗ − Γ ) ⋅ g V ( P ) where:
κ ( Γ ) ≥ 0 \kappa(\Gamma) \geq 0 κ ( Γ ) ≥ 0 D Ω ⊣ R \mathcal{D}_\Omega \dashv \mathcal{R} D Ω ⊣ R ρ ∗ = φ ( Γ ) \rho_* = \varphi(\Gamma) ρ ∗ = φ ( Γ ) φ-operator )( ρ ∗ − Γ ) (\rho_* - \Gamma) ( ρ ∗ − Γ ) g V ( P ) = c l a m p ( P − P c r i t P o p t − P c r i t ) g_V(P) = \mathrm{clamp}\!\bigl(\frac{P - P_{\mathrm{crit}}}{P_{\mathrm{opt}} - P_{\mathrm{crit}}}\bigr) g V ( P ) = clamp ( P opt − P crit P − P crit ) derivation )
Commutators and Anticommutators
Notation Definition [ A , B ] [A, B] [ A , B ] A B − B A AB - BA A B − B A { A , B } \{A, B\} { A , B } A B + B A AB + BA A B + B A
Interiority Operators (Dimension E)
See Interiority Dimension and Exp Category .
Notation Meaning ρ E \rho_E ρ E Reduced density matrix of the Interiority dimension λ i \lambda_i λ i Eigenvalue of Γ \Gamma Γ ∣ q i ⟩ \vert q_i\rangle ∣ q i ⟩ Eigenvector of Γ \Gamma Γ [ ∣ q ⟩ ] [\vert q\rangle] [ ∣ q ⟩] Equivalence class in P ( H E ) \mathbb{P}(\mathcal{H}_E) P ( H E ) P ( H E ) \mathbb{P}(\mathcal{H}_E) P ( H E ) Projective space of qualitiesd F S d_{\mathrm{FS}} d FS Fubini-Study metric
Fubini-Study metric:
d F S ( [ ∣ ψ ⟩ ] , [ ∣ ϕ ⟩ ] ) = arccos ( ∣ ⟨ ψ ∣ ϕ ⟩ ∣ ) ∈ [ 0 , π / 2 ] d_{\mathrm{FS}}([|\psi\rangle], [|\phi\rangle]) = \arccos(|\langle\psi|\phi\rangle|) \in [0, \pi/2] d FS ([ ∣ ψ ⟩] , [ ∣ ϕ ⟩]) = arccos ( ∣ ⟨ ψ ∣ ϕ ⟩ ∣ ) ∈ [ 0 , π /2 ] Consciousness Measures
See Self-observation for full definitions.
Measure Formula Range Integration Φ \Phi Φ Φ ( Γ ) = ∑ i ≠ j ∣ γ i j ∣ 2 ∑ i γ i i 2 \Phi(\Gamma) = \dfrac{\sum_{i \neq j} \lvert\gamma_{ij}\rvert^2}{\sum_i \gamma_{ii}^2} Φ ( Γ ) = ∑ i γ ii 2 ∑ i = j ∣ γ ij ∣ 2 [ 0 , + ∞ ) [0, +\infty) [ 0 , + ∞ ) Differentiation D diff D_{\text{diff}} D diff D diff ( Γ ) = exp ( S v N ( ρ E ) ) D_{\text{diff}}(\Gamma) = \exp(S_{vN}(\rho_E)) D diff ( Γ ) = exp ( S v N ( ρ E )) [ 1 , 7 ] [1, 7] [ 1 , 7 ] Reflection R R R R ( Γ ) = R canonical = 1 7 P ( Γ ) R(\Gamma) = R_{\text{canonical}} = \dfrac{1}{7P(\Gamma)} R ( Γ ) = R canonical = 7 P ( Γ ) 1 P = T r ( Γ 2 ) P = \mathrm{Tr}(\Gamma^2) P = Tr ( Γ 2 ) 1 − ∥ Γ − I / 7 ∥ F 2 P 1 - \dfrac{\|\Gamma - I/7\|_F^2}{P} 1 − P ∥Γ − I /7 ∥ F 2 Q φ = 1 − ∥ Γ − φ ( Γ ) ∥ F 2 / P Q_\varphi = 1 - \|\Gamma - \varphi(\Gamma)\|_F^2 / P Q φ = 1 − ∥Γ − φ ( Γ ) ∥ F 2 / P [ 0 , 1 ] [0, 1] [ 0 , 1 ] Consciousness C C C C ( Γ ) = Φ × R C(\Gamma) = \Phi \times R C ( Γ ) = Φ × R [T] (T-140); D diff ≥ 2 D_{\text{diff}} \geq 2 D diff ≥ 2 [ 0 , + ∞ ) [0, +\infty) [ 0 , + ∞ ) Free will F r e e d o m ( Γ ) \mathrm{Freedom}(\Gamma) Freedom ( Γ ) [T] F r e e d o m ( Γ ) = dim ker ( H Γ ) + 1 \mathrm{Freedom}(\Gamma) = \dim\ker(\mathcal{H}_\Gamma) + 1 Freedom ( Γ ) = dim ker ( H Γ ) + 1 H Γ = ∂ 2 F / ∂ Γ 2 \mathcal{H}_\Gamma = \partial^2 \mathcal{F}/\partial\Gamma^2 H Γ = ∂ 2 F / ∂ Γ 2 finite-dimensional definition . ∞-categorical motivation: π 0 ( M a p ( Γ , T ) non-trivial ) \pi_0(\mathrm{Map}(\Gamma, T)^{\text{non-trivial}}) π 0 ( Map ( Γ , T ) non-trivial ) { 1 , … , 7 } \{1, \ldots, 7\} { 1 , … , 7 } Freedom entropy S freedom S_{\text{freedom}} S freedom S freedom = log ( Freedom ( Γ ) ) S_{\text{freedom}} = \log(\text{Freedom}(\Gamma)) S freedom = log ( Freedom ( Γ )) [ 0 , log 7 ] [0, \log 7] [ 0 , log 7 ]
Self-Modelling Operator
See Formalisation of operator φ for a full description.
CPTP channel (Completely Positive Trace-Preserving):
φ ( Γ ) = ∑ m K m Γ K m † \varphi(\Gamma) = \sum_m K_m \Gamma K_m^\dagger φ ( Γ ) = m ∑ K m Γ K m † Completeness condition (trace preservation):
∑ m K m † K m = I \sum_m K_m^\dagger K_m = I m ∑ K m † K m = I Convergence to fixed point Γ ∗ = φ ( Γ ∗ ) \Gamma^* = \varphi(\Gamma^*) Γ ∗ = φ ( Γ ∗ )
∥ φ n ( Γ 0 ) − Γ ∗ ∥ F ≤ k n ⋅ ∥ Γ 0 − Γ ∗ ∥ F , k ∈ [ 0 , 1 ) \|\varphi^n(\Gamma_0) - \Gamma^*\|_F \leq k^n \cdot \|\Gamma_0 - \Gamma^*\|_F, \quad k \in [0, 1) ∥ φ n ( Γ 0 ) − Γ ∗ ∥ F ≤ k n ⋅ ∥ Γ 0 − Γ ∗ ∥ F , k ∈ [ 0 , 1 ) Interiority Hierarchy
See Interiority hierarchy for formal conditions and proofs.
Level Notation Condition n-truncation L0 I n t ( S ) \mathrm{Int}(S) Int ( S ) ∃ ρ E \exists \rho_E ∃ ρ E τ ≤ 0 \tau_{\leq 0} τ ≤ 0 L1 P G ( S ) \mathrm{PG}(S) PG ( S ) r a n k ( ρ E ) > 1 \mathrm{rank}(\rho_E) > 1 rank ( ρ E ) > 1 τ ≤ 1 \tau_{\leq 1} τ ≤ 1 L2 Cognitive qualia R ≥ R th R \geq R_{\text{th}} R ≥ R th Φ ≥ Φ th \Phi \geq \Phi_{\text{th}} Φ ≥ Φ th D diff ≥ 2 D_{\text{diff}} \geq 2 D diff ≥ 2 τ ≤ 2 \tau_{\leq 2} τ ≤ 2 L3 Network consciousness R ( 2 ) ≥ R th ( 2 ) R^{(2)} \geq R^{(2)}_{\text{th}} R ( 2 ) ≥ R th ( 2 ) τ ≤ 3 \tau_{\leq 3} τ ≤ 3 L4 Unitary consciousness lim n → ∞ R ( n ) > 0 \lim_{n \to \infty} R^{(n)} > 0 lim n → ∞ R ( n ) > 0 P > 6 / 7 P > 6/7 P > 6/7 τ ≤ ∞ \tau_{\leq \infty} τ ≤ ∞
Threshold values (all proved mathematically [T] , threshold justifications ):
Threshold Value Status R th R_{\text{th}} R th 1 / 3 1/3 1/3 [T] Theorem (K = 3 K=3 K = 3 Φ th \Phi_{\text{th}} Φ th 1 1 1 [T] Theorem (T-129: unique self-consistent value) R th ( 2 ) R^{(2)}_{\text{th}} R th ( 2 ) 1 / 4 1/4 1/4 [T] Theorem (L3 threshold) X th ( n ) X^{(n)}_{\text{th}} X th ( n ) 1 / ( n + 1 ) 1/(n+1) 1/ ( n + 1 ) [T] Universal formula
Stress Tensor
See Viability for a full description.
σ s y s ( Γ ) = [ σ A , σ S , σ D , σ L , σ E , σ O , σ U ] T ∈ R 7 \sigma_{\mathrm{sys}}(\Gamma) = [\sigma_A, \sigma_S, \sigma_D, \sigma_L, \sigma_E, \sigma_O, \sigma_U]^T \in \mathbb{R}^7 σ sys ( Γ ) = [ σ A , σ S , σ D , σ L , σ E , σ O , σ U ] T ∈ R 7 Viability condition:
∥ σ s y s ( Γ ) ∥ ∞ < 1 \|\sigma_{\mathrm{sys}}(\Gamma)\|_\infty < 1 ∥ σ sys ( Γ ) ∥ ∞ < 1 Viability margin:
m a r g i n ( Γ ) = 1 − ∥ σ s y s ( Γ ) ∥ ∞ > 0 \mathrm{margin}(\Gamma) = 1 - \|\sigma_{\mathrm{sys}}(\Gamma)\|_\infty > 0 margin ( Γ ) = 1 − ∥ σ sys ( Γ ) ∥ ∞ > 0 Grothendieck Topology
See Grothendieck topology and Categorical formalism .
Bures metric (canonical form):
d B ( Γ 1 , Γ 2 ) = arccos ( T r Γ 1 Γ 2 Γ 1 ) = arccos ( F ) d_B(\Gamma_1, \Gamma_2) = \arccos\left(\mathrm{Tr}\sqrt{\sqrt{\Gamma_1}\Gamma_2\sqrt{\Gamma_1}}\right) = \arccos(\sqrt{F}) d B ( Γ 1 , Γ 2 ) = arccos ( Tr Γ 1 Γ 2 Γ 1 ) = arccos ( F ) Fidelity:
F i d ( Γ 1 , Γ 2 ) = ( T r Γ 1 Γ 2 Γ 1 ) 2 \mathrm{Fid}(\Gamma_1, \Gamma_2) = \left(\mathrm{Tr}\sqrt{\sqrt{\Gamma_1}\Gamma_2\sqrt{\Gamma_1}}\right)^2 Fid ( Γ 1 , Γ 2 ) = ( Tr Γ 1 Γ 2 Γ 1 ) 2 F i d \mathrm{Fid} Fid F F F experience functor F : D e n s i t y M a t → E x p F: \mathbf{DensityMat} \to \mathbf{Exp} F : DensityMat → Exp F F F
Two forms of the Bures metric
UHM uses both forms depending on context:
Form Formula Application Angular d B a n g l e = arccos ( F ) d_B^{angle} = \arccos(\sqrt{F}) d B an g l e = arccos ( F ) Geometric theorems (emergent time ) Chord d B c h o r d = 2 ( 1 − F ) d_B^{chord} = \sqrt{2(1-\sqrt{F})} d B c h or d = 2 ( 1 − F ) Computations, ΔF , specification
Relation: d B c h o r d = 2 ( 1 − cos ( d B a n g l e ) ) ≈ 2 ⋅ d B a n g l e d_B^{chord} = \sqrt{2(1 - \cos(d_B^{angle}))} \approx \sqrt{2} \cdot d_B^{angle} d B c h or d = 2 ( 1 − cos ( d B an g l e )) ≈ 2 ⋅ d B an g l e
Bures ball:
B B ( Γ , r ) = { Σ ∈ C : d B ( Γ , Σ ) < r } B_B(\Gamma, r) = \{\Sigma \in \mathcal{C} : d_B(\Gamma, \Sigma) < r\} B B ( Γ , r ) = { Σ ∈ C : d B ( Γ , Σ ) < r } Bures covering: Family { Φ i : Γ i → Γ } i ∈ I \{\Phi_i: \Gamma_i \to \Gamma\}_{i \in I} { Φ i : Γ i → Γ } i ∈ I Γ \Gamma Γ
∀ ϵ > 0 , ∃ δ > 0 : B B ( Γ , δ ) ⊆ ⋃ i ∈ I Φ i ( B B ( Γ i , ϵ ) ) \forall \epsilon > 0, \exists \delta > 0: \quad B_B(\Gamma, \delta) \subseteq \bigcup_{i \in I} \Phi_i(B_B(\Gamma_i, \epsilon)) ∀ ϵ > 0 , ∃ δ > 0 : B B ( Γ , δ ) ⊆ i ∈ I ⋃ Φ i ( B B ( Γ i , ϵ )) Site: Pair ( C , J B u r e s ) (\mathcal{C}, J_{Bures}) ( C , J B u res ) J B u r e s J_{Bures} J B u res
Classifier from topology:
Ω = O ( C , d B ) \Omega = \mathcal{O}(\mathcal{C}, d_B) Ω = O ( C , d B ) Special Notation
Notation Meaning ∥ ⋅ ∥ F \lVert\cdot\rVert_F ∥ ⋅ ∥ F Frobenius norm: ∥ A ∥ F = T r ( A † A ) = ∑ i j ∣ a i j ∣ 2 \lVert A\rVert_F = \sqrt{\mathrm{Tr}(A^\dagger A)} = \sqrt{\sum_{ij} \lvert a_{ij}\rvert^2} ∥ A ∥ F = Tr ( A † A ) = ∑ ij ∣ a ij ∣ 2 ∥ ⋅ ∥ ∞ \lVert\cdot\rVert_\infty ∥ ⋅ ∥ ∞ Supremum norm: ∥ x ∥ ∞ = max i ∣ x i ∣ \lVert x\rVert_\infty = \max_i \lvert x_i\rvert ∥ x ∥ ∞ = max i ∣ x i ∣ d B ( ⋅ , ⋅ ) d_B(\cdot, \cdot) d B ( ⋅ , ⋅ ) Bures metric F i d ( ⋅ , ⋅ ) \mathrm{Fid}(\cdot, \cdot) Fid ( ⋅ , ⋅ ) F ( ⋅ , ⋅ ) F(\cdot, \cdot) F ( ⋅ , ⋅ ) Fidelity; F i d \mathrm{Fid} Fid F F F B B ( Γ , r ) B_B(\Gamma, r) B B ( Γ , r ) Bures ball of radius r r r Γ \Gamma Γ J B u r e s J_{Bures} J B u res Bures coverage function (Grothendieck topology) Θ ( ⋅ ) \Theta(\cdot) Θ ( ⋅ ) Heaviside function δ i j \delta_{ij} δ ij Kronecker delta T r ( ⋅ ) \mathrm{Tr}(\cdot) Tr ( ⋅ ) Matrix trace A † A^\dagger A † Hermitian conjugate C o h E \mathrm{Coh}_E Coh E E-coherence (HS-projection π E \pi_E π E [T] , ∈ [ 1 / 7 , 1 ] \in [1/7, 1] ∈ [ 1/7 , 1 ] = ∥ π E ( Γ ) ∥ H S 2 / ∥ Γ ∥ H S 2 = \|\pi_E(\Gamma)\|_{\mathrm{HS}}^2 / \|\Gamma\|_{\mathrm{HS}}^2 = ∥ π E ( Γ ) ∥ HS 2 /∥Γ ∥ HS 2 master definition , HS-projection , CC reference IDP Information Distinguishability Principle [D] (T16) — definition , built into A1+A2: distinguishability via J Bures J_{\text{Bures}} J Bures φ coh \varphi_{\text{coh}} φ coh Coherence-preserving self-modelling — generalised φ-operator preserving coherences (Fano channel )
| κ ( Γ ) \kappa(\Gamma) κ ( Γ ) κ ( Γ ) = κ bootstrap + κ 0 ⋅ C o h E \kappa(\Gamma) = \kappa_{\text{bootstrap}} + \kappa_0 \cdot \mathrm{Coh}_E κ ( Γ ) = κ bootstrap + κ 0 ⋅ Coh E D diff D_{\text{diff}} D diff Γ \Gamma Γ I / N I/N I / N
| P crit P_{\text{crit}} P crit = 2 / N = 2 / 7 = 2/N = 2/7 = 2/ N = 2/7 theorem |
| d B c h o r d d_B^{chord} d B c h or d d B c h o r d = 2 ( 1 − F ( ρ , σ ) ) d_B^{chord} = \sqrt{2(1 - \sqrt{F(\rho, \sigma)})} d B c h or d = 2 ( 1 − F ( ρ , σ ) )
Categorical Notation
See Categorical formalism for a full description.
Notation Meaning C \mathcal{C} C Primitive category of UHM — sole primitiveD e n s i t y M a t \mathbf{DensityMat} DensityMat Category of density matrices E x p \mathbf{Exp} Exp Category of experiential states H o l \mathbf{Hol} Hol Category of Holons T T T Terminal object — ∀ Γ , ∃ ! f : Γ → T \forall\Gamma, \exists! f: \Gamma \to T ∀Γ , ∃ ! f : Γ → T F : D e n s i t y M a t → E x p F: \mathbf{DensityMat} \to \mathbf{Exp} F : DensityMat → Exp Experience functor C P T P \mathrm{CPTP} CPTP Completely Positive Trace-Preserving channels M o r ( ρ 1 , ρ 2 ) \mathrm{Mor}(\rho_1, \rho_2) Mor ( ρ 1 , ρ 2 ) Morphisms between objects ⊗ \otimes ⊗ Tensor product (composition of Holons) E x p ∞ \mathbf{Exp}_\infty Exp ∞ ∞-groupoid of experience E x p ∞ d i s c \mathbf{Exp}^{disc}_\infty Exp ∞ d i sc Discrete ∞-groupoid for N < ∞ N < \infty N < ∞ S h ∞ ( E x p ) \mathbf{Sh}_\infty(\mathbf{Exp}) Sh ∞ ( Exp ) ∞-topos of ∞-sheaves over ExpΩ E x p ∞ \Omega\mathbf{Exp}_\infty Ω Exp ∞ Loop space — emergent history D b ( X ) D^b(X) D b ( X ) Derived category of sheaves on XP e r v ( X ) \mathbf{Perv}(X) Perv ( X ) Category of perverse sheaves T H \mathcal{T}_H T H ∞-topos of Holons with HoTT as internal logicS h ∞ ( C ) \mathrm{Sh}_\infty(\mathcal{C}) Sh ∞ ( C ) ∞-topos of sheaves on category C \mathcal{C} C Ω 7 \Omega^7 Ω 7 M a p ( Γ , T ) \mathrm{Map}(\Gamma, T) Map ( Γ , T ) Morphism space in an ∞-category (mapping space) π n ( X ) \pi_n(X) π n ( X ) n-th homotopy group of space X X X ≃ \simeq ≃ Weak homotopy equivalence Ω \Omega Ω Subobject classifier — unified source of L, L_k, τχ S \chi_S χ S Characteristic morphism of subobject S: χ S : Γ → Ω \chi_S: \Gamma \to \Omega χ S : Γ → Ω L k L_k L k Lindblad operators : L k = P k = ∣ k ⟩ ⟨ k ∣ L_k = P_k = \lvert k\rangle\langle k\rvert L k = P k = ∣ k ⟩ ⟨ k ∣ derivation ). Notation L k = χ S k L_k = \sqrt{\chi_{S_k}} L k = χ S k P = P \sqrt{P} = P P = P L 0 \mathcal{L}_0 L 0 Linear Liouvillian (without regeneration): L 0 = − i [ H eff , ⋅ ] + ∑ k D L k \mathcal{L}_0 = -i[H_{\text{eff}},\cdot] + \sum_k D_{L_k} L 0 = − i [ H eff , ⋅ ] + ∑ k D L k T-39a [T] ; unique attractor I / 7 I/7 I /7 L Ω \mathcal{L}_\Omega L Ω Full logical Liouvillian : L Ω = L 0 + R \mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R} L Ω = L 0 + R ρ Ω ∗ ≠ I / 7 \rho^*_\Omega \neq I/7 ρ Ω ∗ = I /7 ▹ \triangleright ▹ Temporal modality on Ω; τ n = ▹ n ( n o w ) \tau_n = \triangleright^n(\mathrm{now}) τ n = ▹ n ( now ) D Ω ⊣ R \mathcal{D}_\Omega \dashv \mathcal{R} D Ω ⊣ R Dissipation–regeneration adjunction ; κ 0 = ∥ N a t ( D Ω , R ) ∥ \kappa_0 = \|\mathrm{Nat}(\mathcal{D}_\Omega, \mathcal{R})\| κ 0 = ∥ Nat ( D Ω , R ) ∥ (МП) Minimal representation principle (historical condition, now [T] T11–T13): among equivalent BIBD( 7 , 3 , λ ) (7,3,\lambda) ( 7 , 3 , λ ) λ = 1 \lambda = 1 λ = 1 b = 7 b=7 b = 7 [T] . Bridge to P1+P2 (КГ) Canonical grouping (historical): categorically natural mechanism for grouping atoms of Ω into composite blocks. Replaced by the weaker (МП), which in turn has been proved as theorem T11–T13
Coherence Cybernetics Notation
See Coherence Cybernetics for a full description.
Notation Meaning V \mathcal{V} V Viability domain V I T \mathrm{VIT} VIT Viability Integrity Tensor κ bootstrap \kappa_{\text{bootstrap}} κ bootstrap Minimum regeneration rate: κ bootstrap = ω 0 / 7 \kappa_{\text{bootstrap}} = \omega_0/7 κ bootstrap = ω 0 /7 [D] scale; resolves the bootstrap paradox κ 0 \kappa_0 κ 0 Categorical norm: $\kappa_0 = \omega_0 \cdot κ ( Γ ) \kappa(\Gamma) κ ( Γ ) Effective regeneration rate: κ ( Γ ) = κ bootstrap + κ 0 ⋅ C o h E ( Γ ) \kappa(\Gamma) = \kappa_{\text{bootstrap}} + \kappa_0 \cdot \mathrm{Coh}_E(\Gamma) κ ( Γ ) = κ bootstrap + κ 0 ⋅ Coh E ( Γ ) [T] C o h E \mathrm{Coh}_E Coh E E E E [T] : C o h E ( Γ ) = ∥ π E ( Γ ) ∥ H S 2 ∥ Γ ∥ H S 2 = γ E E 2 + 2 ∑ i ≠ E ∣ γ E i ∣ 2 T r ( Γ 2 ) \mathrm{Coh}_E(\Gamma) = \dfrac{\|\pi_E(\Gamma)\|_{\mathrm{HS}}^2}{\|\Gamma\|_{\mathrm{HS}}^2} = \dfrac{\gamma_{EE}^2 + 2\sum_{i \neq E}\lvert\gamma_{Ei}\rvert^2}{\mathrm{Tr}(\Gamma^2)} Coh E ( Γ ) = ∥Γ ∥ HS 2 ∥ π E ( Γ ) ∥ HS 2 = Tr ( Γ 2 ) γ EE 2 + 2 ∑ i = E ∣ γ E i ∣ 2 canonical formula (master definition , HS-projection )P E P_E P E E-sector purity (42D): P E = T r ( ρ E 2 ) P_E = \mathrm{Tr}(\rho_E^2) P E = Tr ( ρ E 2 ) ρ E = T r − E ( Γ ) \rho_E = \mathrm{Tr}_{-E}(\Gamma) ρ E = Tr − E ( Γ ) theoretical construction , defined only in the extended 42D formalism (H = C 42 \mathcal{H} = \mathbb{C}^{42} H = C 42 C o h E ≈ P E \mathrm{Coh}_E \approx P_E Coh E ≈ P E structural hypothesis [H] (details ) P crit P_{\text{crit}} P crit Critical purity = 2 / 7 ≈ 0.286 = 2/7 \approx 0.286 = 2/7 ≈ 0.286 theorem θ i \theta_i θ i Stress component thresholds H eff H_{\text{eff}} H eff Effective Hamiltonian: H eff ( τ ) = H 6 D + ⟨ τ ∣ H int ∣ τ ⟩ O H_{\text{eff}}(\tau) = H_{6D} + \langle\tau\vert H_{\text{int}}\vert\tau\rangle_O H eff ( τ ) = H 6 D + ⟨ τ ∣ H int ∣ τ ⟩ O g V ( P ) g_V(P) g V ( P ) V-preservation gate: c l a m p ( P − P c r i t P o p t − P c r i t , 0 , 1 ) \mathrm{clamp}\!\bigl(\frac{P - P_{\mathrm{crit}}}{P_{\mathrm{opt}} - P_{\mathrm{crit}}}, 0, 1\bigr) clamp ( P opt − P crit P − P crit , 0 , 1 ) P > P c r i t P > P_{\mathrm{crit}} P > P crit derivation ) Θ ( Δ F ) \Theta(\Delta F) Θ ( Δ F ) Heaviside function of the free-energy change Δ F \Delta F Δ F g V ( P ) g_V(P) g V ( P ) ρ ∗ \rho_* ρ ∗ = Γ target = \Gamma_{\text{target}} = Γ target Unique stationary state of L Ω \mathcal{L}_\Omega L Ω ρ ∗ = φ ( Γ ) = lim τ → ∞ e τ L Ω [ Γ ] \rho_* = \varphi(\Gamma) = \lim_{\tau\to\infty} e^{\tau\mathcal{L}_\Omega}[\Gamma] ρ ∗ = φ ( Γ ) = lim τ → ∞ e τ L Ω [ Γ ] ω 0 \omega_0 ω 0 Fundamental clock frequency — parameter of the computational approximation; see κ₀ D K L D_{\mathrm{KL}} D KL Kullback–Leibler divergence: D K L ( p ∥ q ) = ∑ i p i log ( p i / q i ) D_{\mathrm{KL}}(p \| q) = \sum_i p_i \log(p_i / q_i) D KL ( p ∥ q ) = ∑ i p i log ( p i / q i )
Dimension Indices (Measurement Protocol)
Empirical indices for measuring Γ projections in AI systems. See Measurement protocol for a full description.
Index Dimension AI metric Formula I A I_A I A Articulation Mutual information input↔latent I A = I ( input ; latent ) / H ( input ) I_A = I(\text{input}; \text{latent}) / H(\text{input}) I A = I ( input ; latent ) / H ( input ) I S I_S I S Structure Jacobian rank I S = r a n k ε ( J f ) / min ( d out , d in ) I_S = \mathrm{rank}_\varepsilon(J_f) / \min(d_{\text{out}}, d_{\text{in}}) I S = rank ε ( J f ) / min ( d out , d in ) I D I_D I D Dynamics Lyapunov exponent I D = max i λ i Lyap I_D = \max_i \lambda_i^{\text{Lyap}} I D = max i λ i Lyap I L I_L I L Logic Layer commutators I L = 1 − ∥ [ f i , f j ] ∥ F / ( ∥ f i ∥ ⋅ ∥ f j ∥ ) I_L = 1 - \|[f_i, f_j]\|_F / (\|f_i\| \cdot \|f_j\|) I L = 1 − ∥ [ f i , f j ] ∥ F / ( ∥ f i ∥ ⋅ ∥ f j ∥ ) I E I_E I E Interiority Differentiation (entropy) I E = D diff approx = exp ( S v N ( ρ attn ) ) I_E = D_{\text{diff}}^{\text{approx}} = \exp(S_{vN}(\rho_{\text{attn}})) I E = D diff approx = exp ( S v N ( ρ attn )) see dimension E I O I_O I O Ground Noise robustness I O = 1 − ∥ ∇ ϵ h ∥ F I_O = 1 - \|\nabla_\epsilon \mathbf{h}\|_F I O = 1 − ∥ ∇ ϵ h ∥ F I U I_U I U Unity Effective Φ (integration) I U = Φ eff = λ 2 ( L attn ) / λ max ( L attn ) I_U = \Phi_{\text{eff}} = \lambda_2(L_{\text{attn}}) / \lambda_{\max}(L_{\text{attn}}) I U = Φ eff = λ 2 ( L attn ) / λ m a x ( L attn ) see dimension U
Relation to Γ: Diagonal elements γ i i ≈ I i 2 \gamma_{ii} \approx I_i^2 γ ii ≈ I i 2
Additional Applied Symbols
Notation Meaning G G G Quasi-functor A I S t a t e → D e n s i t y M a t \mathbf{AIState} \to \mathbf{DensityMat} AIState → DensityMat J P J_P J P Coherence flow: J P = d P / d τ J_P = dP/d\tau J P = d P / d τ ε functor \varepsilon_{\text{functor}} ε functor Upper bound on the error of quasi-functor G G G P norm P_{\text{norm}} P norm Normalised purity: ( P − P crit ) / ( 1 − P crit ) (P - P_{\text{crit}}) / (1 - P_{\text{crit}}) ( P − P crit ) / ( 1 − P crit ) r \mathbf{r} r Generalised Bloch vector: Γ = I / N + ∑ k r k λ k / 2 \Gamma = I/N + \sum_k r_k \lambda_k / 2 Γ = I / N + ∑ k r k λ k /2
Octonionic Notation
See Structural derivation via octonions for a full description.
Notation Meaning O \mathbb{O} O Octonion algebra — 8-dimensional normed division algebra over R \mathbb{R} R I m ( O ) \mathrm{Im}(\mathbb{O}) Im ( O ) Imaginary part of the octonions; dim ( I m ( O ) ) = 7 \dim(\mathrm{Im}(\mathbb{O})) = 7 dim ( Im ( O )) = 7 e 1 , … , e 7 e_1, \ldots, e_7 e 1 , … , e 7 Imaginary units of the octonions; e i 2 = − 1 e_i^2 = -1 e i 2 = − 1 e i e j = − e j e i e_i e_j = -e_j e_i e i e j = − e j e i i ≠ j i \neq j i = j G 2 G_2 G 2 A u t ( O ) \mathrm{Aut}(\mathbb{O}) Aut ( O ) G 2 ⊂ S O ( 7 ) G_2 \subset SO(7) G 2 ⊂ SO ( 7 ) P G ( 2 , 2 ) \mathrm{PG}(2,2) PG ( 2 , 2 ) Fano plane — projective plane over F 2 \mathbb{F}_2 F 2 [ x , y , z ] [x, y, z] [ x , y , z ] Associator: [ x , y , z ] = ( x y ) z − x ( y z ) [x, y, z] = (xy)z - x(yz) [ x , y , z ] = ( x y ) z − x ( yz ) H ( 7 , 4 ) H(7,4) H ( 7 , 4 ) Hamming code: 4 information + 3 check bits; connection to PG(2,2) P1 Theorem [T]: space of internal degrees of freedom ≅ I m ( A ) \cong \mathrm{Im}(\mathbb{A}) ≅ Im ( A ) A \mathbb{A} A P2 Theorem [T]: non-associativity ([ x , y , z ] ≠ 0 [x, y, z] \neq 0 [ x , y , z ] = 0 x , y , z x, y, z x , y , z
Status of octonionic notation [I]
The correspondence e i ↔ e_i \leftrightarrow e i ↔ interpretation [I]. Mathematical operations on O \mathbb{O} O { A , S , D , L , E , O , U } \{A,S,D,L,E,O,U\} { A , S , D , L , E , O , U } open problem .
Gap Dynamics and Fano Structure
Symbols related to the Gap operator , Gap thermodynamics and Fano selection rules .
Notation Meaning G ^ \hat{G} G ^ Gap operator : G ^ = I m ( Γ ) ∈ s o ( 7 ) \hat{G} = \mathrm{Im}(\Gamma) \in \mathfrak{so}(7) G ^ = Im ( Γ ) ∈ so ( 7 ) P F a n o P_{\mathrm{Fano}} P Fano Fano predictive channel : P F a n o ( Γ ) = 1 3 ∑ p Π p Γ Π p P_{\mathrm{Fano}}(\Gamma) = \tfrac{1}{3}\sum_p \Pi_p \Gamma \Pi_p P Fano ( Γ ) = 3 1 ∑ p Π p Γ Π p Π p \Pi_p Π p Projector onto the 3-dimensional subspace of Fano line p p p p = 1 , … , 7 p = 1, \ldots, 7 p = 1 , … , 7 α ∗ \alpha^* α ∗ Optimal self-modelling parameter: α ∗ = argmin F [ P α ; Γ ] \alpha^* = \operatorname{argmin} F[P_\alpha;\, \Gamma] α ∗ = argmin F [ P α ; Γ ] T e f f T_{\mathrm{eff}} T eff Effective Gap temperature : T e f f = ( Γ 2 / κ 0 ) ⋅ k B ⋅ T p h y s T_{\mathrm{eff}} = (\Gamma_2 / \kappa_0) \cdot k_B \cdot T_{\mathrm{phys}} T eff = ( Γ 2 / κ 0 ) ⋅ k B ⋅ T phys ξ F \xi_F ξ F Fano correlation length: ξ F ∼ 160 pc \xi_F \sim 160\;\text{pc} ξ F ∼ 160 pc Θ M \Theta_M Θ M Winding theta-function with Fano character Z Φ ( s ) Z_\Phi(s) Z Φ ( s ) Epstein zeta-function with Fano character B ( b ) B^{(b)} B ( b ) Bilinear form on ( S 1 ) 21 (S^1)^{21} ( S 1 ) 21 N F N_F N F Number of uncorrelated Fano modes: N F ∼ 6,8 × 10 23 N_F \sim 6{,}8 \times 10^{23} N F ∼ 6 , 8 × 1 0 23 r = κ / Γ 2 r = \kappa / \Gamma_2 r = κ / Γ 2 Dimensionless viability parameter — ratio of regeneration rate to decoherence rate t = T e f f / T c t = T_{\mathrm{eff}} / T_c t = T eff / T c Dimensionless temperature — reduced to the critical value T c T_c T c
Spectral Geometry and Bimodular Construction
Symbols related to the bimodular construction of SM representations (T-178–T-181).
Notation Meaning H F H_F H F Finite Hilbert space of the spectral triple as an ( A int , A int ∘ ) (A_{\text{int}}, A_{\text{int}}^\circ) ( A int , A int ∘ ) KO-dim KO-dimension = 6 = 6 = 6 J J J J 2 = 1 J^2 = 1 J 2 = 1 J D = D J JD = DJ J D = D J J γ = − γ J J\gamma = -\gamma J J γ = − γ J D int D_{\text{int}} D int Dirac operator of the internal space; its eigenvalues determine the fermion mass ratios [T-180 [T]]
Related documents:
Glossary — definitions of termsMathematical apparatus — formal definitionsComputational implementation — Python codeCoherence matrix — definition of Γ \Gamma Γ Evolution — equation d Γ ( τ ) / d τ d\Gamma(\tau)/d\tau d Γ ( τ ) / d τ Emergent time — derivation of τ from the structure of ΓViability — measure P P P P crit P_{\text{crit}} P crit Self-observation — measures R R R Φ \Phi Φ D diff D_{\text{diff}} D diff C C C Interiority hierarchy — levels L0→L1→L2→L3→L4Categorical formalism — functor F F F E x p ∞ \mathbf{Exp}_\infty Exp ∞ Formalisation of operator φ — CPTP channelsStructural derivation via octonions — P1+P2 → O \mathbb{O} O Gap dynamics — Gap operator G ^ \hat{G} G ^ Gap thermodynamics — T e f f T_{\mathrm{eff}} T eff Fano selection rules — P F a n o P_{\mathrm{Fano}} P Fano Π p \Pi_p Π p Bimodular construction — SM representations from bimodules of the spectral triple (T-178–T-181)