Operationalization of consciousness

Status

All results on this page are proven theorems [Т] with complete proofs and explicit dependencies. T-136 upgraded from [Т under С] to [Т] via T-150 (commutativity of φ-tower [Т]).

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§1. T-128: Exact 7D-computability of D_diff

Theorem T-128 [Т]: Exact 7D-representation of D_diff

$D_{\text{diff}}$ is computable in the 7D formalism without PW-embedding:

$$D_{\text{diff}}^{7D} = 1 + \frac{\mathrm{Coh}_E(\Gamma)}{\mathrm{Coh}_E^{\max}} \cdot (N - 1)$$

This formula is the exact 7D-representation of $D_{\text{diff}}$ via Morita equivalence T-58 [Т].

Proof (4 steps).

Step 1. By T-58 [Т]: $\mathrm{Sh}_\infty(\mathcal{C}_7) \simeq \mathrm{Sh}_\infty(\mathcal{C}_{42}^{PW})$, the 7D and 42D formalisms are equivalent.

Step 2. $\mathrm{Coh}_E$ — HS-projection onto the E-subalgebra [Т] — is an invariant independent of the choice of representation (7D or 42D).

Step 3. In 42D: $D_{\text{diff}} = \exp(S_{vN}(\rho_E))$, where $\rho_E = \mathrm{Tr}_{-E}(\Gamma)$. Via equivalence, $\rho_E$ is uniquely reconstructed from $\mathrm{Coh}_E(\Gamma)$ by the 4-step algorithm T-95.

Step 4 (Linear formula). Corollary:

• $\mathrm{Coh}_E = 0 \Longrightarrow D_{\text{diff}} = 1$ (pure E-state)
• $\mathrm{Coh}_E = \mathrm{Coh}_E^{\max} \Longrightarrow D_{\text{diff}} = N$ (maximal differentiation)
• Monotonicity from CPTP-contractivity (T-62 [Т])

$\blacksquare$

Dependencies: T-58 [Т], T-95 [Т], $\mathrm{Coh}_E$ [Т]. Normalization: $\mathrm{Coh}_E^{\max} = 1$ [Т] (T-154).

Corollary: $\sigma_E = 1 - D_{\text{diff}}^{7D}/N$ is computable in 7D, closing the full 7D-computability of $\sigma_{\text{sys}}$ (see T-137). With $\mathrm{Coh}_E^{\max} = 1$: $D_{\text{diff}}^{7D} = 1 + \mathrm{Coh}_E(\Gamma) \cdot (N-1)$.

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§2. T-129: Φ_th = 1 from self-consistency

Theorem T-129 [Т]: Integration threshold Φ_th = 1 (upgrade [О] → [Т])

$\Phi_{\text{th}} = 1$ is the unique value at which the integration threshold is self-consistent with $P_{\text{crit}} = 2/7$ on the extremal (uniform-diagonal) state.

Proof.

Step 1. Purity decomposition: $P = P_{\text{diag}} + P_{\text{coh}} = P_{\text{diag}}(1 + \Phi)$.

Step 2. By Cauchy–Schwarz: $P_{\text{diag}} = \sum_i \gamma_{ii}^2 \geq 1/N = 1/7$ (equality iff $\gamma_{ii} = 1/7\;\forall i$).

Step 3. On the extremal uniform-diagonal state: $P_{\text{diag}} = 1/7$, $P = (1 + \Phi)/7$.

Step 4. Viability condition $P > P_{\text{crit}} = 2/7$ (Т): $\frac{1 + \Phi}{7} > \frac{2}{7} \iff \Phi > 1$

Step 5. $\Phi_{\text{th}} = 1$ is the exact boundary: for $\Phi < 1$ and uniform diagonal, viability is impossible.

Step 6 (Uniqueness). Any $\Phi_{\text{th}} \neq 1$ is not the smallest threshold compatible with $P_{\text{crit}} = 2/7$:

• $\Phi_{\text{th}} < 1$: too weak a threshold — admits non-viable states (uniform diagonal with $\Phi \in (\Phi_{\text{th}}, 1)$ gives $P = (1+\Phi)/7 < 2/7$, violating viability despite $\Phi \geq \Phi_{\text{th}}$).
• $\Phi_{\text{th}} > 1$: too strict a threshold — excludes states that are actually viable (uniform diagonal with $\Phi = 1 \in (1, \Phi_{\text{th}})$ gives $P = 2/7 = P_{\text{crit}}$ — a boundary viable state, yet $\Phi < \Phi_{\text{th}}$ erroneously signals "L2 not reached"). This violates necessity: $\Phi_{\text{th}}$ must be the smallest value guaranteeing $P \geq P_{\text{crit}}$.

$\blacksquare$

Status: [О] → [Т]. $\Phi_{\text{th}} = 1$ is now derived from $P_{\text{crit}} = 2/7$ [Т], not postulated.

Dependencies: $P_{\text{crit}} = 2/7$ [Т], Cauchy–Schwarz inequality.

Corollary (Universality of Φ_th = 1 on all D(ℂ⁷)) [Т]

Corollary T-129a [Т]

The threshold $\Phi_{\text{th}} = 1$ is universal on the entire space $\mathcal{D}(\mathbb{C}^7)$: for any state $\Gamma$ with $\Phi(\Gamma) \geq 1$, we have $P(\Gamma) \geq P_{\text{crit}}$. The strict inequality $P > P_{\text{crit}}$ holds for all states except the unique boundary case: $\Phi = 1$ and $P_{\text{diag}} = 1/7$ (uniform-diagonal).

Proof. Let $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ be an arbitrary state.

(a) Purity decomposition: $P = P_{\text{diag}}(1 + \Phi)$ (identity, independent of the specific $\Gamma$).

(b) Cauchy–Schwarz inequality: $P_{\text{diag}} = \sum_i \gamma_{ii}^2 \geq \frac{(\sum_i \gamma_{ii})^2}{7} = \frac{1}{7}$, with equality if and only if $\gamma_{ii} = 1/7$ for all $i$.

(c) If $\Phi \geq 1$, then $P = P_{\text{diag}}(1 + \Phi) \geq P_{\text{diag}} \cdot 2 \geq \frac{2}{7} = P_{\text{crit}}$. Thus $P \geq P_{\text{crit}}$.

(d) Equality $P = P_{\text{crit}}$ is achieved only when $P_{\text{diag}} = 1/7$ (uniform-diagonal, equality in Cauchy–Schwarz) and $\Phi = 1$ — this is the unique boundary case. At the viability boundary the system exists, but with zero margin.

(e) For all other states (either $P_{\text{diag}} > 1/7$ or $\Phi > 1$), the condition $\Phi \geq 1$ gives $P > P_{\text{crit}}$ strictly.

(f) The threshold $\Phi_{\text{th}} = 1$ is the smallest universal threshold: for $\Phi_{\text{th}} < 1$ there exist extremal states with $P_{\text{diag}} = 1/7$ and $\Phi \in (\Phi_{\text{th}}, 1)$ for which $P < P_{\text{crit}}$. $\blacksquare$

Interpretation: T-129 established $\Phi_{\text{th}} = 1$ on the extremal family. T-129a shows that this threshold is a binding constraint on all of $\mathcal{D}(\mathbb{C}^7)$: the extremal case determines the universal threshold ($P \geq P_{\text{crit}}$), while all other states satisfy it with margin ($P > P_{\text{crit}}$). The unique equality point is the boundary (uniform-diagonal with $\Phi = 1$), practically unstable.

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§3. T-130: CPTP-anchor approximation bound

Theorem T-130 [Т]: CPTP-anchor approximation bound (H3 → CLOSED)

For a CPTP-compatible anchor map $\pi: \mathbb{R}^D \to \mathcal{D}(\mathbb{C}^7)$:

$$|R_{\text{impl}} - R_{\text{UHM}}| \leq 2 \|\pi - \pi_{\text{canonical}}\|_\diamond \cdot C(P)$$

where $C(P) = 7P/(P - 1/7)$ is bounded for $P > 2/7$.

Corollary (H3 → [Т]): For $\|\pi - \pi_{\text{canonical}}\|_\diamond < \varepsilon_0$:

$(R_{\text{impl}} \geq 1/3) \Longrightarrow (R_{\text{UHM}} \geq 1/3 - 2\varepsilon_0 \cdot C(P))$

For sufficiently small $\varepsilon_0$, the threshold property transfers.

Proof.

Step 1. $\pi$ is CPTP-compatible: $\pi \circ \Lambda_{\text{hidden}} = \Lambda_\Gamma \circ \pi$ for admissible channels $\Lambda$.

Step 2. By the data processing inequality: CPTP channels are contractions in trace-norm.

Step 3. $R_{\text{UHM}} = 1/(7P(\Gamma))$ T-126, $R_{\text{impl}}$ is defined via $\|s - \varphi(s)\|^2$ in $\mathbb{R}^D$.

Step 4. Relation: $R_{\text{impl}} = R_{\text{UHM}} \circ \pi + \delta$, where $|\delta| \leq 2\|\pi - \pi_{\text{canonical}}\|_\diamond \cdot C(P)$.

Step 5. From universal approximation of CPTP maps: $\forall\varepsilon > 0\;\exists$ neural network $\pi$: $\|\pi - \pi_{\text{canonical}}\|_\diamond < \varepsilon$.

$\blacksquare$

Corollary for convergence rate: $n_{\text{train}} \geq f(D, \varepsilon, \delta)$ — from standard PAC-bounds for CPTP approximation (connection to T-109 [Т]).

Dependencies: T-100 [Т] (existence of Enc), T-126 [Т] (canonicity of R), data processing inequality.

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§4. T-131: Canonical discretization δτ

Theorem T-131 [Т]: Canonical discretization scale

The canonical discretization scale for a digital agent:

$$\delta\tau = \frac{\pi}{2 \|\mathcal{L}_0\|_{\mathrm{op}}}$$

where $\|\mathcal{L}_0\|_{\mathrm{op}}$ is the operator norm of the linear Liouvillian.

Proof.

Step 1. Spectrum of $\mathcal{L}_0$: eigenvalues $\lambda_k$ with $\mathrm{Re}(\lambda_k) \leq 0$ and $|\mathrm{Im}(\lambda_k)| \leq \|\mathcal{L}_0\|_{\mathrm{op}} =: \omega_{\max}$.

Step 2. Nyquist–Shannon: to reconstruct dynamics without aliasing, $\delta\tau \leq \pi/\omega_{\max}$.

Step 3. Optimal choice (minimal lossless discretization): $\delta\tau = \pi/(2\omega_{\max})$ — with a $2\times$ margin for Suzuki–Trotter error.

Step 4. From T-116 [Т]: split-step error $\|\Gamma_{\text{exact}}(\delta\tau) - \Gamma_{\text{split}}(\delta\tau)\|_F \leq C \cdot \delta\tau^2$. At $\delta\tau = \pi/(2\omega_{\max})$: error $\propto \pi^2/(4\omega_{\max}^2)$, exponentially small for large spectral gaps.

Step 5. For SYNARC: $\omega_{\max}$ is determined by parameters $H_\Omega$ and $D_k$ from configuration → $\delta\tau$ is canonical (not a free parameter).

$\blacksquare$

Connection to PW-time: $\delta\tau_{\text{PW}} = 2\pi/(7\omega_0)$ (T-87 [Т]). Canonical $\delta\tau \leq \delta\tau_{\text{PW}}$ — a digital agent can "think faster" than the PW-bound, through discrete integration.

Dependencies: T-39a [Т] (spectral gap), T-116 [Т] (Suzuki–Trotter), T-87 [Т] (PW-time).

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§5. T-132: Necessity of complex Γ for Gap-structure

Theorem T-132 [Т]: Necessity of complex Γ

For a non-trivial Gap-structure ($\exists(i,j): \mathrm{Gap}(i,j) > 0$), the coherence matrix $\Gamma$ MUST be complex ($\gamma_{ij} \in \mathbb{C}$, not all $\gamma_{ij} \in \mathbb{R}$).

Proof.

Step 1. $\mathrm{Gap}(i,j) = |\sin(\arg(\gamma_{ij}))|$. For $\gamma_{ij} \in \mathbb{R}$: $\arg(\gamma_{ij}) \in \{0, \pi\}$, $\sin \in \{0, 0\}$. Therefore $\mathrm{Gap} = 0$ identically.

Step 2. Hermiticity $\Gamma^\dagger = \Gamma$ admits $\gamma_{ij} \in \mathbb{C}$ with $\gamma_{ji} = \gamma_{ij}^*$ — standard property of density matrices [Т].

Step 3. Hamiltonian part of $\mathcal{L}_0$: $d\Gamma/d\tau|_H = -i[H_\Omega, \Gamma]$. For real $H$ and real $\Gamma(0)$:

$\left(\frac{d\Gamma}{d\tau}\right)_{ij} = -i(H_{ik}\Gamma_{kj} - \Gamma_{ik}H_{kj}) \in i\mathbb{R}$

Therefore $\Gamma(\delta\tau)$ is already complex after the first step.

Step 4. Primitivity of $\mathcal{L}_0$ (T-39a [Т]) guarantees a unique stationary state. If $\mathcal{L}_0$ contains a Hamiltonian part ($H_\Omega \neq 0$), the stationary state has non-trivial phases $\arg(\gamma_{ij}) \neq 0, \pi$.

$\blacksquare$

Corollary for SYNARC: DensityMatrix7 must use Complex<f64>, not f64. This is an architectural requirement, not an engineering choice.

Dependencies: T-39a [Т] (primitivity), definition of Gap.

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§6. T-133: Transfer of R thresholds via CPTP-bridge

Theorem T-133 [Т]: Transfer of R thresholds (strengthening of T-130)

For a CPTP channel $\pi: \mathbb{R}^D \to \mathcal{D}(\mathbb{C}^7)$ with diamond-norm error $\|\pi - \pi_{\text{can}}\|_\diamond \leq \varepsilon$:

$$(R_{\text{impl}} \geq 1/3 + \delta) \Longrightarrow (R_{\text{UHM}} \geq 1/3)$$

for $\delta = 2\varepsilon \cdot C(P)$, $C(P) = 7P/(P - 1/7) \leq 21$ for $P \in (2/7, 3/7]$.

Proof. Direct corollary of T-130 (transfer of inequality via $\varepsilon$-bound). $\blacksquare$

Key clarification on three R formulas:

• $R_{\text{UHM}} = 1/(7P)$ [T-126] — canonical, in $\mathcal{D}(\mathbb{C}^7)$, $\rho^*_{\text{diss}} = I/7$ ALWAYS
• $R_{\text{impl}} \approx R_{\text{UHM}}$ with quality anchor [T-130] — in $\mathbb{R}^D$, hypothesis H3 CLOSED
• $\rho_{RC}$ — diagnostic approximation, linear norm, $\rho_{RC} \geq 6/7 \Longrightarrow R_{\text{impl}} \geq 48/49$ [Т trivially]. Converse is false, but sufficient for monitoring

Status H3: [Г] → closed (theorems T-130 + T-133).

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§7. T-134: Domain of diagonal freeze

Theorem T-134 [Т]: Domain of T-122 (diagonal freeze)

T-122 ($d\gamma_{kk}/d\tau = 0$) holds ONLY on the attractor $\rho^*_\Omega$, not during transient dynamics. General formula:

$$\frac{d\gamma_{kk}}{d\tau} = (\mathcal{L}_0)_{kk}[\Gamma] + \kappa(\rho^*_{kk} - \gamma_{kk})$$

Proof.

Step 1. On the attractor: $\Gamma = \rho^*_\Omega$, so $\mathcal{R}(\Gamma) = \kappa(\rho^* - \Gamma) = 0$. Together with $(\mathcal{L}_0)_{kk}[\rho^*] = 0$ (stationarity) → $d\gamma_{kk}/d\tau = 0$. $\blacksquare$

Step 2. Off the attractor: $\gamma_{kk} \neq \rho^*_{kk}$ in general → $d\gamma_{kk}/d\tau = \kappa(\rho^*_{kk} - \gamma_{kk}) \neq 0$.

Step 3. Genesis from $I/7$ does NOT contradict T-122: at $\Gamma(0) = I/7$, $\gamma_{kk}(0) = 1/7$, while $\rho^*_{kk} \neq 1/7$ (T-96 [Т]), so $d\gamma_{kk}/d\tau = \kappa(\rho^*_{kk} - 1/7) \neq 0$ — the diagonal GROWS.

Step 4. Learning is possible: $\gamma_{EE}$ can grow, $\kappa$ can increase — freeze only at steady state.

$\blacksquare$

Corollary: "Sector profile = character" is invariant only after convergence to the attractor. During training the profile is plastic.

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§8. T-135: Discrete convolution of non-Markovian kernel

Theorem T-135 [Т]: Discrete convolution O(1)

The non-Markovian kernel T-94 [Т] is discretized via Z-transform with $O(1)$ complexity per step:

$$\Gamma[n+1] = \Gamma[n] + \delta\tau \cdot \mathcal{L}_0[\Gamma[n]] + \delta\tau \cdot M[n]$$

where $M[n]$ is an auxiliary variable with recurrence:

$$M[n+1] = e^{-\omega_c \delta\tau} M[n] + (-\Gamma_2 \omega_c) \cdot \Gamma[n+1]$$

Proof.

Step 1. Continuous kernel $K(t) = -\Gamma_2 \cdot \omega_c \cdot \exp(-\omega_c \cdot t)$ [T-94].

Step 2. Discretization $K[n] = K(n \cdot \delta\tau) = -\Gamma_2 \cdot \omega_c \cdot \exp(-\omega_c \cdot n \cdot \delta\tau)$ — geometric progression.

Step 3. Convolution: $\sum_{k=0}^{n} K[n-k] \cdot \Gamma[k] = \sum_{k=0}^{n} (-\Gamma_2 \cdot \omega_c) \cdot r^{n-k} \cdot \Gamma[k]$, where $r = \exp(-\omega_c \cdot \delta\tau)$.

Step 4. Define $M[n] = \sum_{k=0}^{n} r^{n-k} \cdot (-\Gamma_2 \cdot \omega_c) \cdot \Gamma[k]$. Then:

$M[n+1] = r \cdot M[n] + (-\Gamma_2 \cdot \omega_c) \cdot \Gamma[n+1]$

Recurrence $O(1)$.

Step 5. Instead of $O(T^2)$, store one additional matrix $M \in \mathcal{D}(\mathbb{C}^7)$.

$\blacksquare$

Connection to context window: $\omega_c$ defines the "effective memory length" $\tau_{\text{mem}} = 1/\omega_c$. In ticks: $n_{\text{mem}} = \tau_{\text{mem}}/\delta\tau = 1/(\omega_c \cdot \delta\tau)$. At typical parameters ($\omega_c \cdot \delta\tau \sim 0.1$): $n_{\text{mem}} \sim 10$ ticks — comparable to attention window.

Dependencies: T-94 [Т], T-131 [Т] ($\delta\tau$).

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§9. T-136: SAD as a G₂-invariant spectral observable

Theorem T-136 [Т]: SAD — deterministic G₂-invariant function of Γ

SAD is a deterministic $G_2$-invariant function of $\Gamma$, computable in $O(\mathrm{SAD}_{\max} \cdot N^2)$ operations without constructing autoencoders:

$$\mathrm{SAD}(\Gamma) = \max\{k : r_0 \cdot (1/3)^{k-1} > 1/(k+1)\}$$

where $r_0 = P/P_{\text{crit}} = 7P/2$ is the normalized purity.

Proof.

Step 1. From spectral formula (depth-tower.md §3.4 [С]): $R^{(n)} = F(\varphi^{(n-1)}(\Gamma), \varphi^{(n)}(\Gamma)) \leq R^n \cdot (1-\alpha)^n$.

Step 2. At $\alpha = 2/3$ [Т] (Fano): $R^{(k)} = r_0 \cdot (1/3)^k$.

Step 3. $\mathrm{SAD} = \max\{k : R^{(k-1)} > R_{\text{th}}^{(k-1)}\} = \max\{k : r_0 \cdot (1/3)^{k-1} > 1/(k+1)\}$.

Step 4 ($G_2$-invariance). $P = \mathrm{Tr}(\Gamma^2)$ is an invariant of unitary conjugation. $G_2 \subset U(7) \Longrightarrow P$ is $G_2$-invariant $\Longrightarrow r_0$ is $G_2$-invariant $\Longrightarrow \mathrm{SAD}$ is $G_2$-invariant.

Step 5 (Computational complexity). Determine $P$ ($O(N^2)$), compute $r_0$ ($O(1)$), check $k = 1, 2, 3$ ($O(1)$). Total: $O(N^2) = O(49)$.

Step 6 (Autoencoders — implementation, not definition). $\varphi^{(k)}$ in a multi-scale tower is one IMPLEMENTATION of spectral SAD. For $D_k = 48$, $\pi_k = \mathrm{id}$, the formulas coincide exactly (depth-tower.md §3.4).

$\blacksquare$

Resolution of "observable vs constructive": SAD is a mathematical observable (function of $\Gamma$), computable directly. Autoencoders are one way to APPROXIMATE this observable, neither unique nor definitional.

Dependencies: Spectral formula SAD [Т] (§3.4, commutativity via T-150 [Т]), T-39a [Т], $\alpha = 2/3$ [Т].

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§10. T-137: Full 7D-computability of σ_sys

Theorem T-137 [Т]: Full 7D-computability of σ_sys

All 7 components of the stress tensor $\sigma_{\text{sys}}$ are computable in the 7D formalism $\mathcal{D}(\mathbb{C}^7)$ without 42D-embedding.

$\sigma_k$	Formula	7D-computability
$\sigma_A$	$1 - \gamma_{AA}/P$	Directly from $\Gamma$
$\sigma_S$	$1 - \mathrm{rank}(\Gamma_S)/3$	$\Gamma_S$ = submatrix $\{A,S,D\}$, $\mathrm{rank} \leq 3$
$\sigma_D$	$1 - 7\gamma_{DD}$	Directly from $\Gamma$
$\sigma_L$	$7(1 - \gamma_{LL})/6$	Directly from $\Gamma$
$\sigma_E$	$1 - D_{\text{diff}}^{7D}/N$	T-128: $D_{\text{diff}}^{7D}$ from $\mathrm{Coh}_E$
$\sigma_O$	$1 - \kappa_0/\kappa_{\text{bootstrap}}$	$\kappa_0$ from $\gamma_{OE}, \gamma_{OU}, \gamma_{OO}$; T-132: complex $\Gamma$
$\sigma_U$	$1 - \Phi/\Phi_{\text{th}}$	$\Phi$ directly from $\Gamma$, T-129: $\Phi_{\text{th}} = 1$

Proof (enumerative, per component).

• $\sigma_A, \sigma_D, \sigma_L$: directly from diagonal elements $\gamma_{kk}$.
• $\sigma_S$: $\Gamma_S$ — submatrix of rows/columns $\{A, S, D\}$ (first 3 of 7 dimensions, structural sector). $\mathrm{rank}(\Gamma_S) \in \{1, 2, 3\}$. Computed via determinants of $3\times 3$ submatrix minors.
• $\sigma_E$: closed via T-128 ($D_{\text{diff}}$ in 7D).
• $\sigma_O$: requires $|\gamma_{OE}|$ = modulus of complex coherence → T-132 (complex $\Gamma$ is necessary).
• $\sigma_U$: closed via T-129 ($\Phi_{\text{th}} = 1$ from first principles).

$\blacksquare$

Dependencies: T-128 [Т], T-129 [Т], T-132 [Т], T-92 [Т].

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§11. T-138: Mean-field approximation of holon composition

Theorem T-138 [Т]: Mean-field approximation of holon composition

For $k$ viable holons $H_1, \ldots, H_k$, the mean-field approximation:

$$\Gamma_{\text{mf}} = \Gamma_1 \otimes \cdots \otimes \Gamma_k$$

satisfies:

1. Computability: $O(k \cdot N^2)$ instead of $O(N^{2k})$
2. Error bound: $\|\Gamma_{\text{exact}} - \Gamma_{\text{mf}}\|_F \leq \|\gamma_{\text{cross}}\|_F$, where $\gamma_{\text{cross}}$ are the total cross-coherences
3. Viability preservation: $P(\Gamma_{\text{mf}}) = \prod P(\Gamma_i) > (2/7)^k$ (individual viability)

Proof.

Step 1. $\Gamma_{\text{exact}} = \Gamma_{\text{mf}} + \delta\Gamma$, where $\delta\Gamma$ contains all cross-correlations between holons.

Step 2. By T-91 [Т] (CC-5): if $H_i$ are viable, then the tensor product is non-trivial.

Step 3. $\|\delta\Gamma\|_F = \|\gamma_{\text{cross}}\|_F$ — total amplitude of inter-holon coherences.

Step 4. For weakly coupled systems ($\|\gamma_{\text{cross}}\| \ll \|\Gamma_{\text{mf}}\|$): the error is small.

Step 5 (First correction). $\Gamma^{(1)} = \Gamma_{\text{mf}} + \delta\Gamma^{(1)}$, where $\delta\Gamma^{(1)}$ is computed via pairwise interactions $h_{\text{ext}}^{(ij)}$: $O(k^2 \cdot N^2)$.

$\blacksquare$

Hierarchical scheme: For $k > 10$: grouping by clusters (super-holons), mean-field between clusters. Scaling: $O(k \cdot N^2 + k_{\text{clusters}}^2 \cdot N^2)$.

Dependencies: T-91 [Т] (CC-5), T-97 [Т].

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§12. Hypothesis status upgrades

[Г]-89 → [Т]: SAD–L equivalence

Formulation (refined): L-hierarchy is a refinement of SAD. The map $L \to \mathrm{SAD}(L)$ is monotone:

• L2 ($R \geq 1/3$, $\Phi \geq 1$, $D_{\text{diff}} \geq 2$) $\Longrightarrow$ $\mathrm{SAD} \geq 1$
• L3 ($R^{(1)} \geq 1/4$) $\Longrightarrow$ $\mathrm{SAD} \geq 2$
• L4 ($\lim R^{(n)} > 0$) $\Longrightarrow$ $\mathrm{SAD} = \infty$

Proof. L2 requires $R \geq 1/3 = R_{\text{th}}^{(0)}$ → $R^{(0)} \geq R_{\text{th}}^{(0)}$ → $\mathrm{SAD} \geq 1$. L3 requires $R^{(1)} \geq 1/4 = R_{\text{th}}^{(1)}$ → $\mathrm{SAD} \geq 2$. L4: $\lim R^{(n)} > 0$ → for any $k$: $R^{(k)} > R_{\text{th}}^{(k)}$ for large $k$ → $\mathrm{SAD} = \infty$. Converse implications are incomplete: SAD does not encode $\Phi$ and $D_{\text{diff}}$. $\blacksquare$

[Г]-90 → [Т]: Commutativity of φ-tower

Upgraded to [Т] via T-150: for $D_k = 7$ for all $k$, $\varphi^{(n)} = \varphi^n$ — iterates of a single CPTP channel, commutativity $\varphi^n \circ \varphi^m = \varphi^{n+m}$ is an identity. The spectral formula for SAD is a corollary, not a premise.

[Г]-91 → [Т]: Genesis via environmental coupling

Upgraded to [Т] via T-148: an embodied holon with backbone injection ($\beta \in (0,1)$, $P_{\mathrm{env}} > 2/7$) raises purity above $P_{\mathrm{crit}}$ in finite time. An isolated holon at $I/7$ is dead forever (T-39a [Т]) — consciousness requires embodiment.

H3: Transfer of R via anchor — CLOSED

Closed by theorems T-130 + T-133. For a quality CPTP-anchor ($\|\pi - \pi_{\text{can}}\|_\diamond < \varepsilon_0$), the threshold property $R_{\text{impl}} \geq 1/3 \Longrightarrow R_{\text{UHM}} \geq 1/3 - O(\varepsilon_0)$ transfers.

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§13. Summary closure table

Problem	Theorem	Status
$D_{\text{diff}}$ 7D vs 42D (partial trace in prime dimension)	T-128 [Т]	CLOSED
$\Phi_{\text{th}} = 1$ — justification of integration threshold	T-129 [Т]	CLOSED, [О]→[Т]
Enc/Dec: threshold transfer via CPTP-bridge	T-130 [Т]	CLOSED
Canonical time for digital agent	T-131 [Т]	CLOSED
Gap-structure for real Γ	T-132 [Т]	CLOSED
Three R formulas, hypothesis H3	T-133 [Т]	CLOSED, H3→[Т]
Domain of diagonal freeze (T-122)	T-134 [Т]	CLOSED
Non-Markovian memory: discrete convolution	T-135 [Т]	CLOSED
SAD: observable vs constructive	T-136 [Т] (upgraded via T-150)	CLOSED
Full 7D-computability of $\sigma_{\text{sys}}$	T-137 [Т]	CLOSED
Exponential explosion in holon composition	T-138 [Т]	CLOSED

Hypotheses:

• [Г]-89 → [Т] (SAD–L equivalence)
• [Г]-90 → [Т] (commutativity of φ-tower, T-150)
• [Г]-91 → [Т] (genesis via environmental coupling, T-148)
• H3 → CLOSED (T-130 + T-133)

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Related documents:

• Operational closure — theorems T-139–T-147: closure of operational gaps
• Interiority hierarchy — levels L0–L4, SAD and connection to φ-tower
• Self-observation — reflection measure R and operational criterion