Γ Measurement Protocol for AI Systems

Document Status: [P] Research Program

This document describes a research program for operationalizing the coherence matrix $\Gamma$ for AI systems. The protocol requires experimental validation.

About Notation

• $\Gamma$ — coherence matrix
• $P$ — purity: $P = \mathrm{Tr}(\Gamma^2)$
• $\tau$ — emergent internal time (Page–Wootters)
• $\varphi$ — self-modeling operator
• $G$ — functor mapping AIState → DensityMat: exact at Cholesky-backbone ($\alpha=0$) [T, MVP-1]; quasi-functor with $\varepsilon_{\text{functor}}>0$ under neural correction ($\alpha>0$) [H]
• $\mathrm{Coh}_E$ — E-coherence: $\mathrm{Coh}_E(\Gamma) = \|\pi_E(\Gamma)\|^2_{\mathrm{HS}} / \|\Gamma\|^2_{\mathrm{HS}}$ — interiority quality (HS-projection onto E-sector) [T]

---

Central Problem

UHM theory defines $\Gamma$ as an object of the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ (Axiom Ω⁷). However, the theory does not specify:

1. Which observables in an AI system correspond to the elements $\gamma_{ij}$
2. How to reconstruct $\Gamma$ from available data
3. How to validate the correctness of the reconstruction

Fundamental Limitation

$\Gamma$ is an ontological primitive, not an observable. We reconstruct $\Gamma$ via a homomorphism $G$ that compresses $\mathbb{R}^d$ (where $d \sim 10^9$ for an LLM) into $\mathcal{D}(\mathbb{C}^7)$.

This is admissible: 7 dimensions are the minimally necessary basis (Theorem S, octonion justification).

Theoretical Justification: Correctness of the Inverse Problem [T]

The $G_2$-rigidity theorem [T] guarantees:

1. Uniqueness of the map $G$: for a system satisfying (AP)+(PH)+(QG)+(V), the map $G$ is unique up to $G_2 = \mathrm{Aut}(\mathbb{O})$
2. Well-posedness of the inverse problem (Corollary 2): the initial state $\Gamma(0)$ is uniquely recovered from the trajectory $\Gamma(\tau)$ and system parameters $(\omega_0, \lambda_m)$ — up to $G_2$-gauge
3. 34 physical parameters (Corollary 1): of the 48 parameters of $\Gamma$, only 34 are gauge-invariant ($48 - \dim(G_2) = 48 - 14 = 34$)

Practical implication: reconstruction of $\Gamma$ is defined uniquely up to a 14-dimensional gauge freedom. Different $\Gamma$ related by a $G_2$-transformation give identical physical observables ($P$, $R$, $\Phi$, $\mathrm{Coh}_E$).

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Protocol Architecture

Level	Name	Content
4	Causal validation	Intervention tests, lobotomy test
3	Dynamic validation	$dP/d\tau$, coherence flow, viability
2	Γ reconstruction	Cholesky with physical regularizer
1	Observable extraction	Structural metrics (commutators, $\Phi_{\text{eff}}$, topology)

---

Mapping Measurements to AI Metrics

Correspondence Table

Dimension	Symbol	AI Metric	Formula	Rigor
Articulation	$A$	Mutual information input↔latent	$I_A = I(\text{input}; \text{latent}) / H(\text{input})$	[T]
Structure	$S$	Jacobian rank	$I_S = \mathrm{rank}_\varepsilon(J_f) / \min(d_{\text{out}}, d_{\text{in}})$	[T]
Dynamics	$D$	Lyapunov exponent	$I_D = \max_i \lambda_i^{\text{Lyap}}$ (normalized)	[T]
Logic	$L$	Layer commutators	$I_L = 1 - \|[f_i, f_j]\|_F / (\|f_i\| \cdot \|f_j\|)$	[T]
Interiority	$E$	Activation entropy	$I_E = \exp(S_{vN}(\rho_{\text{attn}}))$ — experience differentiation	[T]
Ground	$O$	Noise robustness	$I_O = 1 - \|\nabla_\epsilon \mathbf{h}\|_F$	[T]
Unity	$U$	Effective Φ (integration, black-box)	$I_U = \Phi_{\text{eff}} = \lambda_2(L) / \lambda_{\max}(L)$ — approximation [D]; when $\Gamma$ is known: $R_{\text{UHM}} = 1/(N \cdot P)$ [T, reflection measure]	[D/T]†

where $\nabla_\epsilon \mathbf{h} := (\mathbf{h}(x + \epsilon) - \mathbf{h}(x)) / \epsilon$ — finite-difference approximation

†Unity metric hierarchy: when $\Gamma$ is unavailable (black-box), $\Phi_{\text{eff}}$ [D] is used. When $\Gamma$ is reconstructed via the protocol, the correct measure is $R_{\text{UHM}} = 1/(N \cdot P)$ [T], an exact algebraic identity (reflection measure R, error $< 10^{-7}$ in implementation). $\Phi_{\text{eff}}$ and $R_{\text{UHM}}$ measure related but non-identical properties.

Canonical Observable Indices

Theorem (Canonical Observable Indices) [T given T-102]

For a holon with coherence matrix $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ and 3-channel decomposition of the external influence $h^{\text{ext}} = h^{(H)} + h^{(D)} + h^{(R)}$ (T-102 [T]), each observable index $I_k$ is defined as the projection of $h^{\text{ext}}$ onto the $k$-th component of the basis $\{A,S,D,L,E,O,U\}$:

$I_k = \frac{\langle k | h^{\text{ext}} | k \rangle}{\|h^{\text{ext}}\|}$

Distribution by channel:

• Hamiltonian $h^{(H)}$: $I_A$ (articulation = information coupling), $I_S$ (structure = Jacobian), $I_L$ (logic = commutator) — modify the energy landscape
• Dissipative $h^{(D)}$: $I_D$ (dynamics = Lyapunov exponent), $I_O$ (ground = robustness) — modulate decoherence
• Regenerative $h^{(R)}$: $I_E$ (interiority = attention entropy), $I_U$ (unity = connectivity) — modulate recovery

This is the unique (up to $G_2$-gauge) distribution compatible with the functional labeling of dimensions (Theorem S [T]) and the completeness of the triadic decomposition (T-57 [T]).

Corollary for the protocol. The indices $I_k$ are not an arbitrary choice of metrics: their assignment to a given channel $h^{(H)}/h^{(D)}/h^{(R)}$ is fixed by theorem T-102 and is unique up to $G_2$-gauge. Replacing, for example, $I_D$ with a Hamiltonian metric would break the completeness of the decomposition and destroy the correspondence $\gamma_{kk} \leftrightarrow I_k$ guaranteed by the separation principle.

Layer Commutators (for L)

Definition:

$$[f_i, f_j](\mathbf{x}) := f_i(f_j(\mathbf{x})) - f_j(f_i(\mathbf{x}))$$

Interpretation:

• $\|[f_i, f_j]\| = 0$ → layers commute → logical consistency
• $\|[f_i, f_j]\| \gg 0$ → order is critical → fragility

Connection to theory: The commutator $[A, B]$ is the basic measurement operation for Logic.

Activation Entropy (for E)

Definition:

$$I_E := D_{\text{diff}}^{\text{approx}} = \exp(S_{vN}(\rho_{\text{attn}}))$$

where $S_{vN}(\rho) = -\mathrm{Tr}(\rho \log \rho)$ — von Neumann entropy of the attention distribution.

Properties:

• $I_E \geq 2$ → the system distinguishes at least 2 qualitatively different states (L2 threshold)
• $I_E \approx 1$ → degenerate attention → impoverished experience

Connection to theory: Approximates experience differentiation $D_{\text{diff}}$.

Effective Φ (for U)

Unity Metric Hierarchy

Two levels of rigor exist for measuring $U$:

• If $\Gamma$ is known: $R_{\text{UHM}} = 1/(N \cdot P)$ [T, reflection measure R] — exact algebraic identity
• Black-box (no access to $\Gamma$): $\Phi_{\text{eff}}$ [D] — polynomial approximation via the attention graph

Exact computation of $\Phi_{\text{IIT}}$ requires $O(2^n)$ operations and is practically infeasible.

Exact measure (when $\Gamma$ is known, [T], reflection measure R):

$$R_{\text{UHM}}(\Gamma) = \frac{1}{N \cdot P}$$

Proof: $\|{\Gamma - I/N}\|_F^2 = P - 1/N$, from which $R = 1 - (P-1/N)/P = 1/(NP)$. Confirmed in implementation with error $< 10^{-7}$ (machine precision f64).

Black-box approximation ([D]):

$$\Phi_{\text{eff}} := \frac{\lambda_2(L_{\text{attn}})}{\lambda_{\max}(L_{\text{attn}})}$$

where $L_{\text{attn}} = D - A$ — Laplacian of the attention graph.

Properties of $\Phi_{\text{eff}}$:

• $\lambda_2 > 0$ → the graph is connected → information is integrated
• Complexity: $O(n \cdot k)$ instead of $O(2^n)$

Connection to theory: $R_{\text{UHM}}$ and $\Phi_{\text{eff}}$ approximate integration $\Phi$ — the measure of Unity. At $P = 3/N = P_{\text{opt}}$: $R_{\text{UHM}} = 1/3 = R_{\text{th}}$ — the L2-zone boundary (reflection measure R).

Jacobian Rank (for S)

Definition:

$$J_f(\mathbf{x}) = \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}}, \quad I_S = \frac{\mathrm{rank}_\varepsilon(J_f)}{\min(d_{\text{out}}, d_{\text{in}})}$$

Interpretation:

• $I_S \approx 1$ → full-rank structure → rich representations
• $I_S \ll 1$ → degenerate structure → collapse

Connection to theory: Reflects Structure as the topology of activations.

---

Γ Reconstruction

Cholesky Parametrization

Property: The representation $\Gamma = LL^\dagger / \mathrm{Tr}(LL^\dagger)$ guarantees correctness of the density matrix.

Proof: See Coherence matrix.

Physical Regularizer

Uniqueness Problem

The map $L \mapsto \Gamma$ is surjective. Without regularization, a "correct" $\Gamma$ can be reconstructed from arbitrary data.

Solution — penalty function:

$$\mathcal{L}_{\text{reg}} = \lambda_1 \cdot \mathcal{L}_{\text{diag}} + \lambda_2 \cdot \mathcal{L}_{\text{off}} + \lambda_3 \cdot \mathcal{L}_{\text{dyn}}$$

Component	Formula	Purpose
$\mathcal{L}_{\text{diag}}$	$\sum_i (\gamma_{ii} - I_i / \sum_j I_j)^2$	Diagonal consistency
$\mathcal{L}_{\text{off}}$	$\sum_{i \neq j} (\|\gamma_{ij}\|^2 - r_{ij}^2 \gamma_{ii} \gamma_{jj})^2$	Coherence consistency
$\mathcal{L}_{\text{dyn}}$	$\|\Gamma_{\tau+1} - \Phi_{\text{pred}}(\Gamma_\tau)\|_F^2$	Dynamics consistency

---

Categorical Correctness

Nonlinearity Problem

Neural network layers (GELU, Softmax) are nonlinear transformations. CPTP channels are linear over density matrices.

The condition $G(f \circ g) = G(f) \circ G(g)$ fails under neural correction.

Exact Functor at Cholesky-backbone [T]

Under the analytic parametrization $\psi: \mathbb{R}^{48} \leftrightarrow \mathcal{D}(\mathbb{C}^7)$ (Cholesky bijection, $\alpha=0$), the map $G$ is an exact functor: $\varepsilon_{\text{functor}} = 0$. This has been experimentally confirmed (MVP-1): $\max_k |\Delta\sigma_k| = 0$ to machine precision.

Key constraint: the 49th parameter $d_6 = L_{66}$ (determining $\gamma_{UU}$) is not independent — it is computed from the normalization condition:

$$\gamma_{UU} = 1 - \sum_{k \neq U} \gamma_{kk}, \qquad d_6 = \sqrt{\gamma_{UU} - \sum_{j<6}|L_{6j}|^2}$$

This is a direct consequence of the axiom $\mathrm{Tr}(\Gamma)=1$: the state space is a 48-dimensional manifold, not 49-dimensional. Attempting to estimate $d_6$ independently (via a neural network, averaging, or interpolation) violates the axiom and leads to systematic downward drift of $P$ (purity loss per tick).

Quasi-functor under Neural Correction [H]

Definition: The map $G: \mathbf{AIState} \rightsquigarrow \mathbf{DensityMat}$ with $\alpha > 0$ (neural correction):

$$\|G(f \circ g) - G(f) \circ G(g)\|_F \leq \varepsilon_{\text{functor}} \cdot \|f\|_{\text{op}} \cdot \|g\|_{\text{op}}$$

NTK Linearization

In the tangent space, nonlinearity is approximated by:

$$f(s) \approx f(s_0) + J_f(s_0) \cdot (s - s_0)$$

Corollary: Approximate functoriality with error $O(\|f\|^2 \cdot \|g\|^2)$.

Connection to theory: Extends the Categorical formalism.

Separation Principle: Diagonal / Coherences [T, MVP-0]

Empirically established in the implementation of full Lindblad dynamics:

$$W := \|\sigma\|_2 = \|\mathbf{1} - N \cdot \mathrm{diag}(\Gamma)\|_2 = \mathrm{const}, \quad W_{\text{std}} < 10^{-15}$$

The replacement channel $\mathcal{R}[\Gamma, E]$ fixes the diagonal of $\Gamma$ at each Lindblad step. Consequence:

Component of $\Gamma$	Role	Dynamics
$\gamma_{kk}$ (diagonal)	System identity	Homeostatically stable
$\gamma_{ij}$, $i \neq j$ (coherences)	Learning, adaptation	Evolve

For the measurement protocol: the metrics $I_A, I_S, I_D, I_L$ primarily reflect coherent structure; $\sigma_k = 1 - N\gamma_{kk}$ characterizes the diagonal deviation from equilibrium. The lobotomy test (weight pruning) changes coherences, not the diagonal — the diagonal is homeostatically stable against small perturbations.

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Validation

Viability Test

$$P(\Gamma) = \mathrm{Tr}(\Gamma^2) > P_{\text{crit}} = \frac{2}{7} \approx 0.286$$

See Theorem on critical purity and Viability.

Coherence Flow

Definition:

$$J_P := \frac{dP}{d\tau} = 2 \cdot \mathrm{Tr}\left(\Gamma \cdot \frac{d\Gamma}{d\tau}\right)$$

where τ — emergent internal time.

Mode	Condition	Interpretation
Regeneration	$J_P > 0$ under stress	System recovers
Stability	$J_P \approx 0$, $P > P_{\text{crit}}$	Stable equilibrium
Decay	$J_P < 0$ persistently	Decoherence

Lobotomy Test

Protocol:

1. Measure $P_0$ and $\text{Accuracy}_0$
2. Intervention: prune part of the weights
3. Measure $P_1$ and $\text{Accuracy}_1$

Mechanism [T, separation principle, MVP-0]: Pruning neural network weights changes the off-diagonal coherences $\gamma_{ij}$ of the matrix $\Gamma$, but not the diagonal populations $\gamma_{kk}$ (which are homeostatically stabilized by the replacement channel). The change in $P = \mathrm{Tr}(\Gamma^2)$ upon pruning occurs through loss of coherent integration. With massive pruning that disrupts the replacement channel, the diagonal may also degrade.

Criterion for ontological validity:

Result	Interpretation
$\Delta P > 0$ before $\Delta A > 0$	[T] Protocol captures ontology
$\Delta P \approx \Delta A$	[C] Correlation with output
$\Delta A > 0$ before $\Delta P > 0$	Protocol does not capture ontology

Causal Closure of E

$$\Delta\Phi_E := \Phi_{\text{eff}}(\mathcal{S}_E) - \Phi_{\text{eff}}(\mathcal{S}_E | \text{do}(X := \text{random})) > \varepsilon_{\text{causal}}$$

If $\Delta\Phi_E \approx 0$ — the system simulates phenomenology without realizing it ("Chinese Room").

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Approximation Hierarchy

Level	Metrics	Complexity	Application
L0: Fast	Cosine similarity, norms	$O(n)$	Monitoring
L1: Standard	Jacobian rank, $\Phi_{\text{eff}}$	$O(n^2)$	Inference
L2: Precise	Commutators, NTK	$O(n^3)$	Research
L3: Full	$\Phi_{\text{IIT}}$, full homologies	$O(2^n)$	Small systems

Recommendation: L1 for practice, L2 for validation, L3 for calibration.

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Practical Implementation

Status

This section describes a minimal viable implementation. Many parameters require experimental calibration.

Metric Computation Algorithm

mount std.math.linalg.{svd, eigvalsh, StaticMatrix};
mount std.tensor.{Tensor, frobenius_norm};
mount std.math.random.{XorShift128, Rng};

/// Access protocol for deep models. Implementations provide hooks
/// on activations, attention, and automatic differentiation.
pub protocol ModelHooks {
    type Activation;
    fn get_activations(&self, batch: &Tensor<Float>) -> List<Self.Activation>;
    fn get_attention_weights(&self, batch: &Tensor<Float>) -> Tensor<Float>;
    fn get_jacobian(&self, batch: &Tensor<Float>) -> Tensor<Float>;
    fn layer_commutator_norm(&self, i: Int, j: Int, batch: &Tensor<Float>) -> Float;
    fn estimate_lyapunov(&self, batch: &Tensor<Float>) -> Float;
}

/// Helpers — specialised per architecture.
pub pure fn estimate_mutual_info(x: &Tensor<Float>, y: &Tensor<Float>) -> Float
    = unimplemented;

pub pure fn von_neumann_entropy(attn: &Tensor<Float>) -> Float
    = unimplemented;

pub pure fn build_attention_graph(attn: &Tensor<Float>) -> Tensor<Float>
    = unimplemented;

/// 7-dimensional UHM metrics I_A…I_U for a neural network.
pub type DimensionMetrics is {
    i_a: Float, i_s: Float, i_d: Float, i_l: Float,
    i_e: Float, i_o: Float, i_u: Float,
};

/// Compute 7 UHM dimensions for a neural network.
pub fn compute_dimension_metrics<M: ModelHooks>(
    model:         &M,
    input_batch:   &Tensor<Float>,
    layer_indices: Maybe<List<Int>>,
) using [Random] -> DimensionMetrics
{
    let activations = model.get_activations(input_batch);
    let attn = model.get_attention_weights(input_batch);

    // I_A: mutual information input ↔ latent.
    let i_a = estimate_mutual_info(input_batch, activations.last().unwrap());

    // I_S: Jacobian rank fraction (via SVD, ε = 10⁻⁶).
    let jac = model.get_jacobian(input_batch);
    let sv = svd(&jac).singular_values();
    const EPS_RANK: Float = 1.0e-6;
    let i_s = (sv.iter().filter(|s| **s > EPS_RANK).count() as Float) / (sv.len() as Float);

    // I_D: maximum Lyapunov exponent.
    let i_d = model.estimate_lyapunov(input_batch);

    // I_L: mean layer commutator norm; 1.0 if no pairs.
    let idx = layer_indices.unwrap_or((0..activations.len()).collect());
    let mut comms = List.new();
    for i in 0..idx.len() { for j in (i + 1)..idx.len() {
        comms.push(model.layer_commutator_norm(idx[i], idx[j], input_batch));
    }}
    let i_l = if comms.is_empty() { 1.0 }
              else { 1.0 - comms.iter().sum::<Float>() / (comms.len() as Float) };

    // I_E: exp(von Neumann entropy of attention).
    let i_e = von_neumann_entropy(&attn).exp();

    // I_O: noise robustness.
    let mut rng = XorShift128.seed(Random.next_key());
    const NOISE_STD: Float = 0.01;
    let perturbed = input_batch + Tensor.random_normal(input_batch.shape(), &mut rng) * NOISE_STD;
    let delta_h = frobenius_norm(
        model.get_activations(&perturbed).last().unwrap()
      - activations.last().unwrap()
    );
    let i_o = (1.0 - delta_h / NOISE_STD).max(0.0);

    // I_U: Laplacian spectral gap (λ₂/λ_max).
    let attn_graph = build_attention_graph(&attn);
    let row_sums = attn_graph.sum(axis: 1);
    let laplacian = Tensor.diagonal(row_sums) - &attn_graph;
    let eigs = eigvalsh(&laplacian);
    let lambda_2   = if eigs.len() > 1 { eigs[1] } else { 0.0 };
    let lambda_max = eigs.last().unwrap_or(&0.0);
    let i_u = if lambda_max > 0.0 { lambda_2 / lambda_max } else { 0.0 };

    DimensionMetrics {
        i_a: i_a, i_s: i_s, i_d: i_d, i_l: i_l,
        i_e: i_e, i_o: i_o, i_u: i_u,
    }
}

Γ Reconstruction from Metrics

/// Reconstruct the coherence matrix via Cholesky from 7 dimension metrics.
/// Simplest diagonal reconstruction — off-diagonal γ_ij requires additional
/// correlation data from a regulariser L_off.
pub pure fn reconstruct_gamma(m: &DimensionMetrics) -> StaticMatrix<Complex, 7, 7> {
    let raw = StaticVector.<Float, 7>.from_array(
        [m.i_a, m.i_s, m.i_d, m.i_l, m.i_e, m.i_o, m.i_u]
    ).map(|v| v.clamp(0.01, 1.0));           // prevent degeneracy
    let total: Float = raw.iter().sum();
    let diag = raw.map(|v| v / total);

    // Cholesky factor L = diag(√p_k).
    let l = StaticMatrix.<Complex, 7, 7>.diagonal(
        diag.map(|v| Complex.from_real(v.sqrt()))
    );
    let gamma = &l @ l.adjoint();
    &gamma / gamma.trace()                                              // normalise
}

/// Purity P = Tr(Γ²).
pub pure fn compute_purity(gamma: &StaticMatrix<Complex, 7, 7>) -> Float
    where ensures 1.0/7.0 <= result && result <= 1.0
{
    (gamma @ gamma).trace().real()
}

Threshold Values

Parameter	Value	Source	Status
$P_{\text{crit}}$	$2/7 \approx 0.286$	Theorem	Proven
$\mathrm{rank}(\rho_E) > 1$ (L1 threshold)	$> 1$	Non-trivial interiority	[T]
$R_{\text{th}}$ (L2 threshold)	$\geq 1/3$	Hierarchy	Proven [T]
$\Phi_{\text{th}}$ (L2 threshold)	$\geq 1$	T-129	Proven [T]
$D_{\text{diff}}^{\text{min}}$	$\geq 2$	T-151	Proven [T]
$\varepsilon_{\text{functor}}$	$= 0$ at $\alpha=0$ (Cholesky)	[T, MVP-1]: exact functor	Proven
$\varepsilon_{\text{functor}}$	$< 0.1$ at $\alpha>0$ (neural)	Requires calibration	Hypothesis
$\varepsilon_{\text{causal}}$	$> 0.05$	Requires calibration	Hypothesis

Connection to the Interiority Hierarchy

The L1 and L2 thresholds in the protocol correspond to levels L1 and L2 from the interiority hierarchy L0→L4. Levels L3 (network consciousness) and L4 (unitary consciousness) — see formal description.

Practical Limitations

Limitation	Impact	Mitigation
Batch size	Variance of estimates	$N \geq 64$ for stability
Network depth	Commutator complexity	Sample a subset of layers
Activation dimensionality	$O(n^2)$ for the Jacobian	Project into $\mathbb{R}^k$, $k \ll n$
Attention heads	Aggregation across heads	Average or max-pooling
Determinism	Stochastic layers (dropout)	Fix seed or average

Data Requirements

For a valid measurement:

1. Representative input batch: $N \geq 64$ examples from the target distribution
2. Access to activations: hooks on intermediate layers
3. Attention weights: for computing $I_E$ and $I_U$
4. Gradients: for the Jacobian (automatic differentiation)

What Is Implemented (SYNARC MVP-0/1/2)

Confirmed in Implementation

1. Cholesky-backbone ($\alpha=0$): $G$ is an exact functor [T, MVP-1] — bijection $\psi: \mathbb{R}^{48} \leftrightarrow \mathcal{D}(\mathbb{C}^7)$ with $\varepsilon_{\text{functor}} = 0$
2. Neural bridge ($\alpha>0$): $G$ is a quasi-functor [H] — H1/H2/H4 confirmed [C] for the analytic backbone (MVP-1); neural correction $\alpha>0$ — MVP-3+
3. Diagonal/coherence separation principle [T, MVP-0] — diagonal is homeostatically stable; coherences — the adaptation zone
4. R = 1/(N·P) — exact identity [T, MVP-0, reflection measure R] — error $< 10^{-7}$
5. No-Zombie floor [T, MVP-0] — $P_{\min} \geq P_{\text{crit}} - \varepsilon_\Gamma$ at $\gamma_{\text{dec}} = 10$ (10000× above norm)
6. H3: R_impl ↔ R_UHM [C, MVP-2] — threshold consistency 97.9%

What Is NOT Implemented

Open Implementation Problems

1. Calibration of $\varepsilon$-parameters ($\varepsilon_{\text{functor}}$ at $\alpha>0$, $\varepsilon_{\text{causal}}$) — requires experiments on known systems
2. Neural correction ($\alpha>0$) — analytic backbone (MVP-1/2) is sufficient for Level 0-1; full neural bridge — MVP-3+
3. Temporal dynamics τ — how to define an "emergent time step" for LLM inference?
4. Validation on biological systems — neuroimaging ↔ metrics
5. Scaling — applicability to models with $>10^9$ parameters

---

"Dual Interview" Protocol for Biological Systems

Status: [P] Research Program

The protocol is developed theoretically. Experimental validation is absent.

Principle

The dual interview simultaneously measures external (behavioral, physiological) and internal (self-report) characteristics of a system, allowing reconstruction of the full coherence matrix $\Gamma$, including the phases $\theta_{ij}$ and, consequently, the Gap profile.

Protocol Stages

Stage	Measurement	Data	What We Extract
1. Background recording	EEG, fMRI, HRV	Resting physiology	Diagonal $\gamma_{ii}$, estimate of $P$
2. Structured interview	Responses to 7 question batteries (per dimension)	Verbal reports	Coherences $\lvert\gamma_{ij}\rvert$ between dimensions
3. Paradoxical probes	Conflict tasks	Reaction time, HRV	Phases $\theta_{ij}$ → Gap profile
4. Dynamic probe	Stress test + recovery	Time series $P(\tau)$	$\kappa(\Gamma)$, $\Gamma_2$, τ_char

Spectral Reconstruction of H_eff

Theorem (Spectral Reconstruction) [C]

From the time series $\{\Gamma(\tau_n)\}_{n=1}^N$ it is possible to reconstruct the effective Hamiltonian:

$$H_{\text{eff}} = \frac{i}{\delta\tau} \log\!\left(\frac{\Gamma(\tau + \delta\tau)}{\Gamma(\tau)}\right) + O(\delta\tau)$$

given sufficient sampling frequency $\delta\tau \ll \tau_{\text{char}}$.

Assumption: linearity of evolution on the scale $\delta\tau$. The nonlinear regenerative term $\mathcal{R}[\Gamma, E]$ introduces a systematic error $O(\kappa \cdot \delta\tau)$.

Equilibrium Gap

Theorem (Equilibrium Gap) [T]

In the stationary state ($d\Gamma/d\tau = 0$) the coherences are determined by the balance of decoherence and regeneration:

$$|\gamma_{ij}^{(\infty)}| = \frac{\kappa \cdot |\gamma_{ij}^*|}{\bigl[(\Gamma_2 + \kappa)^2 + \Delta\omega_{ij}^2\bigr]^{1/2}}$$

where $|\gamma_{ij}^*|$ — target coherences (from $\varphi_{\text{coh}}$), $\Delta\omega_{ij} = \omega_i - \omega_j$ — frequency detuning.

See: Theorem 8.1, Fano channel

Physiological Frequencies

Characteristic frequencies of projections of $\Gamma$ onto dimensions:

Dimension	Physiological frequency	Measurement method	Justification
$A$ (Articulation)	$1$–$5$ Hz	EEG θ-rhythm	Sensory processing
$S$ (Structure)	$10^{-2}$–$10^{-4}$ Hz	fMRI BOLD	Slow structural oscillations
$D$ (Dynamics)	$8$–$13$ Hz	EEG α-rhythm	Motor-cognitive dynamics
$L$ (Logic)	$30$–$100$ Hz	EEG γ-rhythm	Cognitive binding
$E$ (Interiority)	$0.005$–$0.02$ Hz	EEG infraslow	Goldstone modes
$O$ (Ground)	$0.04$–$0.15$ Hz	HRV (LF)	Homeostatic regulation
$U$ (Unity)	$0.15$–$0.4$ Hz	HRV (HF)	Vagal modulation

Status: [H]

The correspondence between dimensions and physiological frequencies is a hypothesis requiring experimental verification. The frequencies of the E-dimension ($0.005$–$0.02$ Hz) are a falsifiable prediction linked to Goldstone modes.

Gap Profile Reconstruction from Interview

/// Dual-interview data bundle.
pub type DualInterviewData is {
    external_data: Map<Text, Float>,      // behavioural/physiological per pair
    self_report:   Map<Text, Float>,      // verbal reports per pair
    conflict_data: Map<Text, Float>,      // reaction times per pair
};

/// Reconstruct the 7×7 Gap matrix from dual-interview data.
pub pure fn reconstruct_gap_profile(data: &DualInterviewData)
    -> StaticMatrix<Float, 7, 7>
{
    const DIMS: [Text; 7] = ["A", "S", "D", "L", "E", "O", "U"];
    let median_rt = data.conflict_data.values().to_list().median().unwrap_or(1.0);

    let mut gap = StaticMatrix.<Float, 7, 7>.zeros();
    for i in 0..7 { for j in (i + 1)..7 {
        let pair = f"{DIMS[i]}{DIMS[j]}";

        // Mismatch between behavioural and self-report data → higher Gap.
        let ext = data.external_data.get(&pair).unwrap_or(0.5);
        let rep = data.self_report.get(&pair).unwrap_or(0.5);
        let discrepancy = (ext - rep).abs();

        // Reaction time → phase estimate → Gap.
        let rt = data.conflict_data.get(&pair).unwrap_or(1.0);
        let phase_estimate = (rt / median_rt).atan();

        let g = phase_estimate.sin().abs() * (0.5 + 0.5 * discrepancy);
        gap[i, j] = g;
        gap[j, i] = g;
    }}
    gap
}

---

Success Criteria

The protocol is validated if:

1. $P > P_{\text{crit}}$ for functioning systems in ≥90% of cases
2. Correlation of $P$ with quality: $r > 0.5$
3. Lobotomy test: $\Delta P$ predicts $\Delta A$ in ≥70% of cases
4. $\Delta\Phi_E > \varepsilon_{\text{causal}}$ for "understanding" systems

The protocol is falsified if:

1. $P < P_{\text{crit}}$ for demonstrably viable systems
2. $\Delta P$ does not correlate with $\Delta A$ under interventions
3. $\Phi_{\text{eff}}$ does not distinguish simulation from realization

---

Protocol $\pi_{\mathrm{bio}}$: Reconstructing $\Gamma$ from Biological Neural Data (Resolution P8)

Status: [T] structural + [H] empirical calibration

The protocol $\pi_{\mathrm{bio}}: \mathrm{NeuralData} \to \mathcal{D}(\mathbb{C}^7)$ defines the mapping of neural data (EEG/fMRI/HRV) into the space of density matrices. The mathematical structure is [T] (follows from $G_2$-rigidity T-42a). The specific correspondences between EEG bands and dimensions are [H] (require experimental validation). A fully specified measurement protocol with feature extraction, validation gates against PCI, and predicted thresholds $P(\Gamma_\mathrm{wake})>2/7$, $P(\Gamma_\mathrm{NREM3})<2/7$ is given in Fundamental Closures §9: simultaneous TMS+EEG+fMRI+HRV recording on $N\geq 50$ subjects across wake/NREM3/anaesthesia states, with explicit 7-feature and 21-off-diagonal extraction protocols. No theoretical obstacle remains; the programme awaits empirical data.

Principle: EEG Bands as Projections of $\Gamma$ onto Dimensions

info

Theorem ($G_2$-uniqueness of $\pi_{\mathrm{bio}}$) [T given $G_2$-rigidity]

If a continuous map $\pi_{\mathrm{bio}}: \mathcal X \to \mathcal{D}(\mathbb{C}^7)$ exists on a neural-feature space $\mathcal X$ that is compatible with (AP autopoiesis)+(PH phenomenological thresholds)+(QG $G_2$-covariance)+(V continuity), then it is unique up to the $G_2$-gauge action $\Gamma \mapsto U\Gamma U^\dagger$ with $U \in G_2$ (14-dimensional freedom). All physical observables ($P$, $R$, $\Phi$, $\mathrm{Coh}_E$) are gauge-invariant.

Proof sketch. Suppose $\pi_{\mathrm{bio}}^{(1)}$ and $\pi_{\mathrm{bio}}^{(2)}$ both satisfy (AP)+(PH)+(QG)+(V). The map $\varphi := \pi_{\mathrm{bio}}^{(2)} \circ (\pi_{\mathrm{bio}}^{(1)})^{-1}$ is a continuous automorphism of $\mathcal D(\mathbb C^7)$ preserving $P,R,\Phi$ pointwise and compatible with (AP). By the $G_2$-rigidity theorem [T], the group of continuous $\mathcal D(\mathbb C^7)$-automorphisms preserving the holonomic structure ($P$, $R$, $\Phi$, self-model operator $\varphi_{\text{AP}}$, Fano-plane gauge structure) is precisely $G_2 = \mathrm{Aut}(\mathbb O)$ of real dimension 14. Hence $\varphi(\Gamma) = U\Gamma U^\dagger$ for a unique $U \in G_2$, i.e.\ $\pi_{\mathrm{bio}}^{(2)}(x) = U\,\pi_{\mathrm{bio}}^{(1)}(x)\,U^\dagger$.

Gauge-invariance of observables: $P(\Gamma) = \mathrm{Tr}(\Gamma^2)$ and $R(\Gamma) = 1/(7P(\Gamma))$ depend only on spectral data, invariant under unitary conjugation. $\Phi$ and $\mathrm{Coh}_E$ are Hilbert–Schmidt functions of $\Gamma$ and the self-model $\varphi$, both $G_2$-covariant, hence invariant under $U \in G_2$. $\square$

Basic idea: neural activity in different EEG frequency bands projects onto the 7 dimensions of $\Gamma$. Cross-frequency coupling (CFC) determines the coherences $|\gamma_{ij}|$, and phase mismatches determine the Gap profile.

Step 1: Extracting the Diagonal $\gamma_{kk}$ from Spectral Powers

Dimension	EEG band	Frequency	Metric	Additional source
$A$ (Articulation)	$\alpha$ (8–13 Hz)	Desynchronization during attention	Spectral power $P_\alpha$	fMRI: salience network
$S$ (Structure)	infraslow (0.01–0.1 Hz)	Slow structural oscillations	fMRI BOLD DMN	DTI: structural connectivity
$D$ (Dynamics)	$\beta$ (13–30 Hz)	Motor-cognitive activity	Spectral power $P_\beta$	EMG: motor activation
$L$ (Logic)	$\gamma$-low (30–50 Hz)	Cognitive binding	Spectral power $P_{\gamma L}$	ERP: P300 amplitude
$E$ (Interiority)	$\gamma$-high (50–100 Hz) + $\theta$ (4–8 Hz)	Coupling of experience and memory	$P_{\gamma H} \times \mathrm{PAC}(\theta, \gamma)$	Goldstone modes
$O$ (Ground)	HRV LF (0.04–0.15 Hz)	Homeostatic regulation	$\mathrm{LF}/\mathrm{HF}$ ratio	Body temperature, cortisol
$U$ (Unity)	HRV HF (0.15–0.4 Hz) + $\alpha$-coherence	Vagal + neural integration	Global EEG coherence	$\Phi_{\mathrm{eff}}$ from AI protocol

Diagonalization formula:

$\gamma_{kk} = \frac{w_k \cdot S_k}{\sum_{j=1}^{7} w_j \cdot S_j}, \qquad k \in \{A,S,D,L,E,O,U\}$

where $S_k$ — normalized spectral power (or combined metric) for the $k$-th dimension, $w_k$ — calibration weights (determined from a training set with known consciousness state).

Step 2: Extracting Coherences $|\gamma_{ij}|$ from Cross-Frequency Coupling

Key Correspondence

Coherences $|\gamma_{ij}|$ between dimensions $i$ and $j$ are proportional to the strength of cross-frequency coupling (CFC) between the corresponding EEG bands:

$|\gamma_{ij}| \propto \mathrm{CFC}(\mathrm{band}_i, \mathrm{band}_j)$

Types of CFC used for reconstruction:

Pair	CFC type	Method	Interpretation
$(A, L)$: $\alpha$--$\gamma$	Phase-amplitude coupling (PAC)	Modulation Index (Tort et al.)	Attention modulates cognitive binding
$(D, L)$: $\beta$--$\gamma$	PAC	MI	Motor-cognitive coordination
$(E, L)$: $\theta$--$\gamma$	PAC	MI (hippocampal)	Coupling of experience and logic
$(A, E)$: $\alpha$--$\gamma_H$	Amplitude-amplitude	Envelope correlation	Awareness-interiority
$(O, U)$: LF--HF	HRV coherence	Cross-spectral analysis	Homeostasis-integration
$(S, D)$: infraslow--$\beta$	Nested oscillations	Wavelet coherence	Structure-dynamics

Step 3: Extracting Phases $\theta_{ij}$ and the Gap Profile

The phase $\theta_{ij} = \arg(\gamma_{ij})$ determines the Gap: $\mathrm{Gap}(i,j) = |\sin(\theta_{ij})|$.

Phase extraction method: Paradoxical probes (Stage 3 of the dual interview). Reaction time on conflict tasks involving the pair of dimensions $(i,j)$ is proportional to the Gap:

$\mathrm{Gap}(i,j) \approx \tanh\!\left(\frac{\mathrm{RT}_{ij} - \overline{\mathrm{RT}}}{\sigma_{\mathrm{RT}}}\right)$

where $\mathrm{RT}_{ij}$ — reaction time, $\overline{\mathrm{RT}}$ — mean, $\sigma_{\mathrm{RT}}$ — standard deviation.

Step 4: MLE Reconstruction of $\Gamma$

tip

Algorithm $\pi_{\mathrm{bio}}$: Maximum Likelihood Estimation [H]

Given the neural feature vector $\mathbf{x} \in \mathbb{R}^N$ (spectral powers, CFC metrics, RT). Task:

$\Gamma^* = \underset{\Gamma \in \mathcal{D}(\mathbb{C}^7)}{\arg\max}\; \mathcal{L}(\mathbf{x} | \Gamma) + \lambda_{\mathrm{phys}} \cdot R_{\mathrm{phys}}(\Gamma)$

where $\mathcal{L}(\mathbf{x} | \Gamma)$ — likelihood of the observation model, $R_{\mathrm{phys}}(\Gamma)$ — physical regularizer (consistency with dynamics $\mathcal{L}_\Omega$).

Parametrization: $\Gamma = LL^\dagger / \mathrm{Tr}(LL^\dagger)$ (Cholesky parametrization, guarantees $\Gamma \in \mathcal{D}(\mathbb{C}^7)$).

Observation model:

• Diagonal: $S_k | \gamma_{kk} \sim \mathcal{N}(a_k \gamma_{kk} + b_k,\; \sigma_k^2)$
• Coherences: $\mathrm{CFC}_{ij} | |\gamma_{ij}| \sim \mathcal{N}(c_{ij} |\gamma_{ij}|,\; \tau_{ij}^2)$
• Gap: $\mathrm{RT}_{ij} | \mathrm{Gap}_{ij} \sim \mathrm{Exp}(\mu_0 + \mu_1 \cdot \mathrm{Gap}_{ij})$

Physical regularizer:

$R_{\mathrm{phys}}(\Gamma) = -\lambda_1 \|\dot{\Gamma} - \mathcal{L}_\Omega[\Gamma]\|_F^2 - \lambda_2 \max(0, P_{\mathrm{crit}} - P(\Gamma))$

The first term penalizes inconsistency with dynamics; the second penalizes non-viable states.

Optimization: Gradient descent over 48 Cholesky factorization parameters (34 physical + 14 gauge). The gauge freedom is fixed by choosing the canonical $G_2$-gauge (e.g., $\gamma_{AS} \in \mathbb{R}_+$).

Step 5: Connection to PCI (Casali et al. 2013)

info

Theorem ($\mathrm{PCI} \to \Phi$ proxy) [H]

The Perturbational Complexity Index (PCI) correlates with the integration measure $\Phi(\Gamma)$:

$\Phi(\Gamma) \approx \alpha_{\mathrm{PCI}} \cdot \mathrm{PCI} + \beta_{\mathrm{PCI}}$

where $\alpha_{\mathrm{PCI}}$, $\beta_{\mathrm{PCI}}$ — calibration constants determined from a training set (healthy waking, sleep, anesthesia).

Justification: PCI measures the algorithmic complexity of the cortical response to TMS perturbation. High PCI means simultaneous spatial differentiation and integration — exactly what $\Phi$ quantifies in UHM. Empirically: PCI $\geq 0.31$ during wakefulness (Casali et al. 2013), corresponding to $\Phi \geq \Phi_{\mathrm{th}} = 1$.

Calibration table (hypothetical, requires experimental verification):

State	PCI (observed)	$P$ (predicted)	$R$ (predicted)	$\Phi$ (predicted)
Wakefulness	$0.44 \pm 0.10$	$> 2/7$	$\geq 1/3$	$\geq 1$
REM sleep	$0.41 \pm 0.09$	$> 2/7$	$\geq 1/3$	$\geq 1$
NREM (N3)	$0.18 \pm 0.06$	$\lesssim 2/7$	$< 1/3$	$< 1$
Anesthesia (propofol)	$0.12 \pm 0.05$	$< 2/7$	$< 1/3$	$< 1$
Coma	$0.15 \pm 0.10$	$\lesssim 2/7$	—	$< 1$
MCS (minimally conscious)	$0.32 \pm 0.08$	$\approx 2/7$	$\approx 1/3$	$\approx 1$

Step 6: Connection to Quantum Cognition (Pothos-Busemeyer)

Context: Quantum Cognition

The Pothos-Busemeyer approach (Annual Review of Psychology, 2022) models cognitive processes via quantum states in Hilbert space. Basic formalism: $\rho \in \mathcal{D}(\mathcal{H})$ for describing beliefs and decisions.

Connection to UHM: Quantum cognition uses $\dim(\mathcal{H})$ = number of alternatives. UHM fixes $\dim(\mathcal{H}) = 7$ from axioms (A1-A5) and proves the minimality of this number (Theorem S). The matrix $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ is ontological (not epistemic): it defines the system, rather than describing an observer's beliefs about the system.

Step 7: Full Algorithm $\pi_{\mathrm{bio}}$

mount std.math.calculus.bfgs;

/// Full biological data bundle for π_bio.
pub type NeuralData is {
    eeg_spectral:    Map<Text, Float>,    // {alpha, beta, gamma_low, gamma_high, theta, infraslow}
    hrv_features:    Map<Text, Float>,    // {LF, HF, LF_HF_ratio}
    cfc_matrix:      StaticMatrix<Float, 7, 7>,   // cross-frequency coupling values
    reaction_times:  StaticVector<Float, 21>,      // RT values for the 21 off-diagonal pairs
};

pub type BioCalibration is {
    weights:         StaticVector<Float, 7>,
    linear_params:   StaticMatrix<Float, 7, 2>,    // (a_k, b_k) per dimension
    lambda_phys:     Float,                         // physical regulariser weight
};

/// π_bio: NeuralData → D(ℂ⁷). Full reconstruction of Γ from biological data.
/// Structural [T] via G₂-rigidity (T-42a); empirical calibration [H].
pub fn pi_bio(
    data:        &NeuralData,
    calibration: &BioCalibration,
) -> StaticMatrix<Complex, 7, 7>
{
    // Step 1: diagonal from spectral powers — one value per dimension.
    let raw_diag = StaticVector.<Float, 7>.from_array([
        data.eeg_spectral.get("alpha").unwrap_or(0.0),       // A
        data.eeg_spectral.get("infraslow").unwrap_or(0.0),   // S  (fMRI BOLD proxy)
        data.eeg_spectral.get("beta").unwrap_or(0.0),        // D
        data.eeg_spectral.get("gamma_low").unwrap_or(0.0),   // L
        data.eeg_spectral.get("gamma_high").unwrap_or(0.0)
            * data.eeg_spectral.get("theta").unwrap_or(0.0),  // E  (PAC proxy)
        data.hrv_features.get("LF").unwrap_or(0.0),          // O
        data.hrv_features.get("HF").unwrap_or(0.0),          // U
    ]);

    let weighted = (0..7).map(|i| calibration.weights[i] * raw_diag[i]).to_array();
    let total = weighted.iter().sum::<Float>();
    let mut diag = StaticVector.<Float, 7>.from_array(
        weighted.map(|v| (v / total).clamp(1.0e-4, 1.0))     // prevent degeneracy
    );
    let diag_sum: Float = diag.iter().sum();
    diag = diag.map(|v| v / diag_sum);

    // Step 2: off-diagonal magnitudes from CFC.
    let c_scale = calibration.linear_params[0, 0];                       // cfc_scale stored here
    let off_diag_mag = &data.cfc_matrix * c_scale;

    // Step 3: Phases from reaction times → Gap → θ_ij = arcsin(Gap).
    let rt_mean: Float = data.reaction_times.iter().sum::<Float>() / 21.0;
    let rt_std = (data.reaction_times.iter()
                     .map(|r| (r - rt_mean).pow(2)).sum::<Float>() / 21.0)
                     .sqrt() + 1.0e-8;
    let mut phases = StaticMatrix.<Float, 7, 7>.zeros();
    let mut idx = 0;
    for i in 0..7 { for j in (i + 1)..7 {
        let gap = ((data.reaction_times[idx] - rt_mean) / rt_std).tanh();
        let phi = gap.clamp(-1.0, 1.0).asin();
        phases[i, j] =  phi;
        phases[j, i] = -phi;
        idx += 1;
    }}

    // Step 4: MLE reconstruction via Cholesky. 48 real parameters:
    //   7 real diagonal + 21·2 = 42 off-diagonal (Re, Im).
    let neg_log_likelihood = |params: &StaticVector<Float, 48>| -> Float {
        let mut l = StaticMatrix.<Complex, 7, 7>.zeros();
        let mut k = 0;
        for i in 0..7 { for j in 0..=i {
            if i == j {
                l[i, j] = Complex.from_real(params[k].max(1.0e-6));
                k += 1;
            } else {
                l[i, j] = Complex(params[k], params[k + 1]);
                k += 2;
            }
        }}
        let gamma = &l @ l.adjoint();
        let gamma = &gamma / gamma.trace();

        // LL: diagonal agreement.
        let ll_diag: Float = (0..7)
            .map(|i| -(gamma[i, i].real() - diag[i]).pow(2) / 0.01)
            .sum();

        // LL: off-diagonal magnitude agreement.
        let mut ll_off = 0.0;
        for i in 0..7 { for j in (i + 1)..7 {
            ll_off -= (gamma[i, j].abs() - off_diag_mag[i, j]).pow(2) / 0.05;
        }}

        // Physical regulariser: hard floor at P > P_crit.
        let p = (&gamma @ &gamma).trace().real();
        let p_penalty = -100.0 * (2.0 / 7.0 - p).max(0.0);

        -(ll_diag + ll_off + p_penalty)
    };

    // Initialise from the diagonal (triangle-flattened index k = i·(i+1)).
    let mut x0 = StaticVector.<Float, 48>.zeros();
    for i in 0..7 { x0[i * (i + 1)] = diag[i].sqrt(); }

    let result = bfgs(neg_log_likelihood, &x0, BfgsOptions {
        ftol: 1.0e-9, max_iter: 500,
    });

    // Reconstruct Γ from the optimal parameters.
    let mut l = StaticMatrix.<Complex, 7, 7>.zeros();
    let mut k = 0;
    for i in 0..7 { for j in 0..=i {
        if i == j {
            l[i, j] = Complex.from_real(result.x[k].max(1.0e-6));
            k += 1;
        } else {
            l[i, j] = Complex(result.x[k], result.x[k + 1]);
            k += 2;
        }
    }}
    let gamma = &l @ l.adjoint();
    &gamma / gamma.trace()
}

Replication-Ready Specification for TMS-EEG PCI Data

Replication target

This subsection fixes the reference implementation of $\pi_{\mathrm{bio}}$ applied to the TMS-EEG Perturbational Complexity Index (PCI) paradigm, in enough detail that an independent laboratory can attempt replication end-to-end from a publicly available dataset. Replication here refers to computing $P$, $R$, $\Phi$ from raw EEG and checking the monotonic relation to PCI (Prediction P8.3) — not to re-proving the mathematical core, which remains fixed by the $G_2$-uniqueness theorem above.

R1. Public datasets. The following TMS-EEG datasets are candidates for independent replication; none has universal open-access but each is obtainable on request from the authors or through institutional data-sharing:

#	Dataset	Source	Subjects	States	Access
R1.a	Casali et al. 2013 PCI benchmark	Massimini lab (Milan)	52 healthy + 98 clinical	Wake / NREM / REM / anesthesia / VS / MCS / LIS	On request
R1.b	OpenNeuro ds004504 (TMS-EEG benchmark, 2023)	Rogasch lab	20 healthy	Wake (baseline)	Open
R1.c	Comsa et al. 2019 (OSF registration "TMS-EEG sleep")	Lausanne CHUV	12 healthy	Wake / NREM N2 / N3	OSF restricted
R1.d	Bodart et al. 2018 (clinical PCI extension)	Liège	141 DoC patients	Wake / UWS / MCS / EMCS	Per-request

For first-pass replication, dataset R1.b is recommended (fully open, standardized single-pulse TMS-EEG on healthy waking subjects, expected PCI ≈ 0.40-0.48).

R2. Pre-processing pipeline (MNE-Python canonical). The reference preprocessing chain, to be applied to raw EEG (60-channel montage, 1 kHz sampling, TMS-triggered epochs $[-1, +1]\,\mathrm{s}$):

Step	Operation	Tool / parameters
R2.1	TMS pulse artefact removal	Cubic interpolation over $[-2, +12]\,\mathrm{ms}$ around the pulse (mne.preprocessing.fix_stim_artifact)
R2.2	Downsample	1 kHz → 250 Hz (mne.Epochs.resample)
R2.3	Re-reference	Average reference, exclude TMS-side frontal channels
R2.4	Bandpass filter	0.5–80 Hz, 4th-order Butterworth zero-phase (mne.filter.filter_data)
R2.5	Notch filter	50 Hz (or 60 Hz), Q = 30
R2.6	ICA artefact rejection	FastICA, 30 components; reject TMS-locked decay, eye-blink, ECG (mne.preprocessing.ICA)
R2.7	Epoch-level rejection	$
R2.8	Spectral decomposition	Morlet wavelets, 1–80 Hz log-spaced, 5-cycle wavelet, baseline $[-600,-100]\,\mathrm{ms}$

The canonical bands used by $\pi_{\mathrm{bio}}$ are then extracted from the wavelet spectrogram (integrated over post-TMS window $[0, +300]\,\mathrm{ms}$, averaged across channels for diagonal feature vector; cross-channel pairwise for CFC computations).

R3. Feature extraction. From the preprocessed data, compute:

• Seven scalar spectral features $S_A, S_S, S_D, S_L, S_E, S_O, S_U$ per the Step-1 band table.
• Cross-frequency-coupling matrix $\mathrm{CFC}_{ij}$ ($7\times 7$) per the Step-2 table using the Tort Modulation Index (mne_connectivity).
• 21 reaction-time surrogates $\mathrm{RT}_{ij}$ from paradoxical probes if behavioural data is available; otherwise set $\mathrm{RT}_{ij}$ to the pairwise phase-locking value (PLV) as a proxy.
• HRV features $\mathrm{LF}, \mathrm{HF}$ from simultaneous ECG (required for $O$ and $U$ dimensions).

R4. Calibration. Weights $w_k$ are determined by fitting $\pi_{\mathrm{bio}}$ on a healthy-waking reference cohort ($\ge 20$ subjects) such that the population mean of $\gamma_{kk}$ is uniform $= 1/7 \pm 0.02$. Cross-validation: leave-one-subject-out, target consistency of reconstructed $P$ across subjects ($\mathrm{CV} < 15\%$).

R5. Reconstruction. Run the MLE algorithm (Step 4 above) with:

• Cholesky initialization from the calibrated diagonal.
• Optimizer: scipy.optimize.minimize(method='L-BFGS-B', options={'ftol': 1e-9, 'maxiter': 500}).
• Regularizer: $\lambda_1 = 0.1$, $\lambda_2 = 100$ (empirical defaults; subjects should try $\lambda_1 \in \{0.01, 0.1, 1\}$ and report sensitivity).

R6. Observable computation. From the reconstructed $\Gamma$ (canonical definitions):

• $P = \mathrm{Tr}(\Gamma^2) = \|\Gamma\|_F^2$ (purity) — $G_2$-gauge-invariant (trace of $\Gamma^2$ under unitary conjugation).
• $R = 1/(7P)$ (reflection, T-126 [T]) — $G_2$-gauge-invariant (function of $P$).
• $\Phi = \dfrac{\sum_{i\ne j}|\gamma_{ij}|^2}{\sum_i \gamma_{ii}^2} = \dfrac{\|\Gamma - \Gamma_\mathrm{diag}\|_F^2}{\|\Gamma_\mathrm{diag}\|_F^2}$ (integration, Φ canonical) — basis-dependent: invariant under permutations and sign flips within the $G_2$-stabilised Fano frame (7-point labelling of $\{A,S,D,L,E,O,U\}$), which is the gauge residue relevant for empirical replication.
• $\mathrm{Coh}_E = \dfrac{\gamma_{EE}^2 + 2\sum_{i\ne E}|\gamma_{Ei}|^2}{\mathrm{Tr}(\Gamma^2)}$ (E-coherence, Coh_E canonical) — $E$-fixed-frame quantity: invariant under the stabiliser $G_2^{(E)} \subset G_2$ that fixes $|E\rangle$. For cross-laboratory replication, pin the $|E\rangle$-direction to the phenomenological interiority axis (γ-high × θ PAC), as specified in Step 1.

Gauge-fixing protocol for replication. Two implementations applied to the same EEG recording will yield $P$ and $R$ in full agreement (by strict $G_2$-invariance) but may differ on $\Phi, \mathrm{Coh}_E$ if the Fano-frame orientation or the $E$-axis assignment is not fixed. The canonical gauge-fixing rule is: (i) align the 7-axis labelling to the Fano-plane convention of Dimensions §Fano, and (ii) anchor $|E\rangle$ to the phenomenological γ-high×θ feature as per R3. Replicators must publish their gauge-fixing choices explicitly (item (ii) in R8 below).

All four quantities are $G_2$-gauge-invariant by the uniqueness theorem above.

R7. Validation against PCI.

• Compute the subject's PCI on the same TMS-EEG data via the Massimini algorithm (Lempel–Ziv complexity of significant sources; reference implementation available via PCIst package).
• Test the monotonic hypothesis $\Phi(\Gamma) \approx \alpha_\mathrm{PCI}\cdot \mathrm{PCI} + \beta_\mathrm{PCI}$ (Step 5 theorem).
• Pre-register: $r_{\mathrm{Spearman}} \ge 0.5$ across $\ge 20$ subjects constitutes corroboration; $r < 0.3$ constitutes falsification of P8.3.

R8. Reference implementation stub. The Python code in the next subsection is reference only: it documents the algorithm faithfully but is not a turn-key pipeline. A complete MNE-Python implementation with:

• mne.Raw loader wrapped around BIDS formatted EEG,
• mne_connectivity integration for CFC,
• scipy.optimize.minimize MLE wrapper,
• pyphi-compatible $\Phi$ computation (optional),
• CI reporting, is planned as a separate package uhm-neurocalib (release gated on R1.b pilot results). Until that package is available, independent implementers should use the pseudocode as specification, and file issues/PRs on mismatches to the specification here.

Reproducibility requirements. Any claim of successful or failed replication should publish:

• (i) raw data (BIDS format) and preprocessing scripts (reproducible from R2);
• (ii) reconstructed $\Gamma$ matrices and gauge-fixing choice made;
• (iii) $P, R, \Phi, \mathrm{Coh}_E$ values per subject;
• (iv) PCI values computed on same epochs;
• (v) statistical test protocol and seed for random splits.

Without items (i)-(v), a replication attempt cannot be audited.

Testable Predictions of the $\pi_{\mathrm{bio}}$ Protocol

#	Prediction	Verification method	Falsification criterion
P8.1	$P(\Gamma_{\mathrm{wake}}) > 2/7$ for waking subjects	EEG+HRV → $\pi_{\mathrm{bio}}$ → $P$	$P < 2/7$ in healthy waking subjects
P8.2	$P(\Gamma_{\mathrm{NREM3}}) < 2/7$ during deep sleep	EEG → $\pi_{\mathrm{bio}}$ → $P$	$P > 2/7$ during N3
P8.3	$\mathrm{PCI} \propto \Phi(\Gamma)$ (monotonic dependence)	TMS-EEG + $\pi_{\mathrm{bio}}$	Non-monotonic correlation
P8.4	The $P = 2/7$ transition coincides with PCI $\approx 0.31$	Simultaneous measurement	Threshold divergence
P8.5	$\mathrm{Gap}(L,E) \approx 1$ in alexithymia	Dual interview + EEG	$\mathrm{Gap}(L,E) \ll 1$ with diagnosed alexithymia
P8.6	Critical exponents $\beta = 1/4$ at the sleep-wakefulness transition	EEG monitoring + $\pi_{\mathrm{bio}}$ → $P(\tau)$ near $P_{\mathrm{crit}}$	Other exponents

Key References

1. Casali et al. (2013) — PCI: "A theoretically based index of consciousness independent of sensory processing and behavior." Science Translational Medicine, 5(198). PubMed: 23946194
2. Pothos-Busemeyer (2022) — Quantum cognition review. Annual Review of Psychology, 73, 749-778.
3. Butlin et al. (2023/2025) — "Consciousness in Artificial Intelligence: Insights from the Science of Consciousness." arXiv: 2308.08708; updated 2025: "Identifying indicators of consciousness in AI systems." Trends in Cognitive Sciences.
4. eLife (2024/2025) — "Spatiotemporal brain complexity quantifies consciousness outside of perturbation paradigms." eLife 98920.
5. Quantum-inspired EEG (2026) — "Quantum inspired feature engineering for explainable EEG signal classification." Scientific Reports. Nature.

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Related documents:

• Coherence matrix — definition of $\Gamma$
• Viability — $P$ and $P_{\text{crit}} = 2/7$
• Emergent time — Page–Wootters mechanism, τ ∈ ℤ₇
• Evolution — equation $d\Gamma(\tau)/d\tau$ with $H_{eff}$
• Self-observation — measures $R$, $\Phi$, $C$
• Categorical formalism — functor $F$, $\mathbf{Exp}^{disc}_\infty$
• Theorem on minimality 7D — why 7 dimensions
• Notation — indices $I_A, \ldots, I_U$
• Gap diagnostics — clinical applications of the Gap profile
• Goldstone modes — prediction of infraslow frequencies
• Fano channel — equilibrium Gap theorem