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]

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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)

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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.

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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

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Validation constraint: no threshold without ground truth

What the field's one clinically validated measure teaches us

The Perturbational Complexity Index is the only consciousness measure with large-scale clinical validation: TMS-evoked EEG responses, Lempel–Ziv compressed, with an empirical cutoff $\mathrm{PCI}^*=0.31$ derived from a benchmark population and later validated on 719 TMS/hd-EEG sessions across wakefulness, NREM sleep, anaesthesia and disorders of consciousness. Two lessons transfer directly, and the second is a hard limit on what UHM may claim.

Lesson 1 — commensuration is solved by calibration, not by units. PCI is normalised for signal length and amplitude, and its threshold is not derived from theory but fitted to a benchmark population. This is exactly the fix that closes audit A-20: the seven observable indices are defined in incommensurable units, so a $\Gamma$ built from their raw sum is a function of arbitrary definitional choices. The canonical repair is to take each index as a percentile against a declared reference ensemble — the same operation the applied layer already performs when it scores a person against a population. The ensemble must be published together with any number derived from it; a percentile without its reference is not a measurement.

Lesson 2 — a threshold is only as valid as its ground truth, and this bounds UHM. PCI's cutoff is trustworthy because it was calibrated where consciousness was independently known: the same brains awake and under anaesthesia, patients who could later report. That ground truth is what makes the number mean anything. UHM's window $P\in(2/7,3/7]$ is derived (T-124), not fitted — which is a genuine advantage — but derivation fixes the form of the criterion, not the mapping from a given substrate into $\Gamma$. For any system where we have no independent evidence of presence or absence of experience — a fungal network, a slime mould, a language model — the mapping cannot be validated, and a measured $P$ inside or outside the window therefore establishes nothing about experience. It establishes only that the system's functional indices, under a declared coarse-graining, do or do not sit where the theory says a viable holon sits.

Consequently this corpus does not, and will not, assert consciousness or its absence in a non-human substrate on the strength of a $\Gamma$-measurement alone. What such a measurement can honestly do is discriminate states within one system — the paired design that made PCI work — and that is the only design in which our thresholds carry evidential weight outside the human case.

Lesson 3 — measure the functional structure, not the carrier. Complexity read off the raw signal misreads systems whose carrier is atypical: children with Angelman syndrome are awake, volitional and responsive while displaying the hypersynchronous delta EEG normally taken as a signature of unconsciousness. The resolution is that the relevant complexity is not in the amplitude envelope but in the functional organisation. UHM is structurally committed to the same discipline: $\Gamma$ is built from functional indices (integration, differentiation, robustness, sensitivity), never from raw signal statistics, and a protocol that shortcuts to the carrier will inherit the field's known failure cases.

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

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

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

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

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

/// 7-dimensional UHM metrics I_A…I_U for a neural network.
public 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.
public fn compute_dimension_metrics<M: ModelHooks>(
    model:         &M,
    input_batch:   &Tensor,
    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.
public 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.matmul(&l.adjoint());
    &gamma / gamma.trace()                                              // normalise
}

/// Purity P = Tr(Γ²).
public pure fn compute_purity(gamma: &StaticMatrix) -> Float
    where ensures 1.0/7.0 <= result && result <= 1.0
{
    (gamma.matmul(&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

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"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.
public 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.
public 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 core.math.calculus.bfgs;

/// Full biological data bundle for π_bio.
public 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
};

public 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].
public 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 {
        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.matmul(&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.matmul(&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.matmul(&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	$\|\text{max}-\text{min}\| > 120\,\mu\mathrm V$ → drop epoch
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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