Γ Measurement Protocol for AI Systems

Document Status: [Pr] 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. 48 physical parameters (Corollary 1, frame decision D-0910): of the 48 parameters of $\Gamma$, 34 are kinematic $G_2$-invariants ($48 - \dim(G_2) = 48 - 14 = 34$) and the remaining 14 fix the orientation of the functional frame, which the axiomatic dynamics pins; the residual identification freedom is the finite frame group $\Gamma_{\!\text{oct}}$

Practical implication: reconstruction of $\Gamma$ is defined uniquely up to the finite frame group $\Gamma_{\!\text{oct}}$ once the frame is pinned (Step 1 and R6 below). $P$ and $R$ are $G_2$-invariant (functions of the spectrum); $\Phi$ and $\mathrm{Coh}_E$ are frame-pinned observables — two reconstructions can differ on them only if their frame-fixing choices differ, which is why those choices are part of the replication package (R8).

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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	Independent L2 threshold [D] (it read "Proven [T]" until 2026-09-25)
$\varepsilon_{\text{functor}}$	$= 0$ 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: [Pr] 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. Corrected 2026-09-25: the sentence "no theoretical obstacle remains" is withdrawn — as first specified, the estimator contained the tested predicate and the calibration used report-labelled sessions; the conditions under which a test is possible are the position against the substitution argument and its pre-registration SUB-1 … SUB-6.

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 action $\Gamma \mapsto U\Gamma U^\dagger$ of the finite frame group $U \in \Gamma_{\!\text{oct}} \subset G_2$ (kinematic envelope $G_2$, 14-dimensional; dynamical identification $\Gamma_{\!\text{oct}}$ — frame decision D-0910). $P$ and $R$ are $G_2$-invariant; $\Phi$ is invariant under $\Gamma_{\!\text{oct}}$, $\mathrm{Coh}_E$ only under its $192$ elements that keep the $E$-axis, and both change under a generic element of $G_2$ (the statement ascribed $\Gamma_{\!\text{oct}}$-invariance to both until 2026-09-25).

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], a continuous automorphism of $\mathcal D(\mathbb C^7)$ compatible with the octonionic structure is $\Gamma \mapsto U\Gamma U^\dagger$ with $U \in G_2 = \mathrm{Aut}(\mathbb O)$ (the kinematic envelope, of real dimension 14). Since $\varphi$ also preserves $\Phi$ pointwise, and $\Phi$ vanishes exactly on diagonal $\Gamma$, $U$ maps diagonal states to diagonal states and is therefore monomial; the monomial elements of $G_2$ are its signed permutation matrices, i.e. $\Gamma_{\!\text{oct}}$. Hence $\pi_{\mathrm{bio}}^{(2)}(x) = U\,\pi_{\mathrm{bio}}^{(1)}(x)\,U^\dagger$ with $U \in \Gamma_{\!\text{oct}}$. (Until 2026-09-25 the sketch said that the group of automorphisms preserving $P$, $R$, $\Phi$, the self-model operator and the Fano gauge structure "is precisely $G_2$"; $\Phi$ is not $G_2$-invariant, so that group is $\Gamma_{\!\text{oct}}$.)

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 $\mathrm{Coh}_E$ reference the coordinate frame (the diagonal of $\Gamma$ and the $E$-axis): $\Phi$ is invariant under the frame group $\Gamma_{\!\text{oct}}$, which permutes the axes, $\mathrm{Coh}_E$ only under its $192$ elements that keep the $E$-axis (and under the continuous stabiliser $G_2^{(E)} \cong SU(3)$), and neither under a generic $U \in G_2$, which rotates the axes (Lemma G4 of the uniqueness theorem). (The sentence ascribed $\Gamma_{\!\text{oct}}$-invariance to both until 2026-09-25.) Since the axiomatic dynamics pins the frame (Theorem 5.1b), $\Gamma_{\!\text{oct}}$ is exactly the residual freedom of $\pi_{\mathrm{bio}}$. $\square$

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). In a test of $\mathrm{Cons}(S)$ this route is excluded (SUB-3, below): reaction times are behaviour, i.e. inference data; phases then come from the EEG. 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. Corrected 2026-09-25: the second term puts the tested predicate into the estimator — with $\lambda_2 = 100$ every sub-threshold state of the uniform family is reconstructed at $P = 2/7$ exactly (theorem, part (ii)); in any test $\lambda_2 = 0$, and $\lambda_1 = 0$ in confirmatory runs (SUB-2).

Optimization: Gradient descent over the 48 Cholesky factorization parameters (all physical — 34 kinematic invariants + 14 frame-orientation parameters, D-0910). The frame is fixed by the labelling rule of R6 (Fano-frame convention and the $E$-axis anchor), not by a continuous gauge choice; the convention $\gamma_{AS} \in \mathbb{R}_+$ removes only the global phase.

Step 5: Connection to PCI (Casali et al. 2013; Casarotto et al. 2016)

info

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

The Perturbational Complexity Index (PCI) is monotonically related to the integration measure $\Phi(\Gamma)$ (prediction P8.3). A linear form $\Phi(\Gamma) \approx \alpha_{\mathrm{PCI}} \cdot \mathrm{PCI} + \beta_{\mathrm{PCI}}$ with constants fitted on a training set is a calibration, not a bridge: whatever it fits, it cannot test.

Justification: PCI measures the algorithmic complexity of the cortical response to TMS perturbation; a high PCI requires a response that is both integrated and differentiated, and $\Phi$ is UHM's integration measure. The earlier sentence "PCI $\geq 0.31$ during wakefulness, corresponding to $\Phi \geq \Phi_{\mathrm{th}} = 1$" stated a correspondence that nothing derives; it is withdrawn.

What can be derived — the bridge on UHM's side [T]. Three facts fix how UHM's thresholds sit relative to each other, with no neural data:

• $\Phi \geq 1 \Rightarrow P \geq 2/7$ on all of $\mathcal D(\mathbb C^7)$ (T-129a).
• On the uniform-diagonal stratum $P = (1 + \Phi)/7$, $R = 1/(1 + \Phi)$ and $C = \Phi R = \Phi/(1+\Phi)$; the window $P \in (2/7, 3/7]$ is exactly $\Phi \in (1, 2]$, $R \in [1/3, 1/2)$, $C \in (1/2, 2/3]$ (checked in check_core_numbers.py, test_uniform_diagonal_window_is_phi_between_one_and_two).
• The predicate has two exits: $P \leq 2/7$ (too mixed; on the uniform diagonal the same as $\Phi \leq 1$) and $P > 3/7$ (too pure; $R < 1/3$). A low PCI therefore has two possible UHM signatures, not one.

What cannot be derived [✗ if claimed]. A numerical conversion between PCI and $P$ or $\Phi$. PCI is a normalised Lempel–Ziv complexity of a binarised source-activity matrix; $P$ and $\Phi$ are functions of $\Gamma$. The closeness of $\mathrm{PCI}^* = 0.31$ to $2/7 \approx 0.286$ is a coincidence of two unrelated scales and carries no evidential weight. The bridge that can be tested is a concordance of verdicts on the same sessions (P8.4 below): $\mathrm{Cons}(\hat\Gamma)$ against $\mathrm{PCI}_{\max} > \mathrm{PCI}^*$.

Reference data — Casarotto et al. (2016), Table 1 (Ann. Neurol. 80: 718–729, doi:10.1002/ana.24779). Benchmark population: 150 subjects, 540 sets of TMS-evoked potentials; $\mathrm{PCI}^* = 0.31$ from an ROC analysis in which the presence or absence of a subjective report — immediate or delayed — is the ground truth; on this benchmark the cut-off separates the two classes with 100 % sensitivity and 100 % specificity. Values are $\mathrm{PCI}_{\max}$ per subject, median [min–max]:

Condition	Report	Subjects	$\mathrm{PCI}_{\max}$	UHM verdict to be tested
Wakefulness (healthy)	immediate	102	0.53 [0.39–0.70]	Cons
REM sleep	delayed (dream)	8	0.48 [0.36–0.56]	Cons
Ketamine anaesthesia	delayed	6	0.43 [0.36–0.52]	Cons
NREM sleep	none	18	0.25 [0.15–0.31]	¬Cons
Midazolam	none	6	0.30 [0.23–0.31]	¬Cons
Xenon	none	6	0.23 [0.11–0.31]	¬Cons
Propofol	none	6	0.26 [0.23–0.31]	¬Cons

Applied to patients: 36 of 38 in a minimally conscious state had $\mathrm{PCI}_{\max} > \mathrm{PCI}^*$ (sensitivity 94.7 %), and 9 of 43 in a vegetative state did too. (Casali et al. 2013, Sci. Transl. Med. 5(198): 198ra105, doi:10.1126/scitranslmed.3006294, introduced the index.) The table this section carried until 2026-09-25 — "wakefulness $0.44 \pm 0.10$, REM $0.41 \pm 0.09$, NREM $0.18 \pm 0.06$, propofol $0.12 \pm 0.05$, coma $0.15 \pm 0.10$, MCS $0.32 \pm 0.08$", labelled "observed" — has no source in either paper and is withdrawn. The REM and ketamine rows matter most for UHM: consciousness without behaviour, where the verdict cannot be read off reports given at the time.

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;
        }}

        // No viability term (SUB-2, corrected 2026-09-25): the former
        // `-100.0 * (2/7 - P).max(0)` pinned every sub-threshold state at P = 2/7.

        -(ll_diag + ll_off)
    };

    // 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: $\lambda_2 = 0$ and, in the confirmatory run, $\lambda_1 = 0$ (SUB-2); $\lambda_1 \in \{0.01, 0.1, 1\}$ only as a reported sensitivity analysis. The earlier defaults $\lambda_1 = 0.1$, $\lambda_2 = 100$ are withdrawn (2026-09-25): $\lambda_2 = 100$ pins every sub-threshold state of the uniform family at $P = 2/7$.

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).

$P$ and $R$ are $G_2$-invariant; $\Phi$ and $\mathrm{Coh}_E$ are frame-pinned — $\Phi$ invariant under $\Gamma_{\!\text{oct}}$, $\mathrm{Coh}_E$ only under its $192$ elements that keep the $E$-axis (the line said "$\Gamma_{\!\text{oct}}$ only" for both until 2026-09-25) — which is why the gauge-fixing protocol above is part of the replication package (frame decision D-0910).

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 hypothesis [H]).
• 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	Concordance of verdicts: $\mathrm{Cons}(\hat\Gamma)$ agrees with $\mathrm{PCI}_{\max} > 0.31$ on the same sessions, Cohen's $\kappa \geq 0.8$ (SUB-5; until 2026-09-25: "the $P = 2/7$ transition coincides with PCI $\approx 0.31$", a comparison of unrelated scales)	TMS-EEG + $\pi_{\mathrm{bio}}$ with $\theta$ frozen on wakefulness	$\kappa < 0.4$
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

Position against the substitution argument

Kleiner & Hoel (Neurosci. Conscious. 2021(1), niab001; arXiv:2004.03541) separate an experiment's data into prediction data $o_i$, from which a theory predicts experience, and inference data $o_r$ — reports and behaviour — from which the experimenter infers it. If the two are independent (Definition 3.8: for any $o_i, o_i', o_r$ some physically possible variation changes $o_i$ into $o_i'$ and keeps $o_r$), every minimally informative theory is "already falsified" or "every single inference operation is wrong" (Theorem 3.10). If they are strictly dependent ($o_i = f(o_r)$, Definition 4.2), the theory is already falsified or empirically unfalsifiable (Theorem 4.3). Between the two lies a lenient dependency, of which the authors know no instance. The unfolding argument (Doerig et al. 2019) is one case. The predicate $\mathrm{Cons}(S) = (P > 2/7) \wedge (R \geq 1/3) \wedge (\Phi \geq 1) \wedge (D \geq 2)$ is computed from $\hat\Gamma = \pi_\theta(o_i)$, where $\theta$ collects the free parameters of $\pi_{\mathrm{bio}}$ (weights $w_k$, observation-model coefficients, $\alpha_{\mathrm{PCI}}, \beta_{\mathrm{PCI}}$, the regulariser weights $\lambda_1, \lambda_2$).

tip

Theorem (Where $\mathrm{Cons}(S)$ sits between the horns) [T], with part (v) [H]

(i) Calibration is on the strict-dependence horn. As specified above, $\theta$ is fitted on report-labelled sessions (Step 1: "a training set with known consciousness state"; the constants $\alpha_{\mathrm{PCI}}, \beta_{\mathrm{PCI}}$ of Step 5, fitted on "healthy waking, sleep, anesthesia"). On those sessions $\mathrm{Cons}_\theta$ is fitted to reproduce the labels, so its agreement with them tests nothing (Kleiner–Hoel, Theorem 4.3). The same holds for $\mathrm{PCI}^*$ on its own benchmark: fitted to reports, 100 % accurate there by construction.

(ii) The estimator contained the predicate. With the default regulariser of R5 ($\lambda_2 = 100$), the reconstruction returns $\hat P = 2/7$ exactly for every sub-threshold state of the uniform family $\Gamma = I/7 + m(J - I)$ with $m < m_c = 1/\sqrt{294}$: numerically, $m = 0.02$ (true $P = 0.160$) and $m = 0.05$ (true $P = 0.248$) both give $\hat P = 0.2857$, while $\lambda_2 = 0$ returns the true values. Analytically, pinning occurs whenever $\lambda_2 \geq 10\,(1 - m/m_c)$, so any $\lambda_2 \geq 10$ pins the whole family. Prediction P8.2 ($P < 2/7$ in N3) could therefore not be observed, and P8.1 was favoured by the loss itself. (Checked in check_core_numbers.py, test_viability_penalty_pins_every_subthreshold_reconstruction_at_two_sevenths.)

(iii) After freezing, the predicate is on the independence horn. Once $\theta$ is frozen before the test data are seen, $\mathrm{Cons}_\theta$ is a function of $o_i$ alone. Whenever a physically possible variation keeps the reports and moves $\hat\Gamma$ across a threshold — Kleiner and Hoel argue that interventions and unfoldings supply such variations — Theorem 3.10 applies: some possible system falsifies $\mathrm{Cons}$, or report-based inference is wrong for some system with each report. UHM takes the second disjunct for substitutes: it treats reports as evidence only within a declared domain (the "no threshold without ground truth" constraint above), not across all physically possible systems.

(iv) The non-closure exit is closed. Kleiner and Hoel's other way out — experience making a physical difference beyond the physical state — is unavailable: two-aspect monism identifies experience with an aspect of $\Gamma$.

(v) A domain-restricted lenient dependency [H]. Let $D_{\mathrm{nat}}$ be the domain in which reports are the accepted inference: intact adult human brains in natural sleep–wake states and under standard anaesthetics. Inside $D_{\mathrm{nat}}$ the dependence is lenient if (a) one report class occurs with different $\hat\Gamma$ — not strict; $\mathrm{PCI}_{\max}$ already spans 0.39–0.70 among 102 awake subjects — and (b) no member of $D_{\mathrm{nat}}$ with a report has $\mathrm{Cons}_\theta$ false — not independent within $D_{\mathrm{nat}}$. Then neither Theorem 3.10 nor Theorem 4.3 covers tests inside $D_{\mathrm{nat}}$. Condition (b) is empirical and coincides with P8.1, which is why it must be established on sessions disjoint from those that fix $\theta$. This does not answer the substitution argument for systems outside $D_{\mathrm{nat}}$ — feedforward unfoldings, emulations, language models — and UHM makes no consciousness claim there.

Proof. (i) A statistic fitted to labels is, on the fitting set, a function of the labels up to fit error; Definition 4.2 holds there. (ii) For the uniform family the diagonal term is at its optimum, and the off-diagonal loss is $420\,(m' - m)^2$ against the penalty $\lambda_2 \max(0,\, 2/7 - 1/7 - 42 m'^2)$; the derivative of the sum, $840(m' - m) - 84\lambda_2 m'$, is negative on $[m, m_c)$ iff $\lambda_2 > 10(1 - m/m')$ there, i.e. iff $\lambda_2 \geq 10(1 - m/m_c)$, so the minimiser is $m' = m_c$, where $P = 2/7$. The numerical check runs the reference MLE of Step 4. (iii) With $\theta$ fixed, $\mathrm{Cons}_\theta \circ \mathrm{obs}$ depends on $p$ only through $o_i$; Definition 3.8 is the stated premise, and Theorem 3.10 is Kleiner and Hoel's. (iv) By the definition of two-aspect monism. (v) Definitions 3.8 and 4.2 restricted to $D_{\mathrm{nat}}$ fail exactly under (b) and (a). $\square$

The protocol that follows (pre-registration SUB-1 … SUB-6).

• SUB-1. Freeze $\theta$ on wakefulness sessions only (the reference-ensemble normalisation of R4); no NREM, anaesthesia, REM or ketamine label enters the fit.
• SUB-2. $\lambda_2 = 0$ in every confirmatory run. $\lambda_1$ (consistency with $\mathcal L_\Omega$) also carries the theory: $\lambda_1 = 0$ in the confirmatory run, other values only as a reported sensitivity analysis.
• SUB-3. Phases from the EEG (complex phase-locking values, as in §9.3 of the fundamental closures), never from reaction times: reaction times are behaviour, i.e. inference data. $P$, $R$ and $\Phi$ depend only on $|\gamma_{ij}|$ and $\gamma_{ii}$; $D = e^{S}$ depends on the spectrum and hence on the phases.
• SUB-4. Register the verdicts of the table in Step 5 before unblinding. The decisive rows are REM and ketamine (consciousness without behaviour at the time): with $\theta$ frozen on wakefulness they are out-of-sample.
• SUB-5. Concordance with $\mathrm{PCI}^*$ on the same sessions (P8.4 in concordance form): Cohen's $\kappa$ between $\mathrm{Cons}_\theta$ and $\mathrm{PCI}_{\max} > 0.31$; $\kappa \geq 0.8$ corroborates, $\kappa < 0.4$ falsifies [Pr].
• SUB-6. The two exits (Step 5): among sessions with $\mathrm{PCI}_{\max} \leq 0.31$, responses that stay local are predicted to have $\hat\Phi < 1$; responses that spread as a stereotyped global wave, $\hat P > 3/7$ ($\hat R < 1/3$). This compares prediction data with prediction data, so the substitution argument does not touch it — it tests UHM's structure, not its consciousness claim [H].

What separates a system passing $\mathrm{Cons}(S)$ from one with an isomorphic similarity structure [T]. Kawakita, Zeleznikow-Johnston, Tsuchiya & Oizumi (Sci. Rep. 14: 15917, 2024, doi:10.1038/s41598-024-65604-1) aligned colour-similarity structures for 93 colours by Gromov–Wasserstein optimal transport, without labels: GPT-4's structure matched that of colour-neurotypical humans with a matching rate of 91.4 % (GPT-3.5: 11.8 %). In UHM:

• A similarity structure is inference data — judgements, i.e. reports. A system that reproduces it is precisely what a substitution preserves; by (iii) it carries no weight for $\mathrm{Cons}$.
• The verdict and the quality geometry are independent. The Fubini–Study distances between the eigenrays of $\Gamma$ do not depend on its spectrum, and $P = \sum_k \lambda_k^2$ does not depend on the eigenrays. The same geometry is carried by a state with $P = 0.152$ (outside the window) and by one with $P = 0.312$ (inside it), so an isomorphic similarity structure is neither sufficient nor necessary for $\mathrm{Cons}$ (test_cons_verdict_and_quality_geometry_are_independent).
• What $\mathrm{Cons}$ requires is the system's own $\hat\Gamma$, reconstructed from its internal, interventional data by a protocol validated where ground truth exists. For a language model no such validation exists, and the corpus makes no claim ("no threshold without ground truth" above). The 91.4 % result shows that report-level structure can be shared across radically different systems, which is exactly why UHM does not read consciousness off it.

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.
6. Casarotto et al. (2016) — "Stratification of unresponsive patients by an independently validated index of brain complexity." Annals of Neurology 80(5): 718–729. doi:10.1002/ana.24779 ($\mathrm{PCI}^* = 0.31$; Table 1 above).
7. Kleiner & Hoel (2021) — "Falsification and consciousness." Neuroscience of Consciousness 2021(1): niab001. doi:10.1093/nc/niab001; arXiv:2004.03541.
8. Kawakita, Zeleznikow-Johnston, Tsuchiya & Oizumi (2024) — "Gromov–Wasserstein unsupervised alignment reveals structural correspondences between the color similarity structures of humans and large language models." Scientific Reports 14: 15917. doi:10.1038/s41598-024-65604-1.

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