02 · The kernel specification

One library, reused for a person, a team, and an agent. It takes an estimated state and its uncertainty, and returns the readout with the uncertainty carried through. Everything a domain module needs, it gets by calling the kernel; the kernel calls nothing but linear algebra.

§1. Representation of Γ

The kernel's single input type is an estimate object GammaEstimate:

• gamma: a $7\times7$ complex Hermitian matrix, PSD, unit trace — axis order fixed as $[A,S,D,L,E,O,U]$.
• cov: a covariance over the 48 real degrees of freedom (7 populations − 1 trace + 21 magnitudes + 21 phases; the trace constraint removes one population dof), used for confidence propagation. May be null for anchor III.
• anchor: one of MEASURED | AUTOEPHEMERIS | ORACLE (the honesty tag).
• t: timestamp, for trajectory assembly.

Validation and projection. Any incoming gamma is projected to the nearest valid state before use: Hermitize ($\Gamma \leftarrow \tfrac12(\Gamma+\Gamma^\dagger)$), clip negative eigenvalues to zero, renormalise the trace. The projection distance is recorded as an input-quality flag — a large projection means the estimator produced something far from a physical state, which the honesty layer surfaces.

§2. The eleven invariants — exact routines

Each routine is a pure function of gamma (plus, where noted, a second state for increments). All are $O(1)$ in problem size (fixed $7\times7$); the only nontrivial numerics are a Hermitian eigendecomposition and one small Hessian.

Purity $P$. P = trace(gamma @ gamma).real. Range $[1/7, 1]$; $1/7$ is the maximally mixed "heat-death" state $I/7$, $1$ is a pure state. Threshold of viability $P > 2/7$ (T).

Reflection $R$. R = 1.0 / (7*P). Range $[1/7, 1]$; the normalised distance from $I/7$. Consciousness band $R \geq 1/3$ (T). This is the canonical R; do not confuse it with $R_\varphi$ (below) — the two are distinct working forms and the kernel exposes both under their own names.

Self-model quality $R_\varphi$. Requires the self-model map $\varphi$. For the canonical (dissipative) family, phi(gamma) = (1-k)*gamma + k*eye(7)/7 with k = 1 - 1/(7*P); then R_phi = 1 - fro(gamma - phi(gamma))**2 / fro(gamma)**2. Range $[0,1]$. Where a trained target $\rho_\theta$ is available (the two-timescale model), phi_theta(gamma) = (1-k)*gamma + k*rho_theta is used instead — this is what lets practice raise the baseline.

Integration $\Phi$. Computed from the HS-projection functional of dimension-U: the loss of information under the best factorised approximation of Γ across the minimum-information bipartition. Algorithm: for each bipartition of the seven axes, form the product-of-marginals approximation, take the HS distance to Γ; $\Phi$ is the minimum over bipartitions (the "weakest link"), normalised so the consciousness threshold sits at $\Phi = 1$. Sixty-three bipartitions — trivially enumerable.

Differentiation $D_{\mathrm{diff}}$. evals = eigvalsh(gamma); S = -sum(evals * log(evals + eps)); D_diff = exp(S). Range $[1, 7]$; the effective number of occupied dimensions. Threshold $D_{\mathrm{diff}} \geq 2$ (T).

Gap map. For each pair $(i,j)$: Gap[i,j] = abs(sin(angle(gamma[i,j]))), with the vanishing-coherence convention: if abs(gamma[i,j]) < eps_min, set Gap[i,j] = 1 (a channel with no coherence is not "transparent", it is absent — phase is undefined). Returns the 21 upper-triangle values as the transparency chart.

Mandalagram. A pure reshape: the 7 populations diag(gamma).real and the 21 coherences (magnitude, phase) laid on the 7-node / 21-edge chart, with the canonical cell names. No computation, only layout — this is the render model the UI consumes.

Archetype. Threshold the diagonal at $\tau$ (default the mean population) to a 7-bit pattern; decode to one of the sixteen signatures via the fixed lookup (the 16 signatures are the $G_2$-orbit classes of thresholded diagonals). Returns the signature id plus the margin to the nearest neighbouring signature (how robust the classification is).

Mode. Requires two states $\Gamma(t)$, $\Gamma(t')$. Form the increment $\dot\Gamma \approx (\Gamma(t')-\Gamma(t))/(t'-t)$ and decompose it into the triadic components — Hamiltonian (preserving), dissipative (dissolving toward $I/7$), regenerative (restoring toward $\varphi(\Gamma)$). Returns the three fractions summing to one; the "mode" is the dominant fraction.

Meaning. Meaning = P * D_diff * Phi * R_phi — the product form. Returned both as the scalar and as its four factors, so the UI can show which factor limits meaning (the smallest factor is the bottleneck).

Freedom. Build the free-energy Hessian $\mathcal H_\Gamma$ (second derivative of the free-energy functional at Γ over the 48 real dof), take its eigenvalues, count those below a flatness tolerance: Freedom = count(abs(hessian_evals) < tol) + 1 (T). Returns the count plus the flat directions themselves (the eigenvectors) — the actual axes along which the system may move without energy penalty, which the decision-support module reads.

§3. Confidence propagation

Every scalar invariant carries an uncertainty derived from cov by the delta method: for invariant $f(\Gamma)$, var(f) = grad_f.T @ cov @ grad_f, with grad_f computed analytically where closed-form (P, R, R_phi, D_diff, Meaning) and by autodiff/finite-difference otherwise (Phi, Freedom). The kernel returns each invariant as a (value, std, anchor) triple. Anchor III inputs have no cov, so their invariants return std = null and are rendered as "orientational, not measured" — the honesty spine enforced at the numeric layer.

§4. The optical-construction guardrails

Before any readout is released, the kernel runs the three optical-construction checks and attaches the verdicts:

1. Chart type (T-256). Which sub-configuration does this reading actually resolve? A questionnaire estimate resolves populations well and phases poorly; the kernel tags which sectors are trustworthy for this estimate, so downstream code never reads a phase-dependent Gap value off a phase-blind anchor.
2. Anchor licence (T-257). If any external signal fed the estimate, is it on the licensed ledger (solar/lunar/physiological) or a category error (planetary/natal)? Unlicensed external inputs are rejected at ingestion, not silently used.
3. Viability gate. Is the subject actually in a viable regime ($P>2/7$, etc.)? Below the gate, the "consciousness-predicate" readouts are marked not-applicable rather than reported as small numbers — a rock does not have a low $\Phi$, it has none.

These verdicts are not advisory; a domain module that ignores them cannot compile against the API (§5), because the readout type is parameterised by the guardrail outcome.

§5. The internal API

The kernel exposes one entry point and a typed readout:

readout = kernel.evaluate(estimate: GammaEstimate,
                          prev: GammaEstimate | null) -> Readout

Readout fields: invariants (the eleven (value, std, anchor) triples; mode present only if prev given), mandalagram (render model), archetype (id + margin), guardrails (the three §4 verdicts), quality (the input-projection flag from §1). Every field that can be uncertain carries its uncertainty; every field carries its anchor. There is no call that returns a bare number without its provenance — the API makes the dishonest output unrepresentable.

Substrate-agnosticism. evaluate never inspects where gamma came from beyond the anchor tag. The same call serves a human self-audit, an EEG pipeline, and an AI-telemetry adapter — the platform property of 00 §2, enforced by keeping the kernel blind to substrate.

§6. Testing and numerical discipline

The kernel ships with an oracle suite (the same discipline the corpus uses): analytic invariants checked against random states to $\sim10^{-12}$; the $R_\varphi = 1-(1-R)^3$ identity for the canonical family as a cross-check; boundary states ($I/7$, pure states) hitting the exact threshold values; and property tests (monotonicity of Meaning in each factor, $G_2$-invariance of every scalar under a random $G_2$ rotation of the axes). A build that fails any oracle does not ship — the readout's authority rests on the kernel being provably the corpus's functions, not approximations of them.

Where this leads. 03 · Anchors and estimation specifies how a real GammaEstimate is produced from each of the three evidence types, with the full self-audit instrument and the measurement-bridge mathematics.