The Fano fingerprint: polar rates, sum rules, and line tomography

Ask a Fano-wired system how fast each of its twenty-one coherences dies, and it can only answer with seven numbers. The other fourteen degrees of freedom are not hidden — they are forbidden, and the forbidden directions are a sign-twist of $\mathfrak{g}_2$. What a system cannot do is as much of a fingerprint as what it does.

Document status

The core is Theorem T-226 [Т]: elementary linear algebra and finite geometry over the exact channel forms of the corpus; every proof is complete, and every claim has been machine-verified in exact (symbolic) arithmetic. The dynamical premise — the per-coherence decay rate formula — is the canonical [Т] form of the Γ-dynamics. The identification of the mathematical rates with laboratory decoherence rates is [И], as everywhere in the corpus. Technology items are [С]/[О]. Statuses are marked in place.

How this document is organized. The document is self-contained given school linear algebra; the coding-theoretic and finite-geometric background, if wanted, is built from scratch in the Σ-calculus primer §2. §1 states the known premise — the exact per-coherence decay rates — and poses the question: as the seven line rates vary, what part of the 21-dimensional rate space is reachable? §2 proves the lemma chain; §3 assembles Theorem T-226 with all seven parts. §4 identifies the forbidden subspace with a sign-twist of $\mathfrak{g}_2$ — the second, dynamical apparition of the Lie shadow of the Σ-calculus. §5 draws the protection reading. §6 gives the falsifiable prediction and the technology package; §7 records the machine verification; §8 is the status summary.

§1. The premise and the question

The corpus fixes the dissipative half of the Γ-dynamics by the seven Fano projectors: one Lindblad channel per line $\ell_p = \{p,\, p{+}1,\, p{+}3\} \pmod 7$ of $\mathrm{PG}(2,2)$, with a nonnegative rate $\gamma_p$ per line (Gap dynamics §2, Evolution). The exact channel form — the closed-form exponential established in the v2.2 canon — gives every off-diagonal coherence $(i,j)$, $i \neq j$, a pure exponential decay $e^{-r_{ij} t}$ with

$$r_{ij} \;=\; \frac{1}{6} \sum_{p\,:\, |\ell_p \cap \{i,j\}| = 1} \gamma_p .$$

In words: a line damages a coherence only if it separates the pair — contains exactly one of its endpoints. For the uniform assignment $\gamma_p = \bar\gamma$ this reproduces the corpus constant $r = \tfrac{2}{3}\bar\gamma$, consistent with the discrete Lüders multiplier $1/3$ per full cycle (T-110) and the cycle-time identity $\bar\gamma\tau_{\mathrm{cycle}} = \tfrac{3}{2}\ln 3$.

The seven $\gamma_p$ are the system's microscopic dials; the twenty-one $r_{ij}$ are what a rate experiment sees. So the natural — and, it turns out, sharply answerable — questions are:

As $\gamma \in \mathbb{R}^7$ ranges over all line-rate assignments, which vectors $r \in \mathbb{R}^{21}$ occur? Which linear identities must every realizable $r$ satisfy? Can $\gamma$ be reconstructed from $r$, exactly and stably? And do the answers detect the Fano wiring itself?

§2. The lemma chain

Throughout, $G := \sum_p \gamma_p$ is the total flux and $T_m := \sum_{p\,:\, m \in \ell_p} \gamma_p$ is the point flux of the axis $m$ — the sum of the rates of the three lines through $m$.

Lemma Φ.1 (line-sum identity)

For every Fano line $\ell$: $\displaystyle\sum_{m \in \ell} T_m = G + 2\gamma_\ell$.

Proof. $\sum_{m\in\ell} T_m = \sum_p \gamma_p\,|\ell_p \cap \ell|$. The line $\ell$ meets itself in $3$ points and every other line in exactly $1$ point (two distinct lines of a projective plane meet in one point). Hence the sum equals $3\gamma_\ell + \sum_{p \neq \ell}\gamma_p = 3\gamma_\ell + (G - \gamma_\ell) = G + 2\gamma_\ell$. $\blacksquare$

Lemma Φ.2 (general pair form)

For an arbitrary $3$-uniform wiring (any family of $3$-element "lines", not necessarily Fano),

$$r_{ij} \;=\; \tfrac{1}{6}\bigl(T_i + T_j - 2 S_{ij}\bigr), \qquad S_{ij} := \sum_{p\,:\,\{i,j\} \subseteq \ell_p} \gamma_p .$$

Proof. Inclusion–exclusion on the endpoint count: the lines meeting $\{i,j\}$ in exactly one point are those through $i$, plus those through $j$, minus twice those through both. $\blacksquare$

Lemma Φ.3 (polar identity)

In the Fano wiring, let $\lambda = \lambda(i,j)$ be the unique line through the pair $\{i,j\}$ and let $k$ be its third point. Then

$$r_{ij} \;=\; \frac{G - T_k}{6}.$$

Proof. By Fano incidence $S_{ij} = \gamma_\lambda$ (exactly one line through the pair). Lemma Φ.1 for $\lambda = \{i,j,k\}$ gives $T_i + T_j = G + 2\gamma_\lambda - T_k$. Substituting into Lemma Φ.2: $r_{ij} = \tfrac{1}{6}(G + 2\gamma_\lambda - T_k - 2\gamma_\lambda) = (G - T_k)/6$. $\blacksquare$

Definition (polar partition)

For a pair $\{i,j\}$ write $\pi(i,j) :=$ the third point of $\lambda(i,j)$ — its polar point. The polar class of an axis $k$ is $P_k := \{\ell \setminus \{k\} : \ell \ni k\}$ — the three pairs completing the three lines through $k$. The seven classes have three pairs each and partition all $\binom{7}{2} = 21$ pairs. This is the point–pair polarity of $\mathrm{PG}(2,2)$: the same triple structure that underlies the octonion product ($e_i e_j = \pm e_{\pi(i,j)}$) and the sixteen-archetype grammar of the Γ-canon.

Lemma Φ.4 (image and injectivity)

The linear map $M : \gamma \mapsto r$ has rank $7$; its image is exactly the space of functions constant on each polar class.

Proof. By Lemma Φ.3, $r_{ij} = \rho_{\pi(i,j)}$ with $\rho_k := (G - T_k)/6$, so the image lies in the class-constant subspace, which has dimension $7$. For equality and injectivity it suffices to invert: Lemma Φ.5 reconstructs $\gamma$ linearly from $\rho$, so $M$ is injective and the image is the full class-constant space. $\blacksquare$

Lemma Φ.5 (line tomography)

Let $N$ be the $7 \times 7$ point–line incidence matrix ($N_{kp} = 1$ iff $k \in \ell_p$; for the cyclic labeling it is the circulant with symbol $\{0,1,3\}$, $\det N = 24$). Then

$$N^{-1} \;=\; \tfrac{1}{2}N^{\mathsf T} - \tfrac{1}{6}J,$$

and the line rates are recovered from the seven polar values $\rho_k$ by

$$G = \tfrac{3}{2}\sum_k \rho_k, \qquad T_k = G - 6\rho_k, \qquad \boxed{\;\gamma_p \;=\; 3\Bigl(\tfrac{1}{2}\sum_{k}\rho_k \;-\; \sum_{k \in \ell_p}\rho_k\Bigr).\;}$$

Proof. $(N^{\mathsf T} N)_{pq} = |\ell_p \cap \ell_q|$ equals $3$ on the diagonal and $1$ off it, i.e. $N^{\mathsf T}N = 2I + J$; and $JN = 3J$ since every line has three points. Hence $(\tfrac12 N^{\mathsf T} - \tfrac16 J)N = \tfrac12(2I+J) - \tfrac12 J = I$. Summing $\rho_k = (G-T_k)/6$ over $k$ and using $\sum_k T_k = 3G$ gives $\sum_k \rho_k = \tfrac{2}{3}G$, i.e. the $G$-recovery; then $T = G\mathbf{1} - 6\rho$ and $\gamma = N^{-1}T = \tfrac12 N^{\mathsf T}T - \tfrac16 (\mathbf 1^{\mathsf T}T) \mathbf 1 = \tfrac12\sum_{k\in\ell_p}T_k - \tfrac12 G$, which simplifies to the boxed formula. $\blacksquare$

Lemma Φ.6 (conditioning)

$M^{\mathsf T} M = \tfrac{1}{6}(I + J)$. Consequently the singular values of $M$ are $\tfrac{2}{\sqrt 3}$ (once, on the total-flux direction) and $\tfrac{1}{\sqrt 6}$ (multiplicity six), and the condition number is $2\sqrt2 \approx 2.83$.

Proof. $(36\,M^{\mathsf T}M)_{pq}$ counts pairs meeting $\ell_p$ and $\ell_q$ in exactly one point each. For $p = q$: one endpoint on the line, one off it — $3 \cdot 4 = 12$ pairs. For $p \neq q$, with $z$ the intersection point: either one endpoint in $\ell_p \setminus z$ and the other in $\ell_q \setminus z$ ($2 \cdot 2 = 4$ pairs), or one endpoint equal to $z$ — which lies on both lines, so the other endpoint must avoid both ($7 - 5 = 2$ pairs); total $6$. Hence $36\,M^{\mathsf T}M = 12I + 6(J - I) = 6(I+J)$. The eigenvalues of $\tfrac16(I+J)$ are $\tfrac{8}{6} = \tfrac43$ on constants and $\tfrac16$ on the six-dimensional complement. $\blacksquare$

Lemma Φ.7 (selector converse)

For a $3$-uniform wiring of $7$ lines on $7$ axes, the fourteen polar equalities (§3(ii)) hold identically in $\gamma$ iff every pair of axes lies on exactly one line — i.e. iff the wiring is a $2$-$(7,3,1)$ design, hence the Fano plane.

Proof. ($\Leftarrow$) is Lemma Φ.3. ($\Rightarrow$) The total pair-coverage satisfies $\sum_{\{i,j\}} S^{\#}_{ij} = 7\binom{3}{2} = 21$, where $S^{\#}_{ij}$ is the number of lines through the pair; so if the coverage is not identically $1$, some pair is covered $0$ times. For such a pair Lemma Φ.2 gives $r_{ij} = \tfrac16(T_i + T_j)$ — a functional of $\gamma$ supported on the six lines through $i$ or $j$, while any pair with coverage $1$ has the four-line polar form. Two linear functionals with different coefficient vectors cannot be identically equal, and a direct check shows no assignment of the $21$ pairs into $7$ equality-triples survives (in the explicit one-line rewiring of §7, seven of the fourteen equalities already fail identically). A $2$-$(7,3,1)$ design is the Fano plane up to relabeling — the uniqueness chain of Theorem Σ, Lemmas Σ.1–Σ.2. $\blacksquare$

§3. Theorem T-226 (the Fano fingerprint)

Theorem T-226 [Т]. Let the dissipative Γ-dynamics carry line rates $\gamma \in \mathbb{R}^7_{\geq 0}$ on the Fano wiring, and let $r \in \mathbb{R}^{21}$ be the exact per-coherence decay rates. Then:

(i) Polar law. $r_{ij} = \rho_{\pi(i,j)}$ where $\rho_k = (G - T_k)/6$: the twenty-one rates take at most seven values, constant on the polar classes; a rate depends on the pair only through its polar point.

(ii) Fourteen sum rules (polar equalities). The realizable set is exactly the $7$-dimensional subspace of class-constant vectors. Equivalently: within each polar class the three rates coincide —

$$r_{ij} = r_{i'j'} \quad\text{whenever}\quad \pi(i,j) = \pi(i',j'),$$

fourteen independent linear identities that every Fano-wired system must satisfy for every value of the line rates.

(iii) Line tomography. The map $\gamma \mapsto r$ is injective, and inverts in closed form: measuring the seven polar values $\rho_k$ yields $\gamma_p = 3\bigl(\tfrac12 \sum_k \rho_k - \sum_{k \in \ell_p} \rho_k\bigr)$ exactly (Lemma Φ.5). The per-line dissipation strengths are observables, not fit parameters.

(iv) Conditioning. $M^{\mathsf T}M = \tfrac16(I+J)$; singular values $2/\sqrt3$ and $1/\sqrt6$ ($\times 6$); condition number $2\sqrt2$. Rate noise propagates into $\gamma$ with amplification at most $\sqrt6 \approx 2.45$ — the tomography is numerically benign.

(v) Exact gap and cooldown. On the coherence sector, the dissipative generator has spectrum $\{-\rho_k\}$ with multiplicity $6$ per polar class (three pairs, real and imaginary parts; classes with equal $\rho_k$ merge); its spectral gap is

$$\Delta \;=\; \min_k \rho_k \;=\; \frac{G - \max_k T_k}{6},$$

so the neurogenesis cooldown bound of T-39a becomes fully explicit: $\tau_{\mathrm{cool}} \geq 1/\Delta = 6/(G - \max_k T_k)$. Uniform rates give $\Delta = \tfrac23\bar\gamma$.

(vi) Selector. The fourteen polar equalities hold identically in $\gamma$ iff the wiring is the Fano plane (Lemma Φ.7). The sum rules are therefore not bookkeeping but a fingerprint: they detect the combinatorial grammar of the dynamics itself, independently of the octonionic track — an operational companion to the diagnosability selector of Theorem Σ.

(vii) Two-sided budget. Measuring rates costs $7$ numbers, not $21$; the other $14$ dimensions are consistency checks for free. Dually, the seven $\rho_k$ determine and are determined by the seven $\gamma_p$ — the rate spectrum and the line spectrum are linearly equivalent charts of the same dial space.

Proof. Parts (i)–(iv) are Lemmas Φ.3–Φ.6; (v) follows from the exact channel form (each coherence is an eigenvector of the dissipative factor with eigenvalue $-r_{ij}$, populations are preserved) together with (i); (vi) is Lemma Φ.7; (vii) restates (ii)–(iii). $\blacksquare$

Remark (what is [Т] and what is [И]). The theorem is exact linear algebra over the [Т] channel forms. Its empirical use — reading measured decoherence rates of a candidate septarchitecture as the $r_{ij}$ — inherits the usual [И] identification of axes with measured channels, as in every prediction of the corpus.

§3a. A worked example, end to end

Abstract theorems earn their keep on concrete numbers. Take the seven line rates to be simply $\gamma_p = p + 1$ (in whatever units the dissipation is measured), $p = 0, \dots, 6$ — deliberately non-uniform, so that all the structure is visible. Then $G = 1 + 2 + \cdots + 7 = 28$.

Step 1 — point fluxes. The axis $k$ lies on the three lines $\ell_k, \ell_{k-1}, \ell_{k-3}$ (indices mod 7), so $T_k = \gamma_k + \gamma_{k-1} + \gamma_{k-3}$:

axis $k$	0	1	2	3	4	5	6
lines through $k$	$\ell_0,\ell_6,\ell_4$	$\ell_1,\ell_0,\ell_5$	$\ell_2,\ell_1,\ell_6$	$\ell_3,\ell_2,\ell_0$	$\ell_4,\ell_3,\ell_1$	$\ell_5,\ell_4,\ell_2$	$\ell_6,\ell_5,\ell_3$
$T_k$	$13$	$9$	$12$	$8$	$11$	$14$	$17$
$\rho_k = \tfrac{28 - T_k}{6}$	$\tfrac{5}{2}$	$\tfrac{19}{6}$	$\tfrac{8}{3}$	$\tfrac{10}{3}$	$\tfrac{17}{6}$	$\tfrac{7}{3}$	$\tfrac{11}{6}$

Consistency checks land exactly: $\sum_k T_k = 84 = 3G$ and $\sum_k \rho_k = \tfrac{56}{3} = \tfrac{2}{3}G$ (Lemma Φ.5's $G$-recovery).

Step 2 — the twenty-one rates are these seven numbers. Every coherence $(i,j)$ decays at $\rho_{\pi(i,j)}$. For instance the pair $\{1,3\}$ lies on $\ell_0 = \{0,1,3\}$, its polar point is $0$, so $r_{13} = \rho_0 = \tfrac{5}{2}$ — and the same value is shared by the other two pairs polar to $0$: $\{4,5\}$ (the rest of $\ell_4 = \{4,5,0\}$) and $\{2,6\}$ (the rest of $\ell_6 = \{6,0,2\}$). One pair from each of the three lines through the polar point; three rates per point, seven values in all, fourteen equalities for free.

Step 3 — tomography, run backwards. Reconstruct $\gamma_0$ from the seven polar values: $\tfrac12\sum_k \rho_k = \tfrac{28}{3}$, the line $\ell_0 = \{0,1,3\}$ has polar sum $\rho_0 + \rho_1 + \rho_3 = \tfrac{5}{2} + \tfrac{19}{6} + \tfrac{10}{3} = 9$, and

$$\gamma_0 \;=\; 3\Bigl(\tfrac{28}{3} - 9\Bigr) \;=\; 28 - 27 \;=\; 1. \checkmark$$

All seven $\gamma_p$ come back exactly this way (machine-checked in rational arithmetic) — the dials are read off the rates with no fitting.

Step 4 — gap, cooldown, protection. The largest point flux is $T_6 = 17$, so the slowest rate is $\rho_6 = \tfrac{11}{6}$: the spectral gap is $\Delta = \tfrac{11}{6}$ and the exact cooldown constant is $\tau_{\mathrm{cool}} = 1/\Delta = \tfrac{6}{11}$ (in the same units). The three longest-lived coherences are the polar class of axis $6$: the pairs $\{3,4\}, \{1,5\}, \{0,2\}$ — sitting on the three strongest lines of the example, yet themselves untouched by them: guarded, vicariously, by the very flux that flows through their polar point. That is the protection reading of §5 in numbers.

§4. The $\mathfrak{g}_2$-shadow of the forbidden space

The Σ-calculus records the Lie shadow $\mathfrak{so}(7) = \mathfrak{g}_2 \oplus \mathrm{Im}\,\mathbb{O}$, the split $21 = 14 + 7$ of the generator algebra. The fingerprint produces the same split in the observable rate space — and the two are the same split, up to an explicit sign twist.

Identify $\mathbb{R}^{21}$ with the space of off-diagonal patterns indexed by pairs. Let $\varphi$ be the associative $3$-form of the octonion frame aligned with the corpus labeling ($e_i e_j = \varepsilon_{ij}\, e_{\pi(i,j)}$ on lines, $\varepsilon_{ij} = \varphi_{ij\pi(i,j)} \in \{\pm1\}$), and let $W = \mathrm{diag}(\varepsilon_{ij})$ be the orientation twist of pair space.

Proposition C-226 [Т]. Let $c : \mathbb{R}^{21} \to \mathbb{R}^7$ be the unsigned polar collapse ($(c\,r)_k = \sum_{\pi(i,j)=k} r_{ij}$) and $c_\varphi : \Lambda^2\mathbb{R}^7 \to \mathbb{R}^7$ the $\varphi$-contraction $\omega \mapsto \sum_{i<j} \varphi_{ijk}\,\omega_{ij}$. Then $c_\varphi = c \circ W$. Consequently $W$ carries the standard $G_2$-decomposition $\mathfrak{so}(7) \cong \Lambda^2\mathbb{R}^7 = \mathfrak{g}_2 \oplus \iota_\varphi(\mathrm{Im}\,\mathbb{O})$ — with $\mathfrak{g}_2 = \ker c_\varphi$ and $\iota_\varphi(v) = v \lrcorner \varphi$ — onto the fingerprint decomposition

$$\mathbb{R}^{21} \;=\; \underbrace{\{\text{forbidden directions}\}}_{W(\mathfrak g_2),\ \dim 14} \;\oplus\; \underbrace{\{\text{realizable rates}\}}_{W(\iota_\varphi(\mathrm{Im}\,\mathbb O)),\ \dim 7}.$$

Proof. $c_\varphi = c \circ W$ holds entry by entry, since $\varphi_{ijk} \neq 0$ exactly when $k = \pi(i,j)$, with value $\varepsilon_{ij}$; the kernel/image correspondence follows because $W$ is an involution. The identification $\mathfrak g_2 = \ker c_\varphi$ and the complement $\iota_\varphi$ are the standard $G_2$-module decomposition of $\Lambda^2\mathbb{R}^7$. $\blacksquare$

Reading [И]. The fourteen dimensions a rate experiment can never see are the sign-twisted $\mathfrak{g}_2$: the symmetry algebra of the theory is exactly the diagnostically dark subspace of its own decoherence spectrum. And the duality is polar: the Σ-compression pyramid of T-225 aggregates coherences by lines (content monitoring), while the dynamics collapses rates by polar points. Lines and points are exchanged by the self-duality of $\mathrm{PG}(2,2)$ — the measurement pyramid and the decay spectrum are Fano-dual faces of the same $21 = 14 + 7$.

§5. The protection reading

Two structural facts fall out of the polar law and deserve their own sentences.

Intra-line immunity [Т]. A line never damages its own coherences: if both endpoints of a pair lie on $\ell$, then $|\ell \cap \{i,j\}| = 2$ and $\ell$ contributes nothing to $r_{ij}$. This is Shield I of topological protection surfacing at the rate level: the dissipative grammar is structurally incapable of eroding the coherence of its own syndrome triples.

Polar guardianship [Т math, И reading]. $r_{ij} = (G - T_{\pi(i,j)})/6$ is anti-monotone in the polar point's flux: strengthening the three lines through an axis $k$ slows the decay of the three coherences polar to $k$. Each axis guards not its own coherences but the polar ones — protection in $\mathrm{PG}(2,2)$ is always vicarious. The best-protected coherences of a Γ-state are those polar to its highest-flux axis — and those same coherences set the spectral gap (T-226(v)): the strongest guardian's wards decay slowest, so maximal protection and slow late-time convergence are two faces of one number. The worked example of §3a shows both faces concretely.

§6. Falsifiability and technology

Prediction Σ-P2 (candidate for the Pred registry). In any rate-resolved Γ-tomography of a candidate septarchitecture, the measured pairwise decoherence rates must satisfy the fourteen polar equalities within experimental error, for every dissipation regime — the identities are parameter-free. A stable, reproducible violation in even one polar class falsifies the Fano wiring of the dissipative dynamics directly, independently of the octonionic track. Conversely, verifying the fourteen identities across two or more distinct dissipation regimes (different $\gamma$) is strong evidence for the wiring, by the selector (T-226(vi)): a single one-line rewiring already breaks seven of the fourteen identically.

Technology package.

1. Rate-monitoring budget [С]. Seven polar values instead of twenty-one pairwise rates — a threefold reduction, with the remaining fourteen dimensions repurposed as built-in consistency alarms. This composes with the Σ-compression pyramid of T-225: content monitoring compresses by lines, rate monitoring by polar points — together they instrument both Fano-dual faces of the coherence matrix at $7 + 7$ observables.
2. Line-resolved dissipation spectroscopy ("Fano tomography") [С]. The boxed formula of Lemma Φ.5 turns per-line dissipation strengths $\gamma_p$ into direct observables with noise amplification $\leq \sqrt6$ (T-226(iv)) — no fitting, no regularization. For the Γ-tomograph tiers this yields the per-line health of the dissipative grammar from the same data that currently yields only aggregate decay.
3. Exact cooldown budget for SYNARC [О]. T-39a's neurogenesis cooldown becomes the explicit runtime constant $\tau_{\mathrm{cool}} = 6/(G - \max_k T_k)$, computable from the current line rates in $O(1)$; the runtime can tighten its cooldown adaptively as the dissipation profile shifts, instead of using the worst-case constant.
4. Design rule [О]. To protect a chosen coherence $(i,j)$ passively, invest flux in the lines through its polar point — not in its own line (which is neutral to it) and not in the remaining four (which erode it). Vicarious protection is a design lever unavailable to architectures without the $\lambda = 1$ grammar.

§7. Machine verification

All claims were verified in exact arithmetic (symbolic $\gamma$, rational linear algebra):

Check	Result
Every pair meets exactly $4$ lines once; $S_{ij} \equiv 1$ line	✓
$r_{ij} - (G - T_{\pi(i,j)})/6 = 0$ symbolically, all $21$ pairs	✓
$\operatorname{rank} M = 7$; image $=$ class-constant space	✓
$N^{-1} = \tfrac12 N^{\mathsf T} - \tfrac16 J$; $\det N = 24$; tomography exact	✓
$M^{\mathsf T}M = \tfrac16(I+J)$; singular values $2/\sqrt3,\ 1/\sqrt6^{(\times 6)}$	✓
$c_\varphi = c \circ W$; both kernels $14$-dimensional	✓
One-line rewiring ($\{0,2,6\} \to \{0,1,6\}$): $7/14$ polar equalities fail identically	✓

The verification scripts follow the M1 discipline of the SYNARC programme: independent implementation, exact arithmetic, and the falsifying counter-model (the rewired plane) checked alongside the theorem. A third, independent-language verification also passes: a Verum program (integer arithmetic only, sigma_wave_check.vr in the SYNARC repository) re-derives the polar law on all 21 pairs, the line-sum tomography identity, and the polar partition — alongside the Turyn–Golay enumerator of the Σ-calculus §8a — reporting ALL PASS.

§8. Status summary

Claim	Status
Lemmas Φ.1–Φ.7	[Т]
Theorem T-226 (i)–(vii)	[Т]
Proposition C-226 ($\mathfrak g_2$-shadow, $c_\varphi = c \circ W$)	[Т] (reading — [И])
Intra-line immunity; polar guardianship	[Т] (guardianship reading — [И])
Prediction Σ-P2 (polar equalities in tomography)	falsifiable, [И] identification
Rate budget $21 \to 7$; Fano tomography; adaptive cooldown	[С]/[О]

Where this leads

• Σ-calculus — the static selector (T-224) and the measurement pyramid (T-225); this document is their dynamical, rate-level companion, in polar duality to the pyramid.
• Gap dynamics §2 — the Fano–Hamming wiring of the Lindblad operators whose exact channel forms feed §1.
• Topological protection, Shield I — intra-line immunity is Shield I at the rate level.
• Γ-canon — the polar (octonionic) triple structure $e_i e_j = \pm e_{\pi(i,j)}$ that indexes the rate classes.
• Measurement protocol — where the polar-rate observables and the $\gamma$-tomography slot into the experimental tiers.
• Status registry — row T-226.