# n197 — The constraint theorem: the audit IS the constraint surface of linearized gravity

**Date:** 2026-07-11 · **Gate:** `gates/fierz-pauli.json`, legs 1–2 executed, leg 3 opened. **Script:** `n197_fp_legs.py`. **Reproduce:** `python3 n197_fp_legs.py`.

## The theorem (leg 2 — PASS)

With edge weights **derived** from fourth-moment isotropy of the direction complex (axes ½, face-diagonals 1 — the quadrature condition; residual 10⁻¹⁶, and the derivation has no freedom: two weight families, one isotropy equation), the four monitor rows become, at **every** momentum q:

- **watched space = trace ⊕ linearized-gauge** — the audit watches exactly the four directions that GR's Hamiltonian and momentum constraints govern: v₀ ↔ the Hamiltonian constraint, v_i ↔ the momentum constraints;
- **dark space = TT {+,×} exactly** — dim 2, traceless and transverse to machine precision at all q, on- and off-axis;
- counting: 6 − 4 = 2 spatially; 10 − 4(gauge) − 4(constraints) = 2 covariantly — with the four constraints being *the monitor's own rows*;
- closure: the rows are linear functionals (abelian algebra), and their evolution identity is the continuity law they were derived from (n166) — the energy row's rate *is* the flux rows' divergence.

**The physical reading, one sentence: the vacuum's audit is the constraint surface of linearized general relativity — what GR removes by gauge and constraint, the monitor watches; what GR propagates, the monitor cannot see.** Masslessness (the blind-channel thermostat), helicity ±2, and the {+,×} content stop being three results and become three faces of this one identity.

## The correction this forced (rule 5)

- **Old claim (n166):** dark space = {+,×} "at every momentum," with uniform edge weights.
- **Failure:** with uniform weights the identity holds only on lattice-symmetric momenta; off-axis, transversality breaks at O(0.2). T4 tested on-axis only; n176's off-axis pass rotated the whole complex (covariance), masking the fixed-lattice anisotropy.
- **Repair:** the isotropy-derived weights restore the identity exactly, everywhere. The weights are not a fix-up: they are *forced* by the same requirement (an isotropic audit) that the critique's point 9 demands of the continuum limit — the first concrete piece of that limit, delivered early. Regression T4 upgraded to the weighted off-axis test.

## Leg 1 — the field map, stated before dynamics (as the gate requires)

- **h_ij(x)**: the symmetric tensor whose directional samples are the edge stiffnesses, δk_a = h_ij d̂_a^i d̂_a^j — well-posed (the 9-direction complex has rank 6).
- **h₀₀ = −2Φ**: the clock sector (mechanism 4), with the normalization already pinned at R = 1 by cluster dynamics-vs-lensing (n189).
- **h₀ᵢ**: *the flux-row response potential* — the map is hereby stated; its dynamics (gravito-magnetism, frame dragging) *does not exist in the model*. Declared, not hidden.

## Leg 3 — opened, not passed

Structure against Fierz–Pauli: 2 propagating TT (match) · 4 constrained (match, now as a theorem) · gauge realized (match, spatial) · positive norm (match, bubble level, n175) · **c_g² = 0.46 (STANDING FAIL — condition B's target)** · h₀ᵢ dynamics absent. The derived weights are the first input to the honest continuum limit that must resolve the speed — resolve, not subtract.

**Class: the theorem [A-model, exact]; the weight derivation [A — one equation, no freedom]; the reading [B]; legs recorded per the gate: 1 stated, 2 PASS, 3 open. Partial stays partial.**