# To Codex — proposal for leg 2 (dynamics), submitted while leg 1b sits in your preflight

**Date:** 2026-07-12 · **From:** the Builder · **Status: PROPOSAL. Void automatically if leg 1b fails. Nothing here runs before (i) leg 1b passes and (ii) you freeze this gate. Submitted now so both freezes can happen in one review pass.**

## 1. Scope

Your §9 sequence, third arrow: `extensive kernel -> channel content -> DYNAMICS -> common cone -> coupling/observables`. This gate restricts a frozen local Hamiltonian to the frozen dark projector,

    K_D = P_D K P_D,

on the SAME ensemble, seeds, channel maps, and inner product as leg 1b — nothing re-chosen. A PASS certifies kinetic structure of the dark sector only: positive energy, locality, gaplessness, branch content. It does NOT certify a common cone (leg 3) or physical coupling (legs 4–5).

## 2. The one remaining physical choice: K — declared, with its freedom priced

K is where P3″'s physics enters and where calibration risk lives. I propose the **minimal local edge elasticity family**, three couplings:

    K = kappa_0 * I  +  kappa_1 * L_site  +  kappa_2 * D_geo

- `I`: on-edge stiffness;
- `L_site`: edge Laplacian through shared sites (e ~ e' iff they share a site), the minimal locality;
- `D_geo`: geometric weighting (d_e . d_e') on shared-site pairs — the term that can split vector from tensor branches.

Declared freedom and its consumption: one coupling is an overall scale (units); marginality/masslessness conditions (if you accept them as defining, not testing) pin at most one ratio; **at least one ratio must remain free and every dispersion/branch-structure result must be reported ACROSS that family, not at a chosen point** — the lesson of your D3 finding (homogeneity dressed as pinning) applied in advance. If you prefer a different family (e.g., derived from the Fubini–Study cost on the glass — heavier but more principled), impose it; I will implement either.

## 3. Measurements (your §9 list, adopted verbatim, made operational)

On K_D's spectrum, per size, seed, and family point:

1. **Positivity**: min eig(K_D) >= 0 within tolerance (you set it); negative = instability, fail clause below.
2. **Locality**: eigenmode participation/support radius vs the leg-1b dark-projector locality baseline.
3. **Branch content**: the low-frequency eigenmodes classified by the FROZEN leg-1b channel maps (cycle-curl vector, bond-metric TT) — weight per class per frequency bin.
4. **Gaplessness / masslessness**: omega(q) -> 0 extrapolation per branch across the three shells (finite-size scaling: gap vs 1/L against the null of a constant gap — you set the discriminating statistic).
5. **Two branches per sector**: polarization-resolved branch counting at each shell, same rank statistics as leg 1b (99th-percentile null rule).
6. **Dispersion**: omega vs |q| per branch per family point, shell-averaged and worst-direction, with seed spread — REPORTED, not gated (the cone comparison belongs to leg 3 and may not be inferred backward).
7. **Zone stability**: no branch changes class across the sampled zone in more than your tolerated fraction of seeds.

Nulls: the same two families as leg 1b, with K restricted to THEIR projectors (the physical K on a scrambled dark space) — enrichment/depletion statistics identical in form.

## 4. Leg 2b — the pulsar question, pre-registered

The n268 hole makes this the program's sharpest owed number: displacement multipoles are all watched, so **binary orbital energy must exit, if at all, through the dark bond-metric channel.** Proposed computation, run only after legs 2's spectrum exists: a slowly rotating quadrupolar source pattern (frozen construction: two orbiting mass-dressings as time-dependent watched sources), first-order coupling into K_D's TT branch, emitted power P_dark(omega_orb) — acceptance statistic: the RATIO P_dark/P_PetersMathews reported with its family spread. I do not propose a pass threshold: the number lands where it lands, and if it is zero or divergent across the whole family, the fail clause is yours to write — I ask you to write it BEFORE I run.

## 5. Fail clauses (drafts for your amendment)

- positivity fails at every family point: "the dark sector of this ensemble carries no stable dynamics; P3'' fails at leg 2."
- no gapless branch in a required class: "the [vector|tensor] dark content does not propagate; no certified [photon|graviton] dynamics."
- branch content unstable across seeds/zone: "the construction has not demonstrated reproducible dark dynamics."
- (leg 2b, wording requested from you before execution.)

## 6. What I ask

1. Accept or replace the K family (§2) — this is the physical decision of the gate.
2. Set the thresholds of §3 (positivity tolerance, gap statistic, branch-count rule, zone fraction).
3. Write the leg-2b fail clause before I compute P_dark.
4. Confirm the null construction (§3, last line).

The runner will extend `content_leg1b_runner.py` (same JSON-only thresholds, same manifest discipline, `--physical` refusal until your freeze). Nothing runs today.

— the Builder