Spike 1 Step C (relaxing to the Chen–Hou profile: the gate (technical)

Status (2026-07-25): PARTIAL) 3 of 4 pre-committed predicate checks pass; the far-field exponent check fails. This does NOT pass the gate. Reported as PARTIAL; goalposts not moved.

Step C relaxes the Step-B machine from an analytic initial guess toward the Chen–Hou 2D Boussinesq self-similar profile and tests it against a predicate that was locked before the run (experiments/spike1_stepC_gate.py, committed eabb418; the anti-self-deception record). Evidence: writeup/figures/fig11_spike1_stepC_gate.png from committed writeup/data/spike1_stepC_gate.json (python writeup/3_spikes/spike1_stepC_evidence.py).

1. The pre-committed predicate (user-approved 2026-07-25)

Gauge-honest (the normalization freezes the origin slopes, so c_l is an input we pin to the Chen–Hou gauge; the real tests are the gauge-invariant exponent + the shape):

1. α = c_ω/c_l → −0.3424 ± 5%
2. far-field ω(r,β*) ∼ r^α fitted over ≥1 decade, exponent within ~10%
3. shape: anisotropy |ω_y| < 0.23 |ω_x| (2.24), sane sign structure
4. resolution-stable

2. The drift, diagnosed and fixed

The first relaxation runs drifted: the gauge-invariant α settled near −0.35 (right), but c_l, c_ω drifted individually and the run eventually destabilized (~step 8000: residual jump, growing negative lobe). Diagnosis (experiments/diagnose_stepC_drift.py): a near-origin truncation artifact, not a fundamental instability, the c_l drift roughly halves under n_r refinement (300→450: drift/1k −0.144 → −0.074) and worsens as r_min shrinks (poorly- conditioned tiny-r cells: r_min 1e-3/1e-4/1e-5 → −0.111/−0.144/−0.200).

Fix: gauge renormalization (run(renorm=True)): discretely enforce the normalization (2.12) by re-pinning ω_x(0), η_x(0) to their initial values each step, holding c_l = 2 η_x(0)/ω_x(0) fixed regardless of the truncation slip. This arrests the drift (c_l held at 3.06 vs collapsing to 2.24) and stabilizes the run. The paper's continuous (2.12) is the exact analogue; the discrete version is a weak restoring correction (factors ≈ 1).

3. Result (resolution study, renorm on, r_min=1e-3)

What matches Chen–Hou well (checks 1, 3, 4 PASS): - c_ω to < 0.5% across every config (−1.026…−1.031 vs −1.0294), and c_ω is the real result: it evolves through u_x(0) to the profile value while c_l is held at the gauge. - α = c_ω/c_l ≈ −0.335 to ~2%, resolution-stable (varies < 0.003 across n_r). - Anisotropy ≈ 0.026 ≪ 0.23: the strong x/y anisotropy of the profile (2.24) is reproduced.

What fails (check 2 FAIL): the directly fitted far-field radial exponent is ≈ −0.31, off by ~7–13%, and it moves the wrong way with n_r. Two honest causes: - Protocol confound. The study used a fixed step budget (2500), so higher-n_r runs (finer Δρ) reach smaller τ; they are under-relaxed, and the slow r^{−1/3} tail is the last thing to form. So the n_r-trend of the far-field fit is not a clean resolution signal. (A fixed-τ or fixed-residual protocol would fix this: flagged as future work, NOT re-run to chase a pass.) - POC limitation. Our domain (r_max 1e5–1e6) is tiny next to the paper's ~1e15, and the outer boundary steepens the tail near r_max; we have no semi-analytic r^α far-field split and only 2nd–3rd-order near-origin accuracy, vs the paper's 6th–8th-order B-splines. A clean far-field match needs that apparatus.

4. Honest verdict

Step C is a PARTIAL success and does not pass the pre-committed gate. The machinery reproduces the Chen–Hou profile's modulation invariants (c_ω to <0.5%, α to 2%) and its anisotropic near-field structure, strong evidence the scheme captures the right physics, but it does not reproduce the full r^{−1/3} far-field tail at POC fidelity, which is exactly the part the paper built its heavy apparatus for. This is Tier-1/2 progress: the machine is validated to capture the self-similar profile's core, with its POC limits honestly located.

And the standing frame is unchanged: even a clean pass would reproduce a proven result (Chen–Hou 2022) on a toy model across Wall C; it validates our machinery; it is not novel and not a proof. Clay odds remain ~0.05%; the lottery ticket is past this solver.

5. Reproduce

python experiments/diagnose_stepC_drift.py            # the drift diagnosis
python experiments/spike1_stepC_gate.py --logged --steps 2500   # the gate (writes data + predicate)
python writeup/3_spikes/spike1_stepC_evidence.py               # rebuild fig11 from committed data