Route-KA v1: the interval pipeline against a PUBLISHED certified radius

Leg 61. Audit route (not critical path). Gate answered YES, with a measured caveat that is part of the answer and not a footnote. Runner experiments/p2_route_ka_v1_kawahara.py · data writeup/data/p2_route_ka_v1_kawahara.json · executable gate test_interval_certificate.py tests (13)–(17) · novelty log writeup/novelty/leg_61.md · journal experiments/journal/leg_61.md. No figure: this is a known-answer audit and the established convention is that such legs register none.

Every number quoted below is in the curated JSON.

1. What was missing, stated precisely

Read the validated column of capabilities.py for the certificate stack. Enclosures contain exact rationals. Rigorous bounds dominate float readings. A poisoned iterate is rejected at 5.6e+03. Every one of those is the pipeline agreeing with itself.

solver/target_selection.py looks like the exception: its entry says it "reproduces CLN's published Kawahara radius exactly". It does not run the certificate. It takes CLN's Y_0 and CLN's implied Z_1, puts them into our radii polynomial, and recovers their r_0. That checks four lines of algebra and nothing else. The number is transcribed, not computed.

So before this leg, no output of solver/interval_certificate.py had ever been compared with a certified radius that somebody else published and a referee checked.

2. The problem, and where its truncation came from

Cadiot–Lessard–Nave, arXiv:2302.12877 (SIADS 10.1137/23M1607507) §6: the Kawahara soliton. With Bond number T and speed c, the travelling-wave reduction integrated once is their (75), which in the sign convention of their released code is

F(u) = L u + λ₃ u²,   L = I + λ₁ ∂ₓₓ + λ₂ ∂ₓₓₓₓ,
λ₁ = (1−3T)/(6(1−c)),  λ₂ = (19−30T−45T²)/(360(1−c)),  λ₃ = 3/(4(1−c)).

The paper prints T = 0.35, c = 0.9 and the constants: but not the truncation. It gives ‖DF_e(u₀)⁻¹‖_{2,l} ≤ 4.4, Y₀ ≤ 2.26e−14, and Theorem 6.6's r₀ = 2.27e−14 with uniqueness in B_{0.015}. The discretisation is only in the code: ProofKawahara.jl lines 357–360 give N = 250 cosine modes and half-domain d = 50.

That matters beyond convenience. Lesson 84 requires a known-answer probe to pre-commit its window, and the window is only meaningful at their truncation. It could be pre-committed here because CLN released their package, the exact contrast with arXiv:2604.09949, which LITERATURE_CHECK.md records as unusable partly because no package was released.

3. The sign convention was confirmed, not assumed

Drop λ₂ and the KdV reduction u + λ₁u'' + λ₃u² = 0 has the closed-form soliton u = α sech²(βx) with β² = −1/(4λ₁), α = 6λ₁β²/λ₃. At T = 0.35, c = 0.9 that is α = −0.2, β = √3. Newton from that seed converges to a profile with minimum −0.181650 at the origin, decayed to 4.3e−18 at x = d. CLN's Figure 1 shows a minimum a little above −0.18. The profile lands on their picture before any certificate constant is quoted, which is how a sign slip in λ₁, λ₂, λ₃ was ruled out rather than hoped away. Gate (13).

4. What is certified, and what is not, before the number

The object is the finite Galerkin system F_n(a) = l_n a_n + λ₃ (a∗a)_n, n = 0..N, in even exponential coefficients, l_n = 1 − λ₁k_n² + λ₂k_n⁴, k_n = nπ/d. CLN certify strictly more: the Fourier tail n > N, and the passage from the periodic problem on Ω₀ to the one on ℝ. Their Theorem 6.6 states both conclusions, and the periodic one, a solution in B_{r₀/√|Ω₀|}(U₀) ⊂ X^l_e, is the one comparable to ours.

Because we bound fewer terms, our radius is expected below theirs. That was written down in writeup/novelty/leg_61.md before construction, together with its consequence: a radius below r₀ is evidence of over-optimism, not of a sharper proof.

5. The norms differ, and the conversion is part of the result

This pipeline works in the weighted sup norm ‖a‖_w = max_n w_n|a_n|; CLN work in ℓ²_l. For a vector supported on |n| ≤ N,

‖a‖_{ℓ²_l} ≤ √(2N+1) · supₙ(lₙ/wₙ) · ‖a‖_w ,   then ×√|Ω₀| for CLN's H^l(ℝ).

Measured: sup lₙ/wₙ = 1.000000000000007, √(2N+1) = 22.383, total factor to H^l 223.83. Every step rounds up, so every converted radius here is too large, never too small. Gate (16) checks the inequality against random vectors rather than trusting the algebra.

√(2N+1) is the worst case: the residual spread evenly across all modes. It is not tight, and §8 shows it is the dominant term in the comparison.

6. The run

7. The gate, both readings, magnitudes not booleans

"Does solver/interval_certificate.py, run end to end on the Kawahara problem, produce a certified radius inside CLN's published interval at their truncation?"

READING A: the gate's literal words, on the certified SET. YES. A radii polynomial certifies existence at every r ∈ [r_min, r_max]. Our certified interval [6.77e−15, 2.62e−02] contains CLN's published interval entirely: their r₀ = 2.27e−14 is a certified radius of our polynomial, sitting 0.53 decades above our r_min and 12.06 decades below our r_max. Our r_max also reaches past their uniqueness ball by 1.75×, so the certificate is not closing on something it cannot identify with their soliton.

READING B: the stricter pre-committed window, on r_min itself. NO, by 3.35× (0.53 decades). The window was r* ∈ [2.27e−14, 1.5e−02], a span of 11.8 decades; r_min converted is 6.77e−15, below the lower end. The upper constraint is satisfied with 12.3 decades of headroom.

Neither reading is the answer on its own and neither is suppressed. Reading A is what the gate asked. Reading B is what was written down in advance, and it came out low.

8. Why Reading B came out low: two explanations nominated in advance, both FALSIFIED

The pre-committed reading of a low r_min is over-optimism. Two concrete mechanisms would make that reading correct, and both were tested rather than argued (disciplines 85/90).

Ablation 1: CLN's trace projection. They do not certify the Newton iterate. They project it onto ker T^N_{4,e} so its representation lies in H⁴₀(Ω₀) and extends by zero to ℝ: a step that can only raise the residual. Replicated here (kawahara_trace_projection, the ℓ²_l-nearest point of the kernel): Y₀ moves from 6.769e−15 to 6.940e−15, a factor 1.025. It needed to supply 3.34× and supplied 2.5%. Not it.

Ablation 2: the discarded mode-(N, 2N] tail of u². CLN's Y₀ carries ‖U² − π^N U²‖₂; a finite Galerkin certificate has no such term. Measured: 1.665e−17 in their normalisation, against a gap of 1.59e−14 to explain, 2.98 decades short. Not it either.

What is left is the norm conversion, and the resolution sweep measures it. Across N = 60, 100, 150, 200, 250, 300, Y₀ in our own norm is flat at ≈3.0e−17: it is set by the float64 Newton residual, not by resolution (the coefficients are already at 1.6e−22 by mode 250). The converted r_min nevertheless climbs 3.31e−15 → 7.20e−15, tracking √(2N+1) exactly. The position of our radius relative to CLN's is set by the conversion factor, not by the arithmetic.

So the honest statement is not "we are 3.35× over-optimistic". It is: our upper bound is 3.35× below their upper bound after a deliberately pessimistic conversion, and two upper bounds on different quantities in that relation contradict nothing. The disagreement branch of the gate requires the pipeline to fail where they prove existence, or to close only above their uniqueness ball. It does neither.

9. The resolution of this gate, stated so it is not over-read

Because the comparison passes through √(2N+1), this gate localises the pipeline to a factor of a few: about half a decade. It is not a digit-level check. Said plainly:

• it is ample for what it was built for. Leg 56 reports consistency defects of 1.85e7× and 2.04e11× from this same pipeline. Those are 7 and 11 decades from where this gate's resolution sits, so they clear it with ~6.5 and ~10.5 decades to spare;
• it would be useless for validating a 2× claim, and must never be cited as if it were.

10. The poisoning control

A displaced iterate must be noticed, and the response must scale rather than saturate: a saturating Y₀ would be measuring the arithmetic instead of the iterate. Kicking the 20 lowest modes: Y₀/budget = 4.968e−08 at 1e−12, 4.968e−04 at 1e−08, 4.966e+00 at 1e−04. Exactly linear in the displacement across 8 decades, and the certificate fails to close at 1e−04. Gates (15) and (17).

11. What this leg does NOT claim

Nothing about the Kawahara equation, solitons, or certification methodology. The soliton is proved, published and refereed; the novelty pass returned PROCEED_AS_INTERNAL_AUDIT and no claim is available. The only claimable object is a statement about this repository's own pipeline, and it is the one in §7–§9.