Route-CP v1 (leg 62) (Cadiot arXiv:2505.03091's scope, settled from the full text

Gate (pre-committed, DIRECTION.md): Does Cadiot arXiv:2505.03091's construction cover an operator whose unbounded part is off-diagonal with a non-decaying tail inverse) i.e. does it already contain leg 58's no-go, or a positive result that contradicts it?

Answer: NO, in the branch's own pre-committed wording: "The gap leg 57's ledger measured is confirmed at full-text depth for the one paper most likely to close it. Bank the located hypotheses as an executable ledger entry; NG may claim novelty against this paper and no further."

Runner experiments/p2_route_cp_v1_cadiot.py → writeup/data/p2_route_cp_v1_cadiot.json. Figure writeup/figures/fig56_route_cp_v1_cadiot.png, rebuilt from that JSON alone by experiments/p2_route_cp_v1_cadiot_evidence.py. Ledger and measurement live in solver/certificate_shapes.py (additive; leg 57's SHAPE_LEDGER is untouched), gated by test_certificate_shapes.py gates 16–25. Every number quoted below is in the JSON.

0. Scope, before anything else

writeup/novelty/leg_62.md, verdict PROCEED_AS_BOOKKEEPING, committed before any of this was built. This leg claims no mathematical novelty of its own. The dominance-hypothesis observation is folklore in print: leg 57's finding, unchanged. What this leg produces is a located, executable scope record plus magnitudes measured against the paper's own examples.

Three things this document does not say:

• not that leg 58's no-go is true. A gap in one paper is not a theorem, and leg 58's gate is a separate question this leg does not touch. This leg caps NG's claim; it does not support it.
• not that the standing ban on re-claiming leg 51's finding at full strength is lifted. The ban's lift condition names this question; answering it does not by itself discharge the ban. The correct bookkeeping change is a narrowing annotation, and that is integration-owned.
• not that Cadiot's paper is deficient. Every clause below is a hypothesis that paper states plainly and discharges on its own examples: as the measurement confirms.

Nothing here moves any link of the L1→L4 chain. Clay stays at ~0.05% behind Walls 1 and 2.

1. Why this leg exists and why leg 57 did not close it

Leg 57 located two sentences in this paper (§2's "the operator L becomes an infinite diagonal matrix L_q", §3's "By construction D is supposed to be diagonally dominant") and correctly concluded that our observation is folklore. It did not establish whether the paper's construction reaches the off-diagonal unbounded part with a non-decaying tail inverse, which is NG's hypothesis, not leg 51's. The standing ban says so verbatim:

lifted by: never, unless a pass resolves whether Cadiot's construction covers a zero diagonal, which is now the live open question, not BDL's

The paper was fetched, not searched: bash Papers/fetch.sh 2505.03091 (egress probe HTTP 200, 1004 KB, 30 pp., extracted with pypdf).

2. CP1: six located clauses, and our operator is outside every one

Each row of CADIOT_SCOPE carries a section/assumption/lemma number and the sentence verbatim. cp_unlocated_rows() is asserted empty (gate 16); a row citing an abstract is inadmissible, which is leg 53's failure mode made executable.

The CLASS clause is the strongest and it is reached before Assumption 1: a Fourier multiplier is diagonal in the Fourier index by construction (§2.2 spells it out, L_q U = (l(n/2q) u_n)_n), and Remark 2.2 says a polynomial l makes L "a linear differential operator with constant coefficients". Our unbounded part is X d/dX = sin θ ∂_θ, whose matrix in the sine basis is bidiagonal with exactly zero diagonal. It is outside the class at the level of the class, not at the level of a hypothesis.

Forward closure. Two genuine forward citations, both read at full text, neither relaxing anything:

• arXiv:2509.17099 (Blanco–Cadiot–Fassler, cites 2505.03091 as [19]), Assumption 1, verbatim: "assume there exists σ₀ > 0 such that |det(l(ξ))| ≥ σ₀ for all ξ ∈ R." This is the systems form, and it is the one that matters: a system is the only route an off-diagonal entry has into this framework, and the systems hypothesis is a non-vanishing determinant of the matrix symbol, checked (their Lemma 2.1) not dropped.
• arXiv:2509.16693 (van der Aalst–Cadiot, cites it as [4]), establishes an explicit positive lower bound on its own symbol. Same hypothesis, discharged by computation.

One independent candidate, the strongest off-diagonal one the search produced: arXiv:2605.03920 (Castro–Gómez-Serrano–Pascual-Caballo, Burgers–Hilbert). Its unbounded part is a transport term. It is not a counterexample, does not cite 2505.03091, and proceeds by Fuchsian ODE theory to a finite-dimensional interval-Newton system: the Chen–Hou pattern again, the shift case certified by abandoning the tail estimate rather than repairing it, on the torus instead of the line.

Not obtained, recorded rather than glossed: Farid–Lancaster, LAA 143:7–17 (1991), Cadiot's [24] and the engine behind Lemma 3.2, paywalled. It did not need to be obtained: the load-bearing hypothesis is reproduced inside Cadiot's own proof (the row above).

3. CP2/CP3: the hypothesis as a number, on the paper's own examples

A hypothesis you can only quote is a sentence. Cadiot states l_min in words for three of his four examples; cadiot_symbol_admissibility computes it from the transcribed symbols and reports the difference.

(The Whitham row reproduces his own sentence: "notice that l(ξ) ≥ l(0) = 1 − c = 0.2 for all ξ ∈ R." The Swift–Hohenberg residual 1.42e−07 is grid resolution at the minimum, which sits at |2πξ| = 1.)

l_min for our operator is 0.0 exactly, and that is not a small number) it is the absence of the quantity. Ratios against it have no referent, so none are reported (discipline 73).

CP3, the one systems example. §5.3 is the only place an off-diagonal entry appears anywhere in the paper, so if the framework reached an off-diagonal unbounded part it would have to be here. It does not:

• off-diagonal entry λ₁λ₂ − 1 = 1/9 = 0.111111, a constant;
• diagonal growth exponent +2.0000;
• |offdiag| / min_i |diag_i| exponent −2.0000, value 2.53e−10 at |ξ| = 1e4.

For our operator the same ratio is flat in k (1/(2μ) at every mode) and infinite at μ = 0. And Cadiot's off-diagonality is in the component index; ours is in the Fourier index: a different axis of the same matrix.

4. CP4/CP5, Lemma 3.2's shift: one finite number, or none at all

This is the load-bearing measurement, and it is a statement about his proof rather than an opinion about his paper. Lemma 3.2 imports Farid–Lancaster's generalized Gershgorin theorem, and to enter it the proof exhibits one s ∈ C, big enough in amplitude, with

|λ_n + s| > r_n / 2      simultaneously at every n,      r_n = Σ_{k≠n} |R_{n,k}|.

For real centres the minimum-modulus such s is purely imaginary, giving the closed form |s| = sqrt( max_n (r_n²/4 − λ_n²)_+ ): that is cadiot_shift_requirement, and it is computed on whatever matrix it is handed.

Linear growth in the truncation means no finite s survives the limit, so Lemma 3.2 cannot be entered at all on our operator) not "the constant is bad", but "the object the proof needs does not exist".

The same contrast in the underlying quantity (fig56 panel A): Cadiot's dominance ratio r_n/|λ_n| decays, with the ladder −0.816, −0.738, −0.659, −0.605 over N = 128, 256, 512, 1024, drifting toward the analytic −1/2 set by his symbol's sqrt growth (it is pre-asymptotic because l = m_T − c subtracts a constant). Ours is flat: exponent +0.0039 at every μ > 0, identically to 13 digits, if anything very slightly increasing; and refused at μ = 0, where all 502 interior rows have an exactly zero diagonal and the ratio has no referent.

CP5 (the control can report the other answer, and it does

The Whitham row above is a positive control run through the identical code path: his symbol l(ξ) = m_T(2πξ) − c (T = 0.5, c = 0.8) exactly on the diagonal, plus a finitely-supported convolution off it) which is what DG(U0) is for his F(u) = M_T u − c u + u².

The convolution is a surrogate and is labelled one. His u₀ is not distributed with the paper. The surrogate is built so it cannot carry the conclusion: its ℓ¹ norm is a dial, and its centre coefficient is set to zero on purpose (a convolution's centre lands on the diagonal, where it only helps his dominance, so zeroing it is the choice that is conservative against his framework).

Lesson 90 says four identical numbers should read as a bug. Here is why they do not:

• the level moves with the operator (|s| = 0.0, 0.28723, 1.98997, 9.99800 as ‖V‖₁ = 0.05, 0.35, 2.0, 10.0;
• the ratio's exponent does not move at all) spread 4.1e−15 across those four, because the numerator is exactly 2‖V‖₁ in the interior, so the exponent belongs to Cadiot's symbol alone;
• and the binding row is the one Assumption 1 is about: |s| = sqrt((r/2)² − l_min²) = sqrt(0.35² − 0.2²) = 0.28723, matching the measurement exactly. The shift his proof needs is set by his own l_min.

5. CP6: three numbers on one dial, kept apart

Two published hypotheses land on the same Λ¹-dissipation dial at different places, and neither is the operator's own hinge. Conflating them is how a hypothesis of a construction gets read as a property of an operator.

• Cadiot Lemma 3.2 admits this family for μ ≥ 0.5, measured: μ = 0.45 diverges (exponent +1.0165), μ = 0.50 needs |s| = 0 exactly. The mechanism is an identity, checked not assumed: r_k = k − 1 exactly for the interior rows of tail_block (max error ≤ 5.7e−14 over μ = 0.25, 0.45, 1.0), against a diagonal μk, so the condition is μk > (k−1)/2.
• BDL assumption (5) admits it only for μ > 1 (δ = 1/(2μ) < 1/2).
• The operator's own hinge (leg 57, a different quantity: the tail inverse's finiteness in M) is μ = 0 exactly.

The factor between the two published thresholds is exactly 2: Gershgorin bounds the whole row sum with a ½, BDL bounds each ratio separately. Both are vacuous at μ = 0, which is the case of interest. This does not contradict leg 57: it measures a different quantity and says so.

That r_k = k − 1 identity is also why the ratio's exponent is μ-independent (+0.0039 at every μ, spread < 1e−9) while its level moves, 3.992, 1.996, 0.998, 0.499 at μ = 0.25, 0.5, 1, 2. A level alone cannot make a tail estimate close.

6. CP7, the gate, and it can answer both ways

cadiot_covers() answers off the located clauses: no, with 6 of 6 clauses failing and 0 of 3 forward citations relaxing anything. Lesson 90 again, the predicate flips to yes two independent ways, both exercised in test_certificate_shapes.py gate 18:

• CP_SYNTHETIC_COVERING_SCOPE, a fictitious clause set that our operator satisfies;
• setting relaxes_the_hypothesis on any CP_FORWARD row.

So no is a property of the located clauses, not of the code.

7. What this establishes, and its ceiling

Establishes. arXiv:2505.03091 does not cover an operator whose unbounded part is off-diagonal with a non-decaying tail inverse, and it fails to on six independent located clauses rather than one. The obstruction is quantitative: Lemma 3.2 needs one finite s serving every mode, and on our operator the required |s| grows linearly in the truncation (exponent +1.005 over M = 128…2048) while on Cadiot's own Whitham operator it is a single number, 0.28723, unchanged across N = 128…1024 and predicted exactly by his own l_min = 0.2. Both forward citations restate the hypothesis; the systems form is |det(l)| ≥ σ₀ > 0.

Ceiling. Four papers is a corpus, not a theorem. Every quote here is transcribed from a full text by a human-equivalent process and is exactly as good as that: the url is on every row so the next pass can check rather than trust. The measurement is float64, no intervals: every number is a measurement of a matrix. The Whitham control's DG(U0) is a surrogate (§5). And a gap in one paper is not a theorem: NG may claim novelty against this paper and no further.