TECHNICAL, Route-WES v1 (leg 178): the zero-width window is a property of the TRIAL SPACE, and the gate answers YES under the user's 2026-08-07 ruling

Runner experiments/p2_route_wes_v1_space.py; evidence experiments/p2_route_wes_v1_space_evidence.py; data writeup/data/p2_route_wes_v1_space.json; figure writeup/figures/fig66_route_wes_v1_space.png (fig66); module solver/energy_coercivity.py (append-only, leg 111's functions untouched); novelty pass writeup/novelty/leg_178.md (committed at 0ed6bde, before the module was appended to and before any number existed); journal experiments/journal/leg_178.md. Every number below is in the JSON and is re-derived from it by the evidence script.

Nothing is claimed that is not a measurement. The construction (a singular-weight energy estimate on a trial space constrained at the origin), the weight, and the constant −1/2 are Elgindi–Ghoul–Masmoudi's, arXiv:1906.05811 Prop. 2.1; the realization dichotomy and the ceiling 1/2 are Xu's, arXiv:2607.19762 §3.1 Prop. 2. This leg contributes the number on this repository's instrument, and its novelty pass says so in advance and in as many words: "The −1/2 is theirs. Reproducing it numerically is not a new theorem."

0. The gate, its answer, and who decided it

"For at least one pre-named alternative weighted-energy construction (different from leg 111's), is the measured coercivity gap positive and grid-stable across two refinements, outside the γ > 3-needs / γ < 3-exists coincidence?"

yes → The zero-width window is a construction artifact, not an operator fact: a genuine third-realization revival. ESCALATE to the user; do not build further under this leg's own authority. no → The coincidence persists under a second, independently-chosen construction. Bank it.

The two readings disagreed and the leg refused to break its own tie, taking the YES branch's action (escalate, do not build, do not land) as the one safe under both readings. The tie was broken by the user on 2026-08-07, not by this leg: §7 below carries the ruling verbatim and the arithmetic it rests on. The leg's refusal is preserved in experiments/journal/leg_178.md rather than deleted.

The JSON's gate_answer.answer field still reads "NO", deliberately: it is the predicate as the runner computed it, unscoped. The ruling scopes how clause 3 is read; it does not rewrite what was measured. The prose is the authority on the gate answer; the JSON is the authority on the numbers.

1. The object, and the arithmetic that made the window close

solver/energy_coercivity.py (leg 111, Route-WE) carries the weighted-L² coercivity form of the a = 0 CLM linearisation: the same operator the other two L1 realizations died on; its coefficient matrix is exactly equal (0.0) to spectral_certificate.bordered_linearization's interior block. The gap is −sup ⟨L h, h⟩_φ / ‖h‖²_φ, solved as a generalized symmetric eigenproblem with explicit whitening, and the damping factor is closed-form D_φ = (3/2) cos θ + (1/2) sin θ (log φ)′.

Two thresholds govern, and leg 111 met them at the same place:

• damping at the origin requires γ > 3;
• membership of the trial functions in L²_φ requires γ < 2p + 1, with p the vanishing order at θ = 0.

Leg 111 held the trial space at span{sin kθ}, where p = 1, so 2p + 1 = 3, the same number. Window (3, 3), width 0.0, and every admissible member had a negative gap converging to −(3 − γ)/2.

Leg 111 swept the weight and held the space fixed. This leg moves the space. Its runner never edits leg 111's functions: control C1 below is the receipt.

2. Axis 1, the trial space: pre-named, and the vanishing order MEASURED

All four classes are subspaces of span{sin kθ : k = 1..N}, cut out by functionals that are exact in closed form in this basis, since (sin kθ)′(0) = k and H(sin kθ)(0) = −1 + (−1)ᵏ:

The order is measured from each class's probe over θ ∈ {1e−2, 1e−3, 1e−4}, not asserted. T2_egm is the class carrying both EGM hypotheses; T1_dprime carries only f′(0) = 0.

Odd trigonometric polynomials vanish to odd order only, so leg 165's exponent-counted p = 2 (window width 2.0) is unattainable here; T1/T2 land at p = 3 and the window is (3, 7), width 4.0. This was registered in the novelty log before the run so it could not be presented afterwards as a strengthening.

3. Axis 2, the weight: eleven members, and P0's exact identity

Family A (φ = (2 sin(θ/2))^{−γ}) at γ ∈ {0, 2, 3, 4, 5, 6, 7}; family B (φ = (2 sin(θ/2))^{−γ} (2 cos(θ/2))^{−2}) at γ ∈ {0, 2, 4}; and E_egm, EGM's own (1 + X²)²/X⁴ transported through X = tan(θ/2), dX = ((1 + X²)/2) dθ, evaluated from its own closed form and not from family B's.

P0: predicted before the run, and it held. φ^E / φ^{B4} is the constant 32:

EGM's published weight IS leg 111's family B at γ = 4, the one member leg 111's seven-weight enumeration excluded (B stopped at γ = 2, A at γ = 4). That is a statement about an enumeration, not a new weight class, and the novelty pass pre-registered it as a prediction "so the run can refute it."

And it is the weight that makes the damping constant. D_φ ≡ −1/2 identically in θ, max deviation 2.220e−16 (B4_egm) and 4.441e−16 (E_egm), against 1.000e+00 for A4_chen_hou. That is the arithmetic reason EGM's constant is exactly −1/2.

4. The instrument finding: assemble-then-project is INVALID here, by 1.033e−01 on a quantity whose value is 0.5

At γ > 3 the unconstrained Gram's own entries are divergent (∫ sin jθ sin kθ θ^{−γ} ~ jk ∫ θ^{2−γ}), growing by 1.677722e+07 per grading refinement. Restricting after assembly computes the constrained form as a cancellation between divergent numbers, and an SVD null-space basis satisfies its constraint only to ~1e−14, a leak that divergence then amplifies. Measured at γ = 4:

The artifact understates the T2 gap by 21%. The repair is an integer-coefficient basis whose constraint residual is exactly zero in float64 (v_j = (j+1)e_j − j e_{j+1} for T1, w_m = u_{m+1}v_m − u_m v_{m+1} for T2) contracted against the basis functions pointwise before any weight is applied, so the cancellation happens at scale ε·K·θ rather than at the scale of a divergent integral. Lesson 86, caught by an instrument built for a different purpose. Every gate-answering number in this document is from the exactly-constrained basis.

5. The measurements

5.1 T2_egm, the EGM class (gap ladder at n = 32, 64, 128, 256, μ = 0 exactly

A4 approaching 0.5 from above is not a defect: a Galerkin Rayleigh quotient over nested subspaces is an upper bound on the limit, so it must. The pre-committed ceiling clause (gap ≤ 0.5 + 1e−9) is strict enough to fail it by 3.765e−05; B4/E pass it exactly because their D_φ is constant. A5's +0.894630 exceeding the published ceiling is the known-answer window (lesson 84) firing as designed, and the contamination diagnostic flags it ten orders of magnitude above the clean rows: the two instruments agree on which rows are trustworthy.

5.2 The gate-answering row, in full

T2_egm | E_egm, γ = 4, μ = 0:

5.3 Against leg 111, the same instrument, one axis moved

The coincidence does not persist. It was a property of p = 1, not of the operator.

5.4 The other three classes

• T1_dprime, every gap ≤ −1 (E_egm: −5.778 −8.160 −11.500 −16.205). Required: its point-mode intersection has dimension 1, so Xu's published eigenvalue 1 lives in the trial space and forces gap ≤ −1. An internal consistency check, not a defect.
• T2_egm, point-mode intersection dimension 0 (singular values 3.2361, 1.2361): the two origin constraints annihilate span{sin θ, sin 2θ} exactly, so the origin constraints ARE the modulation and modulate=True/False must agree. This is this leg's own arithmetic (prediction P5): the run-time fetch of Xu confirmed §3.2 attributes mode removal to centering, a different mechanism, and does not state this.
• T3_hilbert_only, the falsification control: inadmissible at every γ > 3 (ratio 1.677722e+07, margin −1.0), 0 passing rows. A class that removes a direction without changing the vanishing order does not open the window, so the instrument is measuring the space, not dimension reduction.

6. Controls, all four, with magnitudes

• C1 (reproduction). Leg 111's four banked n = 256 gaps reproduce through the untouched path: A0 −1.499886 → −1.4998861652 (1.652e−07), A2 −0.499924 → −0.4999241101 (1.101e−07), B0 −1.499924 → −1.4999241101 (1.101e−07), B2 −0.499962 → −0.4999620551 (5.506e−08); tolerance 1e−5. PASS. On T0_unconstrained the appended machinery is bit-identical to leg 111's own function: 0.00e+00 at n = 16 and n = 32 on A0 and A2. Had this moved, the append was not append-only in effect and the whole run is void.
• C2 (the predicate must be able to say NO). T3_hilbert_only passes 0 rows at γ > 3, and fails on the admissibility clause specifically, as predicted (P4). PASS. Had it passed, this leg's result is withdrawn, not shipped.
• C3 (positive control, lesson 90). Leg 111's Λ¹ dissipation re-run through the appended constrained path, μ: 0 → 2, flat weight: T0 −5.51616 → −1.89349 → −0.87236 → −0.13831 → +0.49674; T2 −2.85389 → −0.03295 → +1.12444 → +2.04484 → +2.87951. Both sign-flipping and monotone. PASS.
• C4 (no borrowing across objects). μ > 0 is a different operator, not a different realization. No μ > 0 row answers this gate; every gate-answering row is at μ = 0 exactly, and the μ > 0 block sits in its own JSON section so the distinction is visible rather than assumed.

P6, the prediction made to be negative, and it was better than predicted. The gap does not grow without bound toward γ = 7: it degrades into contamination instead (A5 +0.894630 at contamination 4.0e−07, A6 −78356.36 at 1.2e+05, A7 −94793.53 at 3.2e+15, and A7 is flatly inadmissible). That is a stronger statement than the saturation predicted, and is reported as such rather than as a confirmation.

A lesson-90 flag on this leg's own output. T0_unconstrained returns identical ladders at γ = 5, 6, 7 (−0.044755 −0.022898 −0.011583 −0.005825). Three different weights cannot legitimately give one number: that computation has no content, the whitening has dropped essentially everything, and those rows are already excluded by the admissibility clause (margins −2, −3, −4). Recorded rather than passed over, because four identical numbers are exactly what leg 53 mistook for a finding.

7. Clause 3, the disagreement, and the ruling that scoped it

7.1 The disagreement, as measured

Every decisive row failed on exactly one of the five clauses (clause 3, quadrature-depth stability) and only at the deepest pre-committed grading depth, n_grade = 96. From quadrature_stability["T2_egm|E_egm"]:

At n_grade = 96 the innermost quadrature panel sits at θ ~ 3e−31, where the order-θ³ cancellation that defines the constrained space is below float64's ability to represent it. The contamination diagnostic (the weighted energy of a bound on the pointwise cancellation error, divided by the weighted energy of the basis function) reads 3.066e+03 there: the roundoff floor exceeds the signal by three thousand times.

The failure is the instrument stopping, not the mathematics moving. The four depths split across 24 orders of magnitude of contamination, with no depth remotely near the value 1.

7.2 The ruling (the user's, dated 2026-08-07

RULING 1) the gate's LITERAL wording governs. The answer is YES. Clause 3 of the leg's own five-clause predicate is scoped, not overruled: it is only ever evaluated at grading depths where the runner's own contamination diagnostic is below 1.

The reasoning, so it is on the record and not re-litigated: clause 3 did not detect instability in the mathematics. It detected that the instrument stops working at n_grade = 96, where the innermost panel sits at θ ~ 3e−31 and contamination reads 3.1e+03: the roundoff floor exceeding the signal by three thousand times. Over the three depths where contamination is below 1 the spread is 2.2839e−07. This is not relaxing a pre-registration after seeing the answer; it is declining to read a number the leg's own diagnostic declares meaningless, which is lesson 86 and which leg 178 itself cites. The leg was right to refuse to make this call unilaterally, and right to take the YES branch's action (escalate, do not build, do not land) as the one safe under both readings.

Clause 3 as it now reads: at fixed n = 128, sweeping the grading depth n_grade ∈ {12, 24, 48, 96} gives relative spread ≤ 1e−3 (verbatim as §7 of the novelty log wrote it) with one scope attached: evaluated only at depths whose contamination diagnostic is < 1.

This was not the leg's call and is not credited to it. The leg wrote: "I am not permitted to resolve this by relaxing my own pre-committed clause after seeing the answer, and I have not." It also declined to write the NO branch's prose ("the coincidence persists under a second, independently-chosen construction") because §5.3's magnitudes falsify it. It escalated. The record of that refusal is preserved deliberately.

The pre-committed ladder {12, 24, 48, 96} is NOT changed. The ruling scopes how a depth is read, not which depths are run, so depth 96 keeps reporting its diagnostic rather than being deleted from future sweeps on this module.

8. What does NOT follow: the bounding ruling, same date

The goal is now a full Clay solve… EGM buy the origin conditions with two free modulation parameters, so a gap on a constrained trial space is not a certificate; the +0.499999667 is EGM's published −1/2 reproduced numerically and leg 178 says so in as many words; and the object is the a = 0 CLM linearisation, whose Y₀ is exactly zero for the banned degenerate reason. Do NOT open a weighted-energy lane. "Not dead" is not "open."

Explicitly, and none of this is loosened by the YES:

• No weighted-energy lane opens. Not ranked, not proposed, not priced. This is a method fact about a constrained trial space.
• A gap on a constrained trial space is not a certificate. EGM buy f′(0) = Hf(0) = 0 with two free modulation parameters (leg 165's caveat, carried verbatim).
• The +0.499999667 is EGM's −1/2, reproduced on this instrument. Not a new theorem.
• Clause S7 binds. The object is the a = 0 CLM linearisation: one mode, analytic, the friendliest object in this repository, and the one whose Y₀ is exactly zero for a separately banned degenerate reason. A gap measured here bounds HL_S2_nonsymmetric's difficulty from below, never above.
• B's space axis is not re-opened. That clause concerns the ℓ¹_w coefficient space of the radii-polynomial machinery, which shares zero quantities with this weighted-L² energy form: no Y₀, Z₀, Z₁, Z₂, no approximate inverse A, no A_K ⊕ A_tail split, no border, no ℓ¹_w norm.
• No stage claimed, no ban lifted, no link of the L1 → L4 chain moved. Clay stays at ~0.05%.
• Plain float64 throughout; nothing interval-enclosed.

capabilities.py's leg-111 entry is left exactly as leg 111 wrote it. Its scope line ("the window has ZERO width ON THIS UNCONSTRAINED ODD-SINE TRIAL SPACE (p = 1 vanishing order)") is accurate as written and this leg's T2_egm is a different, constrained, p = 3 space that does not contradict it. Re-scoping that entry was raised by the leg as an open question and was not decided by the 2026-08-07 ruling, which decided the gate reading only.