The window wasn't zero-width. The trial space was.

Route-WES v1, leg 178. Technical companion: TECHNICAL_P2_WES_V1.md. Data: writeup/data/p2_route_wes_v1_space.json. Runner: experiments/p2_route_wes_v1_space.py. Figure: fig66.

Gate answer: YES, under the user's ruling of 2026-08-07, which broke a tie this leg deliberately refused to break itself. That story is the second half of this post.

The thing leg 111 found, and the one knob it never turned

Sixty-seven legs ago this project built a weighted-energy instrument for the a = 0 CLM linearisation, the friendliest object in the repository, and asked it a simple question. You want the operator to be damped at the origin, and for that you need the weight to be singular enough: exponent γ > 3. You also need your trial functions to actually live in the weighted space, and for that you need the weight to be tame enough: γ < 3.

Same number. The window where both hold has width zero, closing exactly at γ = 3. Every admissible weight in leg 111's seven-member family came back with a negative gap converging to −(3 − γ)/2.

That is a real obstruction, and it looks like a fact about the operator. But leg 111 swept exactly one axis, the weight, and held the other one fixed. Its trial space was span{sin kθ}, whose functions vanish at the origin to order p = 1. The membership threshold is not 3 in general; it is 2p + 1. At p = 1 that is 3, which is the same number as the damping threshold, which is why the window closed.

Change p and the two thresholds stop coinciding. This leg turned that knob.

Constrain the trial space at the origin: someone else's idea, and we say so

The move is not ours and cannot be claimed at any strength. Elgindi, Ghoul and Masmoudi (arXiv:1906.05811, Prop. 2.1) prove a coercivity estimate for this operator family on functions that are odd with f′(0) = Hf(0) = 0, in the weighted space ∫|f|²φ with φ = (1 + y²)²/y⁴, and their conclusion is a gap of 1/2. At a = 0, by their own §1, the object is CLM. The trial space is theirs, the weight is theirs, and the −1/2 is theirs. Reproducing a published theorem numerically is not a new theorem.

What is unpublished, and all this leg reports, is the measurement: leg 111's own gate, re-asked on this repository's own operator and quadrature, on a trial space whose vanishing order is not 1.

Four classes were named in a novelty log committed before the module was touched and before a single number existed:

The orders are measured by log-log slope, not asserted. T3 is the falsification control, and it is why this isn't a tautology: it deletes a direction from the trial space exactly as T1 and T2 do, but it doesn't change the vanishing order, so if the window opened for T3 too, the instrument would be measuring "I removed some directions" rather than "the space changed," and the whole result gets withdrawn instead of shipped.

Two things fell out before the main answer

One: EGM's published weight is a member leg 111's own family excluded. Transport (1 + X²)²/X⁴ through X = tan(θ/2) and it is leg 111's family B at γ = 4, exactly, the pointwise ratio is the constant 32 to a relative spread of 2.0e−15. Leg 111's family B stopped at γ = 2 and its family A at γ = 4. The published weight was one slot past the end of the ladder. This is a statement about an enumeration, not a new weight class, and it was predicted in writing before it was measured so the run had a chance to refute it.

Two: that weight is the one that makes the damping constant. For EGM's weight the damping factor D_φ is identically −1/2 in θ, to 4.441e−16. For Chen–Hou's γ = 4 it swings by 1.0. That is the arithmetic reason EGM's constant is −1/2 rather than something near it.

The answer

On T2_egm, the class carrying both EGM hypotheses, with EGM's own weight, at n = 32, 64, 128, 256:

+0.500000 +0.500000 +0.500000 +0.499999667

Positive. Grid-stable: the last two relative refinement steps are 5.890e−07 and 7.409e−08. Admissible on the constrained space with ratio 1.000000 where leg 111 banked 1.677722e+07 (divergent) on the unconstrained one. Exponent margin +3.0. Under Xu's published ceiling of 0.5, and the local half of the form supplies +0.499999667 while the nonlocal Hilbert half supplies −1.4e−14.

Side by side:

Leg 111's zero-width window is a property of leg 111's trial space, not of the operator. The controls agree: the falsification control T3 passes 0 rows at γ > 3, and the reproduction control reproduces leg 111's four banked values to 1.7e−07.

The part where the leg refused to answer its own question

Here is what makes this leg worth reading twice.

Its own pre-committed pass predicate had five clauses, and on the winning row it scored four out of five. The one that failed was clause 3 (sweep the quadrature grading depth over {12, 24, 48, 96} and demand the answer barely moves) and it failed only at the deepest depth, n_grade = 96, where the gap reads −230.7.

The leg knew exactly why. At depth 96 the innermost quadrature panel sits at θ ~ 3e−31, and the order-θ³ cancellation that defines the constrained space is smaller than float64 can represent there. The runner had built its own contamination diagnostic for precisely this, and it reads:

At depth 96 the roundoff floor exceeds the signal by three thousand times. Over the three depths where contamination is below 1, the spread is 2.2839e−07: inside clause 3's own 1e−3 tolerance by a factor of 4.4e+03.

So: the gate's literal wording answered YES. The leg's own stricter predicate answered NO: 0 of 44 rows, on one clause, at one depth, for a reason the leg could demonstrate was about the arithmetic and not about the mathematics.

And the leg did not resolve that itself. It 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 would not write the NO branch's prose, "the coincidence persists", because its own measurements show it doesn't. So it did the only thing safe under both readings: took the YES branch's action (escalate, don't build, don't land), pushed a branch, and parked.

The tie was broken on 2026-08-07, by the user, not by the leg. The ruling: the gate's literal wording governs, the answer is YES, and clause 3 is scoped rather than overruled; it is only ever evaluated at depths where the runner's own diagnostic says the number means something. In the ruling's words, 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."

That distinction is the whole point, and it is why the leg's refusal is preserved in the record rather than deleted now that there's an answer. A leg that relaxes its own clause after seeing the result has no clause. A user who scopes it, on the record, with the reasoning written down, has made a decision that can be argued with later.

What this does not mean, at some length

The ruling came with a bound attached, and it matters more than the number does.

No weighted-energy lane opens. Not "next", not "promising", not ranked. In the user's words: "Not dead" is not "open."

The reasons are all things this leg said about itself first:

• A gap on a constrained trial space is not a certificate. EGM buy the origin conditions with two free modulation parameters. This leg prices nothing and proposes nothing.
• The +0.499999667 is EGM's −1/2, reproduced. Our own novelty pass says it in as many words: "The −1/2 is theirs. Reproducing it numerically is not a new theorem."
• The object is the a = 0 CLM linearisation: one mode, analytic, the friendliest thing in this repository, and the one whose Y₀ is exactly zero for a degenerate reason that is separately banned. A gap here bounds the real target's difficulty from below, never above.
• Plain float64 throughout. Nothing interval-enclosed. No stage claimed, no ban lifted, no link of the L1 → L4 chain moved. Clay stays at ~0.05%.

What this leg is: a method fact. The threshold coincidence that closed the window was an artifact of holding one axis fixed, and now that is measured rather than assumed. That's a real thing to know, and it is not a route.

One more instrument finding, because it changed the answer by 21%

The first assembled run of this leg was wrong, and the check that caught it was built for a different purpose.

At γ > 3 the unconstrained Gram's own entries diverge, by 1.677722e+07 per grading refinement. If you assemble that Gram and then project onto the constrained subspace, you are computing a cancellation between divergent numbers, and an SVD null-space basis satisfies its own constraint only to 1.73e−14. Divergence amplifies that leak. Measured cost at γ = 4 on T2_egm: gap +0.396742 instead of +0.500000, an error of 1.033e−01 on a quantity whose value is 0.5, understating it by 21%.

The fix is an integer-coefficient basis whose constraint residual is exactly 0.0 in float64, contracted against the basis functions pointwise before any weight is applied. And the proof it didn't quietly become a different computation: on the unconstrained class the new machinery is bit-identical to leg 111's own function, 0.00e+00 difference.

Two separate instrument failures in one leg (the assemble-then-project artifact, and the depth-96 roundoff floor) both caught by diagnostics the leg built for itself, and both reported as magnitudes rather than swept up. That is the part worth keeping.