The in-domain floor for the finite-m margin, and two artifacts on the way to it

2026-08-13/14. Measurements only. Nothing here is kernel-checked, nothing here is evidence about RH, and nothing here closes blocker 2.

Recorded because these numbers were produced in a session and existed nowhere else. PROOF-LEDGER.md logs the general shape five times, a quantity carried across contexts without being re-derived where it is used, and a measurement that lives only in a transcript is the same failure with the transcript as the context.

1. The domain, which is where the first artifact came from

The retention claim is about off-line zero pairs at depth y = |β − ½|, and zeros lie in the critical strip, so y < ½ always. It is a geometric constraint, not a modelling convenience; PREPRINT.md states it and depth_uniform.py's eighteen cells tile (0, ½] exactly.

An evolutionary search run against paper_joint reported a configuration with margin per pair −0.786, spacing 2.086 mean gaps, depth 0.979, a 24-site occupancy, and read it as a counterexample. It reproduces here (−0.782; the difference is step and G). It is not a counterexample: y = 0.979 is outside the domain, and at y > ½ the cosh(y·w) pair term runs away and the instrument is being read outside its calibration.

The same spacing and occupancy, at every valid depth, closes:

The search's guard admitted 0.01 < y < 0.99. Clamped to 0 < y < 0.5, the "massive region of catastrophic resonances" disappears, it is massive, and it is entirely outside the strip.

2. The second artifact: refinement has artifacts of its own

Screening at the evaluator's own resolution (step 0.005, G = 60) produces violations that are not there. sparse-24 (every third of 24 sites), s = 2.0000, y = 0.0237 reads −0.004396 at that setting and closes under any refinement:

That much argues for a two-stage evaluator: screen coarse, re-score any positive hit at finer resolution. Two stages are not enough. A second candidate, every-5, s = 4.0, y = 0.011: went +0.000075 coarse and −0.002288 at step 0.0025 / G = 120, i.e. it appeared only at the refinement stage, and died on the full ladder:

The control is a structurally adjacent configuration that closes throughout, which is what identifies the candidate's −0.002288 as an isolated artifact of one setting rather than a feature. A ladder is required, not a refinement.

3. The floor

The tightest converged in-domain margin found is

+0.000073 at every-5 / every-4, s = 4.0000 mean gaps, y = 0.011, stable from step 0.000625 down.

Two things about it. It is roughly 4.6× tighter than the +0.00034 that an evolutionary run and an independent hand scan agreed on earlier, and it sits at a different resonance, s ≈ 4, not the s ≈ 2 both had converged on, and at a shallower depth. Whether the floor keeps falling with s is unmeasured and is the obvious next probe: a pattern in s would be a candidate lemma, and a floor that does not converge would be a different finding entirely.

4. What this is not

No in-domain violation survives refinement, and that is not evidence that none exists. A failed adversarial search bounds the searcher. docs/25's scoring rule is the relevant one: "nobody has attacked it yet" scores as a penalty, not as survival.

Blocker 2's multi-pair statement remains open, and 7df6ed8 (defect #19) records that the k = 1 closure is not it.