MISSION: ARM B, window asymptotics for the depth-1 damage kernel (support run 5418c63e)

This is a hunt. Nothing in hunts/ is a result, and nothing here is evidence for or against RH (docs/08).

A support run launched by a parent session working hunts/r_c7f779 (run 872d7dce), to answer one bounded question independently and hand the answer back.

The bounded question

hunts/frontier_math/gram_form.py defines ghat(z) = int_{-1/2}^{1/2} cos(sqrt2 t) e^{zt} dt and the damage D(a, s) = -Re ghat(a + i s)^2. A depth-1 damage window is a maximal interval of s on which D(1, s) > 0.

Locate the depth-1 window edges out to s ~ 1e5. Decide whether the window centres minus 2 pi d are (i) linear in d, (ii) bounded and converging to a constant offset, or (iii) something else. Concretely: does every multiple of 2 pi lie inside a depth-1 window for all d? If not, give the first d that does not. If so, give min_d dist(2 pi d, nearest edge) and say whether it is bounded away from 0.

It matters because hunts/r_b9552d/RESULTS-37fb06a9.md §2 derives the atom-reserve ceiling rho* <= 0.153216295 from the measured fact that P = 0 on the critical lattice out to d = 4000. If the windows track 2 pi asymptotically the ceiling is exact; if they drift, the true rho* is slightly below it.

Scope

May write: this directory and one case-log entry in hunts/README.md. May not write: any other hunt directory, zeta/, harness/, lean/, meta/, ontology/, hunts/frontier_math/, or any root markdown file.

The launching brief's deliverables paragraph named hunts/r_c7f779/arm_b/ while its scope paragraph named hunts/support_5418c63e/ and forbade writing into any other hunt directory. The scope paragraph wins, so the work is here and the parent is told where to find it.

What it measures

probe_arm_b.py (numpy + mpmath, ~7 s; --quick in under a second):

• the closed one-term form of ghat, cross-checked against gram_form.damage (read-only import) and against an independent mpmath evaluation of the two-term form at dps = 50;
• the derived 1/s^2, 1/s^3, 1/s^4 expansion of D(1,s), checked against the exact kernel at five ordinates;
• the exact rational identity for D restricted to s in 2 pi Z, and the sign of the cubic that controls it;
• every sign change of D(1, .) on [1e-6, 1e5], by a 0.002 scan plus bisection, with the window/lattice incidence, the centre offsets and the margins;
• the edge-law constants w0, K, W2, against high-precision edge locations at dps = 50 and dps = 60.

Results in RESULTS.md and results.json. The reserved word is not used.