Depth-1 damage windows do not drift: the first correction to D(1,s) is odd so it translates the window rather than widening it, D(1, 2 pi d) > 0 for every integer d >= 1 by an exact cubic with no d_max, and a 0.002 scan of [1e-6, 1e5] finds 31830 sign changes with all 15915 lattice points inside a window.
hunts/support_5418c63e (Record 86 of 98 in chronological sequence)
Guiding Question
Do the windows of D(1, s) > 0 track 2 pi asymptotically or drift, and does every multiple of 2 pi lie inside a depth-1 window for all d >= 1?
Method & Verification
ghat was put over the common denominator z^2+2 to an exact one-term form, D restricted to 2 pi Z reduced to a cubic P(u) with one nonnegative root below (2 pi)^2, and a 0.002 scan plus bisection on [1e-6, 1e5] was checked against mpmath dps=60 edge locations.
Lineage & Relationships
Primary Sources (at pin 8fa46e134)
Editorial Notes
No HANDBACK at the pin. Question filled from the bounded question in MISSION.md. Raw RESULTS H1 is only a title; the headline is taken from the RESULTS status paragraph and the case log. Evidence grade is ordinary mathematical derivation for D(1, 2 pi d) > 0 for all d >= 1; the scan is measured confirmation. d = 0 is the one exception (D(1,0) = -0.9116) and never arises in P because the sum is over p != q. One expansion error (expanding 1/S at 2 pi d instead of keeping the x term) was caught before the number was written down. Disposition maps settled to completed.
Date Provenance
commit 82b80ff41176b13a59beab61e94b527140ddc94f, hunts/support_5418c63e/RESULTS.md, author 2026-08-25T08:46:52-05:00