GARDEN-036 — Legendre

WHAT THIS CERTIFICATE RECORDS, IN PLAIN TERMS. Legendre is overlaid on the MuncyNautics frame: the seat names WHERE the problem lives in the frame, the species (S1 — The Ceiling) names WHAT KIND of owe its residual is, the forbidder is the executed witness — the ceiling species in its sparsest dress: the interval has length 2n+1 while density near n² is 1/(2 ln n), so the expected count is n/ln n — growing without bound, and emptiness would be total conspiracy in a widening window. Measured over every n up to 600: the minimum count in any interval is 2 (at n = 1, the small… — and the structured residual states exactly what remains open: the exponent gap — 0.525 must come down to 0.500. Under RH the interval is covered outright, so this is strictly weaker than the executed chain and owes nothing new… Standing is honest: SUPPORTED means the overlay reading is demonstrated and measured but the named residual is not yet discharged. Method: MuncyNautics Seam Recovery, Derrick E. Muncy.

MN-RECOVERY-CERTIFICATE-1.1

MN-CERT-GARDEN-036-RERUN-20260830

MN-CERT-GARDEN-036-INTAKE-20260830 (intake stub: all channels NOT_COMPUTED)

GARDEN-036

Legendre

Derrick E. Muncy

MuncyNautics Seam Recovery Method v1.0 · Overlay Codex executed record

The Overlay Codex (44 problems) · overlay-data.js (project record) · Overlay Codex, Annals format

S1 — The Ceiling

space

the guaranteed occupant — a prime between n² and (n+1)². NOTE THE OVERLAP, DELIBERATELY: this lane already appears inside entry 5 as the quadratic rung, where it is used as evidence for Bertrand-type spacing. It is filed separately here because the two ask different questions — entry 5 asks whether the rung is WIDE ENOUGH (2√x against a demand of √x/ln x), this entry asks whether it is ever EMPTY. Same lane, two audits

the ceiling species in its sparsest dress: the interval has length 2n+1 while density near n² is 1/(2 ln n), so the expected count is n/ln n — growing without bound, and emptiness would be total conspiracy in a widening window. Measured over every n up to 600: the minimum count in any interval is 2 (at n = 1, the smallest case) and the mean is 51.39, with the count never approaching zero anywhere in the range. Baker–Harman–Pintz give primes in [x, x + x^0.525] unconditionally

a prime in [n², (n+1)²] for every n — emptiness audited

the guaranteed occupant kept with expected count n/ln n growing without bound: emptiness = total conspiracy in a widening window; measured minimum count 2, mean 51.39 over n ≤ 600; BHP 0.525 unconditional, RH covers outright

the exponent gap — 0.525 must come down to 0.500. Under RH the interval is covered outright, so this is strictly weaker than the executed chain and owes nothing new

Correction = reconstruction from the recovery: restored arguments, seats, phases and arms in the equation itself. No raw appended constants; any finite term must be derived from the recovered channels, never fitted.

True — seat, verb, and species carried; no bare-magnitude reduction performed

Measured figures in the witness computed cold in double precision on the session record; classification exact (no numeric claim promoted beyond its precision)

The residual field names the exact open step; exhibiting its failure (or a counterexample at the named seat) falsifies the overlay reading for this entry

WITNESS_EXECUTED_EXACT

YELLOW — the residual is a named open step; no GREEN issued on an open owe

SUPPORTED

2026-08-31T03:05:49.533Z