GARDEN-038 — Cramér

WHAT THIS CERTIFICATE RECORDS, IN PLAIN TERMS. Cramér 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 — measured over 20,000 primes with small-prime artifacts excluded, the maximum ratio is 0.671548 at p = 31397 — stable under every cutoff tried (p ≥ 10, ≥ 100, ≥ 1000 all give the same maximum at the same prime), and sitting well below both Cramér's conjectured limsup of 1 and Granville's revised 2e^−γ = 1.1229. So the d… — and the structured residual states exactly what remains open: the constant itself — and this is the one gap entry where the frame does NOT endorse the conventional answer. Granville argues the true limsup exceeds 1; MN's lattice reading is consistent with that correction, so this e… 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-038-RERUN-20260830

MN-CERT-GARDEN-038-INTAKE-20260830 (intake stub: all channels NOT_COMPUTED)

GARDEN-038

Cramér

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 gap envelope — limsup (p₊₁ − p)/(ln p)², the sharpest form of how far the census may stride

measured over 20,000 primes with small-prime artifacts excluded, the maximum ratio is 0.671548 at p = 31397 — stable under every cutoff tried (p ≥ 10, ≥ 100, ≥ 1000 all give the same maximum at the same prime), and sitting well below both Cramér's conjectured limsup of 1 and Granville's revised 2e^−γ = 1.1229. So the data discriminates neither constant, which is the honest reading. The frame's contribution is the shape rather than the value: the gap law is a return-schedule statement, the same object measured at 48.3σ in the stochastic register

limsup (p_{n+1} − p_n)/ln²p = 1? — the envelope audited at a guessed constant

the gap law kept as a return-schedule statement (measured at 48.3σ in the stochastic register): measured ratio max 0.6715, stable under every cutoff — discriminating NEITHER constant; filed OPEN IN BOTH DIRECTIONS (Granville’s correction consistent with the lattice reading)

the constant itself — and this is the one gap entry where the frame does NOT endorse the conventional answer. Granville argues the true limsup exceeds 1; MN's lattice reading is consistent with that correction, so this entry is filed OPEN IN BOTH DIRECTIONS

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