GARDEN-044 — Mersenne prime infinitude

WHAT THIS CERTIFICATE RECORDS, IN PLAIN TERMS. Mersenne prime infinitude 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 — Lenstra–Pomerance–Wagstaff predict (e^γ/ln 2)·ln ln x = 2.569544·ln ln x Mersenne primes with exponent below x. Measured against the known exponents: 14 actual against 4.97 predicted below 1000 (ratio 2.819), 20 against 5.71 below 10⁴ (ratio 3.506) — running well ABOVE the heuristic, and the excess growing rather than … — and the structured residual states exactly what remains open: the infinitude itself — and note the pairing with entry 43 is the point: convergent expectation closes the Fermat census, divergent expectation opens the Mersenne one. Same heuristic, opposite verdicts, neither proven… 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-044-RERUN-20260830

MN-CERT-GARDEN-044-INTAKE-20260830 (intake stub: all channels NOT_COMPUTED)

GARDEN-044

Mersenne prime infinitude

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

issue

the open census — 2^p − 1 prime for infinitely many p, the counterpart to entry 43 and pointing the opposite way

Lenstra–Pomerance–Wagstaff predict (e^γ/ln 2)·ln ln x = 2.569544·ln ln x Mersenne primes with exponent below x. Measured against the known exponents: 14 actual against 4.97 predicted below 1000 (ratio 2.819), 20 against 5.71 below 10⁴ (ratio 3.506) — running well ABOVE the heuristic, and the excess growing rather than settling. Read honestly this is small-sample behaviour, not evidence of a stronger law; the frame's reading is that the census is open because the divergent sum Σ1/p over exponents keeps demanding occupants, the same protection argument that gives N(T) → ∞

2^p − 1 prime for infinitely many p — the census audited

the open census kept with its divergent demand: Σ1/p over exponents diverges — the same protection that forces N(T) → ∞ keeps demanding occupants; measured 2.8–3.5× above Lenstra–Pomerance–Wagstaff (small-sample, read honestly); paired with 43: same heuristic, opposite verdicts

the infinitude itself — and note the pairing with entry 43 is the point: convergent expectation closes the Fermat census, divergent expectation opens the Mersenne one. Same heuristic, opposite verdicts, neither proven

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