GARDEN-042 — Sierpiński / Riesel numbers

WHAT THIS CERTIFICATE RECORDS, IN PLAIN TERMS. Sierpiński / Riesel numbers is overlaid on the MuncyNautics frame: the seat names WHERE the problem lives in the frame, the species (S5 — The Lane / Carry) names WHAT KIND of owe its residual is, the forbidder is the executed witness — the frame's lane covering in its purest form, and the arithmetic is exact: k = 78557 is covered by {3, 5, 7, 13, 19, 37, 73}, verified here for every n from 1 to 200 with no uncovered exponent. The tiling condition is Σ1/ord₂(p) ≥ 1, and the measured orders are 2, 4, 3, 12, 18, 36 and 9 giving Σ = 1.361111 — comfortabl… — and the structured residual states exactly what remains open: the minimality — proving 78557 is the SMALLEST Sierpiński number requires eliminating five remaining candidates (21181, 22699, 24737, 55459, 67607), each needing one prime found. A finite search, not a theorem: species S… 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-042-RERUN-20260830

MN-CERT-GARDEN-042-INTAKE-20260830 (intake stub: all channels NOT_COMPUTED)

GARDEN-042

Sierpiński / Riesel numbers

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

S5 — The Lane / Carry

cover

the covering system — a FINITE set of prime addresses whose residue classes tile every exponent, so k·2ⁿ+1 is composite for all n

the frame's lane covering in its purest form, and the arithmetic is exact: k = 78557 is covered by {3, 5, 7, 13, 19, 37, 73}, verified here for every n from 1 to 200 with no uncovered exponent. The tiling condition is Σ1/ord₂(p) ≥ 1, and the measured orders are 2, 4, 3, 12, 18, 36 and 9 giving Σ = 1.361111 — comfortably above the threshold, which is WHY the covering closes. Note the naive Σ1/p = 0.846471 is below 1 and would suggest failure: the order, not the prime, is the seat count, and that distinction is the whole mechanism

k·2ⁿ + 1 composite for all n — audited per k

the covering system kept with ORDERS as seat counts: k = 78557 covered by {3,5,7,13,19,37,73}, tiling condition Σ1/ord₂(p) = 1.361111 ≥ 1 (the naive Σ1/p = 0.846 would wrongly predict failure); minimality = five candidates, one prime each — purely computational residual

the minimality — proving 78557 is the SMALLEST Sierpiński number requires eliminating five remaining candidates (21181, 22699, 24737, 55459, 67607), each needing one prime found. A finite search, not a theorem: species S5 with a purely computational residual

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