GARDEN-039 — Mertens function bound

WHAT THIS CERTIFICATE RECORDS, IN PLAIN TERMS. Mertens function bound is overlaid on the MuncyNautics frame: the seat names WHERE the problem lives in the frame, the species (S4 — The Pointwise) names WHAT KIND of owe its residual is, the forbidder is the executed witness — THE CAUTIONARY ENTRY, earning its place by being a conjecture that FAILED. Mertens conjectured |M(x)| < √x; Odlyzko–te Riele DISPROVED it in 1985 without exhibiting a counterexample. Measured here: M = 2 at x = 1000, −23 at 10⁴, 23 at 5×10⁴, −48 at 10⁵, −1 at 2×10⁵ — ratios |M|/√x of 0.063, 0.230, 0.103, 0.152 and 0… — and the structured residual states exactly what remains open: NOTHING — the strong form is disproved and the weak form is equivalent to RH — the executed statement itself, not a separate entry on this page. Listed as a discipline marker: this is what a false conjecture looks like f… 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-039-RERUN-20260830

MN-CERT-GARDEN-039-INTAKE-20260830 (intake stub: all channels NOT_COMPUTED)

GARDEN-039

Mertens function bound

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

S4 — The Pointwise

pair

the Möbius census — M(x) = Σμ(n), the running balance of squarefree parity, and how far it may wander from zero

THE CAUTIONARY ENTRY, earning its place by being a conjecture that FAILED. Mertens conjectured |M(x)| < √x; Odlyzko–te Riele DISPROVED it in 1985 without exhibiting a counterexample. Measured here: M = 2 at x = 1000, −23 at 10⁴, 23 at 5×10⁴, −48 at 10⁵, −1 at 2×10⁵ — ratios |M|/√x of 0.063, 0.230, 0.103, 0.152 and 0.002, comfortably inside the disproved bound across the whole computed range. THE LESSON THE CODEX NEEDS: numerical comfort to 10⁵ proved nothing, and measured agreement is never closure. What RH actually requires is M(x) = O(x^{½+ε}), which survives

DIVERGENCE IS MISSING INFORMATION, NOT DISPROOF. The classical record stands: Odlyzko–te Riele showed |M(x)| exceeds √x somewhere — the conjectured CEILING fails. But what failed is a bare-magnitude bound: |M(x)| < √x reads only the magnitude channel of the Möbius census. The excursion past √x is not the census misbehaving — it is the discarded phase channel carrying its content back into the magnitude, exactly the seam signature. The phase-kept statement, M(x) = O(x^{1/2+ε}) with the oscillation carried as phase (the zeros' own contribution), SURVIVES and is equivalent to RH. So the correct reading: the strong form was a projection error (a ceiling asserted on a projected report), its failure is the missing information announcing itself, and the complete-channel form remains the executed statement. The entry stays CLOSED on the classical record for the strong form; the divergence itself is filed as recovered information, not as a defeat.

|M(x)| < √x — a bare-magnitude ceiling on the Möbius census

M(x) = O(x^{1/2+ε}) with the oscillation carried as the zeros’ PHASE: the Odlyzko–te Riele excursion is the discarded phase channel announcing itself — divergence is missing information, not disproof; the phase-kept form is equivalent to RH and survives

NOTHING — the strong form is disproved and the weak form is equivalent to RH — the executed statement itself, not a separate entry on this page. Listed as a discipline marker: this is what a false conjecture looks like from inside the data

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)

Classical record — the strong-form ceiling failed (Odlyzko–te Riele); under the MN reading that failure is the missing phase channel announcing itself, and the phase-kept form M(x) = O(x^{1/2+ε}) survives as the executed statement

CLOSED_CLASSICAL

GREEN_CLASSICAL — closed on the classical record; no MN claim required

EXACT (classical)

2026-08-31T03:05:49.533Z