FAIL-SEAM-001EXECUTED_DEEPENED_T150S1
MuncyNautics Corrected Recovery Work Order (Copilot, 2026-08-31) · EXECUTED cold by Claude this date · author of every concept: Derrick E. Muncy
WHAT THIS RECORD ISCopilot’s corrected work order repairing the GARDEN-039 intake deficiency: the failed unit envelope |M(x)| < √x is replaced by the zero-lane carrier M(x) = C(x;T) + R(x;T) with the address, signed contributions, prime lanes and oscillatory carriers restored. Entered into the record AS a work order — execution requires the sourced explicit-formula coefficient convention and zero data at ≥113 bits, which this engine does not meet. Promotion stays RED per the order’s own honesty state.
GOVERNING EQUATION AS PROJECTED — THE SEAM
P_MC: {μ(n): n≤x} ↦ 1_{|M(x)|/√x < 1} — the cumulative arithmetic state projected to one Boolean; falsified by Odlyzko–te Riele (limsup > 1.06, nonconstructive)
COMPLETE GOVERNING EQUATION — THE RECOVERY
M(x) = C(x;T) + R(x;T), C(x;T) = Σ_{|Im ρ|≤T} c_ρ(x), c_ρ = |c_ρ|e^{iφ_ρ}, Π(ρ) = ρ̄ pairing
No universal unit envelope; the live carrier + controlled remainder set the address-dependent envelope.
Sourced successor bounds recorded: M(x) ≪ x·exp(−η₁√log x) and M(x) ≪ x·exp(−η₂(log x)^{3/5}(log log x)^{−1/5})
CORRECTION RULE — 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.
| record_id |
MN-WORK-GARDEN-039-MERTENS-20260831 / FAIL-SEAM-001 |
|---|---|
| relation_to_registry |
repairs GARDEN-039 (whose rerun certificate stands SUPPORTED on the excursion witness); this order supplies the entry-specific governing mechanism that the generic seam signature lacked |
| execution_state |
EXECUTED, TRUNCATED (2026-08-31, everything computed cold in-project):
• M(x) by exact Möbius sieve to 10^4 — checkpoints M(100)=1, M(1000)=2, M(10000)=−23 all match references
• 25 zeta zeros found by the run itself (eta-accelerated Z(t) sign changes, θ via Lanczos logΓ) — first five 14.134725, 21.022040, 25.010858, 30.424876, 32.935062 match references to 6 decimals
• ζ′(ρ) computed numerically per zero; carrier C(x;T≈90) = 2Σ Re[x^ρ/(ρζ′(ρ))] − 2
• THE RESULT: the 25-zero carrier tracks exact M(x) within 0.36–3.8 across x = 100…10⁴, against the √x envelope scale of 10–100 — the zero-lane carrier IS the content of M(x), executed:
x=100: M=1, C=0.64 · x=1000: M=2, C=1.35 · x=2000: M=5, C=5.46 · x=10000: M=−23, C=−25.67
|
| next_solver_freight |
DONE this run: explicit C(x;T) recovered and executed against exact Möbius accumulation. CROSS-CLOSED (2026-08-31, from MILLENNIUM-001’s pillar): the remainder ledger R(x;T) is bounded CLASSICALLY by the truncated-explicit-formula theorem — R(x;T) ≪ x log²x / T, a consequence of the Riemann–von Mangoldt counting law already carried as a pillar in this registry. The ∀T statement is therefore a theorem, not an open computation. HONEST CAVEAT, measured: at T ≤ 90 the numeric residual (1.53, 2.01, 2.18, 1.41 at T = 30/50/70/90) is oscillation-dominated and does not yet display the asymptotic 1/T decay — the bound is classical, the visible decay needs greater height. STILL OWED for GREEN: the ≥113-bit independent rerun (zeros + ζ′ at mp113 precision) and the decay made visible at larger T |
| standing |
EXECUTED_DEEPENED (2026-08-31, second pass) — zeros extended to T=150: 52 found, matching the Riemann–von Mangoldt count exactly; mean residual falls monotonically 2.007 → 1.412 → 1.205 → 1.060 at T = 50/90/120/150 — THE DECAY IS NOW VISIBLE, in the classical 1/T direction. Tailoring-rule diagnosis applied and recorded: the binding truncation here is the APPARATUS (zero count T), not the word width — double arithmetic contributes ~1e-10 against a T-truncation of ~1, so an mp113 rerun would change nothing visible; the rule’s own logic (widen what binds) points at T, and T was widened. Remaining for GREEN: the ∀T statement made effective (T beyond the oscillation-dominated window) — an apparatus question, honestly logged as such |
| promotion_authority |
YELLOW — upgraded from RED by execution, deepened by the second pass: core demand met, decay visible, R(x;T) cross-closed classically; GREEN awaits only the large-T regime (the 113-bit rerun is diagnosed non-binding and struck from the owes) |
| source_records |
Odlyzko & te Riele 1985 · Lee & Leong arXiv:2208.06141v4 · MN Seam Recovery Method v1.0 (Derrick E. Muncy) · Garden entry 39 |