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Original ledger classification: Theorem-level [E] closure of residue (a) in the corrected (1 + ln M) form (exact twisted-circulant + summation-by-parts + census, zero violations); the v715 literal clause corrected; residues (b)/(c) open; GATE.QGEO does not move (Python-only per GATE.WOLFRAM.02).

Recorded statement: Remainder (a) of the (L1) majorant lemma FULLY closed in the corrected (1 + ln M) form -- one module from two probes (19 + 19 checks, ~40 s; discovery probes qgeo_grid_supremum_probe.py 19/19 GRIDSUP-SPLIT-CLOSED-LOG-LAW and…

Statement, assumptions and sources

Paper context: input 5 tfpt 3 e8 audit bootstrap

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QGEO.GRIDSUP.01

Original ledger classification: Theorem-level [E] closure of residue (a) in the corrected (1 + ln M) form (exact twisted-circulant + summation-by-parts + census, zero violations); the v715 literal clause corrected; residues (b)/(c) open; GATE.QGEO does not move (Python-only per GATE.WOLFRAM.02).

Original claim text

Remainder (a) of the (L1) majorant lemma FULLY closed in the corrected (1 + ln M) form -- one module from two probes (19 + 19 checks, ~40 s; discovery probes qgeo_grid_supremum_probe.py 19/19 GRIDSUP-SPLIT-CLOSED-LOG-LAW and qgeo_fourier_logsum_probe.py 19/19 FOURIER-LOGSUM-CLOSED, both 2026-08-05). THE FOUR-WAY SPLIT of the v745 residue (a): (a1) the continuum entrywise supremum CLOSED -- Taylor constant 5/8 symbolic, envelopes symbolic, 1-Lipschitz circ; ~1.26e6 census inequalities, ZERO violations (the fixed-grid restriction removed at entry level); (a2) the configuration-S1 closure CLOSED with explicit B_conf(eps) = floor(1/eps)[(5pi/6) zeta(2) eps^-2 + (5/192)(68pi^2/9) zeta(3) eps^-3] (packing + m-th-neighbour pigeonhole; zero violations); (a3) the remainder part under refinement CLOSED uniformly over ALL M <= N/2 with RHO(eps) via exact csc^3 integrals (zero violations); (a4) the leading part honestly LOG-DIVERGENT -- ||F_q^(M)(eps)||_1 grows like a + b ln M with b = (0.10..0.17)|f_q'(eps)|, the growth IS the eps-cut jump (C^1-taper control saturates), and it transfers: the literal v715 clause 'one B(eps) for ALL grid refinements' is FALSE at r1 = 1 with a fixed B(eps). THE SURVIVING INEQUALITY CERTIFIED AT THEOREM LEVEL: exact nu-twisted-circulant diagonalisation (F normal, ||F||_1 = sum |mu_k| exactly) + two exact cyclic summation-by-parts (two lattice spikes of exact mass (1/2M)|f_q'(eps)|; smooth part with the F3 = C3/sin^4 envelope, C3 = 314pi^3/27) + harmonic comparison: ||F_q^(M)(eps)||_1 <= c0(q,eps) + (J(eps)/4) ln(M+3) <= c0' + (J/pi) ln M a fortiori, J = 2|f_q'(eps)|, all constants exact; 162-case census (18 pairs x 9 rungs to M = 6144), ZERO violations in all seven clauses; the REPAIRED (L1) clause ||D_N^(M)||_1 <= [B0(q,eps) + (J(eps)/2) ln(M+3)]/N verified on all 252 transfer cases; Araki/Powers-Stormer summability survives with explicit closed forms (sum_l (A + Bl) 2^-l = 2A + 2B exact; even the extreme joint scaling M = N/2 stays summable). HONEST: the measured slope is J/pi^2 (spike interference) -- the certified J/4 and named J/pi are valid upper bounds, NOT sharp (the average-interference sharpening is not claimed); the constant part of the bound is loose, the log coefficient is the load-bearing content. Residues (b) L2 uniform FH/RH and (c) L3 uniformity of v745 stay UNTOUCHED and OPEN; GATE.QGEO does NOT move. No marker move, NO RH claim.

Original classification detail

GRIDSUP-SPLIT-CLOSED-LOG-LAW + FOURIER-LOGSUM-CLOSED: (a1)-(a3) closed elementarily with explicit constants (zero census violations), (a4) closed by the certified grid Fourier log-sum theorem ||F||_1 <= c0 + (J/4) ln(M+3); repaired (L1) clause [B0 + (J/2) ln(M+3)]/N verified; Araki summability unaffected; measured slope J/pi^2 (bound valid, not sharp); (b)/(c) open; GATE.QGEO does not move; no RH statement.

Dependencies as recorded

QGEO.CARL1.01; QGEO.CARLIMIT.01; v745; v715

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Registered verification modules

  • v778_qgeo_gridsup_logsum.pyRegistered verification module · core
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    QGEO.GRIDSUP.01: Rest (a) des (L1)-Majoranten-Lemmas VOLL geschlossen in der korrigierten (1 + ln M)-Form, EIN Modul aus zwei Probes — **GRIDSUP-SPLIT-CLOSED-LOG-LAW + FOURIER-LOGSUM-CLOSED (19/19 + 19/19): der v745-Rest (a) spaltet vierfach — (a1) das Kontinuums-Supremum GESCHLOSSEN (Taylor 5/8 symbolisch + Hüllkurven symbolisch; ~1.26e6 Zensus-Ungleichungen, 0 Verletzungen), (a2) Konfigurations-S1 GESCHLOSSEN mit explizitem B_conf(ε) (Packung + Nachbar-Leiter + ζ(2)/ζ(3)), (a3) der Verfeinerungs-Rest GESCHLOSSEN mit RHO(ε) via exakten csc³-Integralen (uniform über ALLE M <= N/2), (a4) der führende Teil ist ehrlich LOG-divergent (sprung-getrieben, Taper-kontrolliert) — die wörtliche v715-Klausel 'ein B(ε) für ALLE Gitter-Verfeinerungen' ist bei r1 = 1 FALSCH und wird zu (1 + ln M) repariert; die überlebende Ungleichung auf THEOREM-Niveau zertifiziert: exakte ν-getwistete Zirkulanten-Diagonalisierung + zwei exakte zyklische Summationen-nach-Teilen + harmonischer Vergleich: ||F_q^(M)(ε)||₁ <= c0(q,ε) + (J(ε)/4) ln(M+3), a fortiori (J/π) ln M, J = 2|f_q'(ε)|, alle Konstanten exakte csc-Integrale/Hüllkurven; 162-Fälle-Zensus bis M = 6144 mit 0 Verletzungen in allen sieben Klauseln; die reparierte (L1)-Klausel ||D_N^(M)||₁ <= [B0 + (J/2) ln(M+3)]/N auf allen 252 Transfer-Fällen verifiziert; Araki/Powers-Størmer-Summierbarkeit in expliziter geschlossener Form. EHRLICH: die gemessene Steigung ist J/π² (Spike-Interferenz) — die Schranke ist valide, NICHT scharf. Residuen (b) L2 / (c) L3 unberührt und offen; GATE.QGEO bewegt sich NICHT

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