A registered result in its source context

QGEO.CARL1.01

Original ledger classification: Proof-near [E] sector lemma (exact geometric sums, derived gamma = 2, 39744+108 inequalities at 0 violations) with the named residues typed (Python-only per GATE.WOLFRAM.02).

Original claim text

The (L1) sector lemma of the CAR continuum majorant, proven modulo elementary steps with explicit constants (one module, 11 checks, ~1 s; discovery probe qgeo_car_l1_sector_lemma_probe.py 11/11, verdict L1-SECTOR-PROVEN-MODULO-ELEMENTARY). THE LEMMA: || K_2N^(q) - K_N^(q) ||_{S1(Lambda_eps)} <= B_q(eps)/N for all 6 undressed sectors on the fixed grid -- the occupied-arc mode sums are EXACT geometric sums (no Riemann-sum error; exact summation replaces Euler-Maclaurin), the entire 1/N structure is the sublattice embedding offset, the leading term is the exact matrix (iu/2MN) f_q'(s) (ratio -> 1 at 6e-4), gamma = 2 is DERIVED and consistent with the measured collision exponents, the Taylor-Lagrange envelope covers every entry and every trace norm with 39744 + 108 inequalities at 0 violations, and r1 = 1 is sharp. STATUS TYPED: (L1, undressed sectors, fixed grid) = PROVEN-MODULO-ELEMENTARY with explicit constants; the named residues -- (a) grid-sup [elementary], (b) L2 uniform FH/RH [hard-technical], (c) L3 uniformity [medium] -- are named, not claimed. GATE.QGEO does not move. No marker move. [2026-08-05 LXII (v778): residue (a) is FULLY CLOSED in the corrected (1 + ln M) form -- (a1) the continuum entrywise supremum, (a2) the configuration-S1 closure with explicit B_conf(eps) (packing + zeta(2)/zeta(3)), and (a3) the refinement remainder with RHO(eps) via exact csc^3 integrals are closed elementarily with zero census violations; (a4) the leading part is honestly LOG-divergent under refinement (jump-driven, C^1-taper-controlled), so the literal v715 clause 'one B(eps) for ALL grid refinements' is FALSE at r1 = 1 with a fixed B(eps) and is REPAIRED to ||D_N^(M)||_1 <= [B0(q,eps) + (J(eps)/2) ln(M+3)]/N, certified at theorem level by the grid Fourier log-sum inequality ||F_q^(M)(eps)||_1 <= c0(q,eps) + (J(eps)/4) ln(M+3), J = 2|f_q'(eps)| (exact nu-twisted-circulant diagonalisation + two exact cyclic summation-by-parts + harmonic comparison; 162-case census to M = 6144, zero violations in all seven clauses; repaired clause verified on all 252 transfer cases; Araki/Powers-Stormer summability survives with explicit closed forms). Honest: the measured slope is J/pi^2 (spike interference) -- the certified J/4 and named J/pi are valid upper bounds, NOT sharp. Residues (b) and (c) stay OPEN; GATE.QGEO does not move.]

Original classification detail

L1-SECTOR-PROVEN-MODULO-ELEMENTARY: (L1) proven modulo elementary steps for all 6 sectors -- exact summation instead of EM, leading term at 6e-4, gamma = 2 derived, 39744+108 inequalities 0 violations, r1 = 1 sharp; residues (a) elementary / (b) hard-technical / (c) medium named, not claimed; GATE.QGEO does not move; 2026-08-05 (v778): residue (a) fully closed in the corrected (1 + ln M) form (the literal v715 all-refinements clause repaired; log coefficient certified, measured slope J/pi^2); (b)/(c) stay open.

Dependencies as recorded

QGEO.CARLIMIT.01; GATE.QGEO.01; v715