The charged scaling limit and Type III identification

SEAM.MMST.TYPEIII.CHARGED.01

Original ledger classification: Open (research contract; master scaling-limit theorem; children listed here not back) | Round-3: C1/C2a/TV and scalar C2b subclaims proved [E]; ALG2 implication refuted [X]; CF/DG and ALG-EXH remain Open [O].

Original claim text

THE MASTER SCALING-LIMIT THEOREM FOR THE CHARGED/TYPE-III IDENTIFICATION [O] (registered 2026-08-29, review wave 5). DEMAND: ONE scaling-limit theorem Psi_lambda,N -> Psi_lambda strongly on the finite-energy core + crossed-product continuity, from which B ≅ (E8)_1, mu = 1, type III_1, unique rotational KMS, unique conditional expectation, deck equidistribution, and modular invariance all follow as corollaries. CHILDREN (listed here, not listing back): SEAM.SIMPLECURRENT.GENERATOR.01, SEAM.STATE.RPMIXING.01 continuum leg, ALPHA.QUILLEN.EXACT.01 rigidity premise, AX.P1.01 reading. CITED DOCUMENT: articles/2026-08-29/holomorphic_kms_extension_en.tex (13 pp): unconditional rotational-KMS uniqueness via KLM + Longo-Tanimoto GIVEN the identification; P1 = theorem-conditional on seam identification + beta_angle = 2pi. KILL: the scaling limit fails to exist; uniqueness of rotational KMS fails; type III_1 fails; a competing conditional expectation exists. Status: Open [O]. No marker move on children. EXECUTED UPDATE (2026-08-30, review wave 7, v1006): five lemmas proved in-house -- L3 crossed-product, UGF K_G=pi^2/4 (measured plateau ~1.023), TEL-a N^{-2} with exact isometries (A_TEL=5120(1+pi/2), measured envelope ~3.42), integer D5+A3 pairings, Z4 outerness in the limit + finite-inner correction (256-dim gauge unitary is inner). Residual boxed: TEL-B-EXTERNAL (Hankel C_R; HS max 2.105079358 / fit 2.107565497 / v998 bound 2.11) + ALG-EXH (two Buchholz-Verch estimates). Display stays [O]; no marker move. MONDAY MORNING (2026-08-31, v1016): TEL-B third (certified-tail) route BLOCKED -- N^2 D_N grows 530 -> 19032, remainder UV-supported. TEL-B-EXTERNAL (2026-09-04 reduction): triangle bound via cover identity + sawtooth split; (A) lattice piece certified numerically for all N, ||R^sm_N|| <= 1.7833 (all-mode BV, one float64 TV constant); (B) explicit piece reduced to Sigma|Res_N|^2 <= 0.2769 (measured 0.1550), open items C1 (C* <= 0.1525 enclosure), C2a (|phi_N| <= 0.51/N), C2b (cell oscillation of r); numerically 2.57 < 3, conditional analytic 2.9495 < 3. Status unchanged [O]. N-uniform CAR nuclearity PROVED (nu_N <= exp(16 C_beta); C_beta = (67.97, 29.73, 12.15) for beta = (0.5, 1, 2)) as ALG specialist compactness input; ALG-EXH projector-identification line UNCHANGED. Display stays [O]. EXECUTED UPDATE (2026-09-05, v1022/v1025): TEL-B C1 is interval-certified at C_star<=0.1520453048...<0.1525; C2a is analytically proved with N sup|phi_N|<0.509 for every even N>=16; the native 256-bit/5812-cell TV certificate gives TV_p(F'')<=15.817695962988924...<15.928674 and K3_TV<2.517464500835. The analytic C2b scalar reduction plus Arb arithmetic gives c_potential=0.399224424139106...<0.400. Full C2b remains open exactly at CF<=0.139/sqrt(N) and DG<=0.060/sqrt(N). The old ALG2 Vandermonde-to-generated-algebra implication is refuted by an exact M2 countermodel; FE-GEN or a precise gauge/code-projected algebra is required and ALG1 is conditional on the same generated-sector completeness. MMST/type-III identification remains Open [O]. ROUND-4 CURRENT: v1026 proves the relaxed uniform norm ||R_N||HS<2.995906<3 for every evenN>=16 (CF/DG/CROSS plus validated Fourier aliases). Historical2.11 and finite1.7833 are not promoted. The residual is microscopic one-edge FE-GEN, actual-word/joint-adjoint tails, ALG-EXH and MMST identification; v1030 is conditional. Aggregate [O] unchanged.

Original classification detail

Open [O] + five lemmas in-house (v1006): L3, UGF K_G=pi^2/4, TEL-a N^{-2} isometries, integer D5+A3 pairings, Z4 outerness in the limit. Residual boxed TEL-B-EXTERNAL + ALG-EXH. WAVE-7 2026-08-30. Display stays [O]. | MONDAY MORNING (2026-08-31 v1016): TEL-B narrowed to scalar dyadic A_R<43 (factor-350 headroom; UV-supported remainder); uniform CAR nuclearity proved as specialist input; ALG-EXH unchanged. Display stays [O]. | ROUND3 2026-09-05: TEL-B reduced to CF/DG; C1/C2a/TV closed; ALG2 correction binding. Charged/type-III parent stays [O]. | ROUND-4 CURRENT: v1026 proves the relaxed uniform norm ||R_N||HS<2.995906<3 for every evenN>=16 (CF/DG/CROSS plus validated Fourier aliases). Historical2.11 and finite1.7833 are not promoted. The residual is microscopic one-edge FE-GEN, actual-word/joint-adjoint tails, ALG-EXH and MMST identification; v1030 is conditional. Aggregate [O] unchanged.

Dependencies as recorded

SEAM.SIMPLECURRENT.GENERATOR.01; SEAM.STATE.RPMIXING.01; ALPHA.QUILLEN.EXACT.01; AX.P1.01