Publication history: 2026-06-27

Details

• New module v433 (ALPHA.QUILLEN.PROGRESS.02) — a second honest step on the external target ALPHA.QUILLEN.EXACT.01 (v382); it grounds and connects, it does not close it. The target stays [O](external math, type A, v384); the value \ainv=137.0359992 stays [E]. The earlier honest attempt v391 gave a solvable model on a 1D circle whose \zeta-determinant variation reached only the low Seeley–DeWitt orders a_0,a_1; separately, v342 grounded the quadratic Calderón term in the Gilkey gauge-curvature coefficient a_4 (30/360=\tfrac1{12}, \Omega^2/12). This module supplies the missing link between the two. itemize
• [E]a solvable 4D model reaches the a_4 order: the flat operator \Delta=-\partial^2+m^2 on a 4-torus of volume V has heat trace \operatorname{Tr}e^{-t\Delta}=V/(4\pi)^2\,t^{-2}e^{-tm^2}, with Gilkey coefficients a_0=V/(4\pi)^2, a_2=-Vm^2/(4\pi)^2 and a_4=Vm^4/(2(4\pi)^2); the t^0 (conformal-anomaly) order a_4 is non-zero — exactly the order v391's 1D circle toy could not reach.
• [E]it is the order v342 uses: so v391 (“a \zeta-determinant variation is closed heat-kernel data”) and v342 (“the seam term is the Gilkey a_4”) are unified — the variation is heat-kernel data at the a_4 order, not only the low orders v391 reached.
• [E]the measure is \cthree-built: the universal 4D heat-kernel measure (4\pi)^{-2}=1/(16\pi^2)=(2\cthree)^2=4\cthree^2 in the seam's one-sided 2D-boundary normalisation \cthree=\tfrac12(4\pi)^{-1}; and the v382 split is re-verified at \ainv=137.0359992.
• [O]not closed: computing the actual seam U(1) \zeta-determinant a_4 coefficient (=8b_1\cthree^6) on the \mu_4 seam moduli, and the origin of the cubic \alpha^3 Maxwell/Chern moment, stay open (external math, type A, v384); the exact seam F-normalisation fixing the coefficient value stays [C](v342). A second honest step that grounds and connects, it does not close.

Details

• New module v431 (E8.DEGREE.LADDER.01) — the reverse-audit “unmapped” E_8 Casimir degrees are NOT diffuse overhead but a forced two-family decomposition. The structural complement of the reverse audit (v354/v355) and its sheet/deck complement (v430): \deg(E_8)=6\cdot\mathrm{spine}\{2,3,4,5\}\sqcup(\{2\}\cup\mathrm{det\text{-}ladder}\{8,14,20\}). [E]the two-family split is forced by h(E_8)=30=2\cdot3\cdot5 — the exponents are the \varphi(30)=8 totatives of 30, hence coprime to 6, hence \equiv\pm1\bmod6, so the degrees occupy only the residue classes \{0,2\}\bmod6. [E]the 6k family \{12,18,24,30\}/6=\{2,3,4,5\} is the v91 spine; the 6k{+}2 family \{8,14,20\}=(\det R,\det C,\det L) is the winding line \det M(s,0)=6s+8 of the flavor diamond (v135). [E]“18=6\Nfam” is not a holdout (spine family), and the clean split is special to E_8 among the simply-laced exceptionals (E_6/E_7/D_5 fail). [C]honest v355 line: the arithmetic decomposition is exact, but the functorial claim “6k{+}2 Casimir stratum = flavor sheet operators” needs a representation-theoretic map and stays [C](12,24 admit two canonical readings each) — a structure theorem, not a forced functorial flavor map. It reclassifies the v354 overhead to zero orphan degrees and closes no gate.
• New module v432 (OVERDET.FLOOR.01) — the unconditional improbability floor and the L1\leftrightarrowL2 bridge. The complement of the conditional null model (v100) and the multiply-vs-compress framework (v427/v428): the defensible over-determination number that survives even if a skeptic rejects the declared formula grammar. [E]compression removed (v428), the genuinely multiplicative pieces are the input “8” forced four independent ways plus the foreign witness \ainv\sim137. [C]multiplying only across disjoint pieces (input over-determination \times the \alpha census 1/94500 \times the \phiz-seed cluster counted once \times one downstream witness) gives a floor \le10^{-6} across a grid of generous assignments, nominal \sim10^{-10}. [E]that floor sits \sim20 orders above the v100 conditional 10^{-30.7}; the gap is exactly the evidential value of proving the forms forced, so the \alpha form-derivation (ALPHA.QUILLEN.EXACT.01, v382) and this floor are the same lever. [C]honest scope: bounds only the “is it numerology?” question, not the physics bridge (firewall v187); a conservative floor, not a posterior.
• Surfaces synced in the same change: v431/v432 added to run_all.py, script_registry.csv (master index + website ScriptIndex), status_ledger.csv, tfpt_5_redteam (reverse-audit winbox + Casimir-degree figure) and tfpt_safeguards (the over-determination layer); v431 mirrored in wolfram/tfpt_readouts_extension.wl (v432 is a conservative bound, Python-only).

Details

• New module v430 (E8.OTHERSIDE.AUDIT.01) — the sheet/deck complement of the reverse audit (v354/v355). The reverse audit found five E_8 Casimir degrees \{12,14,18,20,24\} with no primary readout (“hull overhead”); the double cover has an other side (the one-sided \Z_2 collar / conjugate half-spinor S^-), so the sharp adversarial follow-up is “is the leftover where the second sheet lives?” The answer is a disciplined negative. [E]the deck is the degree-2 invariant: the one-sided cover contributes |\Z_2|=2 (the \tfrac12 in \cthree=1/(|\Z_2|\,4\pi)=1/(8\pi), v58/v216), and 2=\min\deg(E_8) is the quadratic/metric — one of the matched primary readouts \{2,8,30\}, never one of the unmapped degrees. [E]the two sheets are the 128-spinor: \dim S^+{=}\dim S^-{=}16, the spinor block 128={\rm rank}\,E_8\cdot\dim S^+=2^{{\rm rank}-1}=\sum\deg, the spinor half of 248=120+128; S^- enters 248 as the conjugate (\overline{16},\overline4) (128=2\cdot64), not a spare singlet. [E]forced-disjoint: the sheet/deck set \{2,16,32,128\} has empty intersection with \{12,14,18,20,24\}, its only degree contact the matched 2. [O]each unmapped degree, “explained” from sheet atoms, needs an unforced coefficient and admits \ge2 readings (the v355 discriminator), so there is no forced sheet\tounmapped-degree identity.
• A structural negative, not a status change. It consolidates the S^--dark-matter downgrade (v227) and the no-spare-singlet WIMP no-go of the frontier E_8 scan (tfpt_4_frontier): the other side is matched conjugate matter, the five unmapped degrees are the hull overhead, and the two do not meet. The only live “other side” question — whether the 128 phase/glue channel hosts a dark sector (v227) — concerns matched structure, not the unmapped five. Closes no gate, upgrades nothing.
• Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv; the reverse-audit Outcome winbox + degree-audit table caption and the Target D winbox of tfpt_5_redteam, the magnitude/phase keybox of tfpt_3_e8_audit_bootstrap, the WIMP-no-go passage of tfpt_4_frontier; the E8 node of the website verification DAG; the Wolfram extension/README/ledger counts and the website verification count; manifests refreshed.