Publication history: 2026-07-21

Details

• v511_celestial_wp5e_delta2_tensor_ledger.py (CELEST.WP5E.DELTA2.01, 41 checks, \sim4 s; integer-scaled weight multisets + Kostant/Weyl alternating sums + sympy/Fraction, no floats; deterministic). D0/D1 structure: 240 roots, class split (52,64,60,64); \mathfrak g_0=\mathfrak d_5\oplus\mathfrak a_3 SEMISIMPLE without \mathfrak u(1) (40{+}12 class-0 roots, no mixed root, root span rank 8); sectors \mathfrak g_1=(16_s,4), \mathfrak g_2=(10,6), \mathfrak g_3=(\mathfrak g_1)^* as multiplicity-free minuscule Weyl-orbit products, each irreducible (64/60/64); the grading is INNER (h=(2,2,2,2,2;0^4) in the \mathfrak g_0 Cartan). D3/D4 Hom tables: bilinear nonzero ONLY at j'=-j (dims 2/1/1/1; all 12 charged pairs zero — the inner-torus theorem at arity 2); trilinear: ALL 15 non-neutral triples \mathrm{Hom}=0 (arity 3), neutral dims (3,2,2,1,1) with all cross-sector trilinears ANTISYMMETRIC in their repeated legs — the unique symmetric survivor is the \mathfrak{su}(4) d-symbol (\mathrm{Inv}(S^3\,45)=0, \mathrm{Inv}(S^3\mathfrak a_3)=1). D5 the new channel: Q_{dd}=\tfrac1{60}(T_3-P_3/4) by Killing-propagator contraction AND the hand route |x^2-(\operatorname{tr}x^2/4)\mathbf 1|^2/60; \Phi_{T_3}(Q_{dd})=\tfrac1{60}\neq0; brackets/tadpoles/d{\times}f die exactly; no T_5 analogue. D6 decision: strict reading unreachable (rank 2 vs 3); generous reading STILL unsolvable (rank 3 vs 4) with the weaker certificate \psi(A_{\mathrm{fix}})=64; relaxed reading EXACTLY solvable with c_d=1920=32\times60. D7 controls: Killing control = v508 replication (A_{\mathrm{fix}}, E_{13}, E_{22}, E_{00}=225E_{22}); SO(16)/D_8 bilinears (1,0,1), trilinears (1,0,1,0), \mathrm{Inv}(S^3\mathrm{adj}_{D_8})=0; false \mathfrak g_0=\mathfrak d_5{+}\mathfrak a_2{+}\mathfrak u(1): (5,2,2)\neq(2,1,1), \mathrm{Inv}(S^3\mathfrak g_0)=6\neq1; false sector content (16_s,\bar4) kills every pairing. Discovery provenance: experiments/tfpt-discovery/celestial_seam_wp5e_delta2_full_tensor_ledger_probe.py (41/41), same day.
• The \delta_1 fence (exploration-level, NOT promoted — verdict UNDECIDED). The twisted BCOV contact-term probe (celestial_seam_wp5e_delta1_bcov_contact_probe.py, 27/27, stays sandbox; v-numbered on demand) contributes its four exact sharpenings as fence text: (i) the strict holomorphic q^0 reading is REFUTED at the \Z_2/Eguchi–Hanson anchor (method boundary, not a kill); (ii) the forced leading (T_5,T_3) ratio is 4{:}3 (sector 2 : sectors 1{+}3); (iii) the SL(2,\Z) orbit decomposition 15=12+3 identifies the contact term as the MODULAR COMPLETION of the AB data (a Harvey–Moore-type \tau-integral), not a free number; (iv) massive coset levels n\ge2 are mandatory. The remaining \delta_1 target is the \tau-integral, evaluated against the \delta_2-sharpened criterion (\psi=64 slice or selection-rule relaxation).
• Ledger + papers + Wolfram + website. Ledger: new row CELEST.WP5E.DELTA2.01 ([E]Hom tables/innerness/Q_{dd}/certificate; [C]the number fences; [O]arity \ge4, cubic GS term, \delta_1); CELEST.SEAM.01 updated (delta-2 executed via v511 — the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T_3 with the weaker certificate \psi=64; delta-1 stands UNDECIDED at exploration level). Papers: tfpt_research_contracts WP5e bullet gains the executed \delta_2 stage + the \delta_1 exploration fence, WP5e-proper targets renamed (\tau-integral, cubic GS term, \varepsilon_1, back-reaction); tfpt_3 celestial section extended to the FOURTEEN-step story (new Step 14 “the full-tensor ledger: the collapse is real, and one cubic door opens”) with the “What remains” block updated. Wolfram: seven mirrors (innerness count + \mathfrak g_0 structure; the bilinear and trilinear Hom tables by exact Kostant alternating sums; the unique symmetric survivor; the SO(16)/false-\mathfrak g_0/ false-sector controls; Q_{dd} by two routes; the \psi certificate + the relaxed solution (-1,8,2;1920)) verified with the active engine the same day (463\to470, ALL WOLFRAM EXTENSION CHECKS PASSED). Registry + run_all (505 modules, suite green: ALL CHECKS PASSED). No gate closes, no marker moves — WP5e proper stays [O].

Details

• What is consolidated (and why now). The P2 state of affairs was spread over five module rounds: the partition corollary (v491: 4=2{+}1{+}1 unique \Rightarrow\gcar=e_2=5), the weight typing (v499: Deligne + Birkhoff–Grothendieck + Mehta–Seshadri force E=\mathcal O(-2){\oplus}\mathcal O(-1)^2, residuals R1/R2 [C]), the residual convergence (v503: QGEO-R1 \equiv P2-R1), the clock rigidity and bit origin (v506/v507: one alignment bit, order fermionically forced, deck class derived) and the freedom theorems (v510: the position half is topology, the edge class excluded). The honest consolidated statement, now in one place: the compiler's dimensionless inputs reduce to four marks (derived from P1-side topology), one square-modulus datum (\tau=i, equivalently: the clock / the CM-fixed half-period / the non-split extension class on the free seam circle), and \pi — plus the [C] identifications R1/R2 of v499. No new claims; conservative wording; no “axiom abolished” language anywhere.
• Papers + ledger + website. tfpt_1_architecture_e8: new keybox “The current reduction state of P2” next to the foundational-postulates keybox, with v491/v499/v506/v507/v510 citations. tfpt_3: mirror bullet “The current reduction state” in the honest-status section. introduction: the anchor paragraph gains the consolidated one-sentence state and the status card row a=(1,1,2) now names the reduction state. Ledger: SYNTHESIS NOTE appended to AX.P2.01 (“reduction state 2026-07-21: four marks + one square-modulus datum (\tau=i) + \pi”), marker Axiom unchanged. Website: papers.ts mirrors of the changed sections. No gate closes, no marker moves.

Details

• v510_seam_bit_freedom.py (SEAM.BIT.FREEDOM.01, 30 checks, <5 s; sympy + exact \mathrm{Cl}(16) Fractions, no floats; deterministic). Part A (dihedral/fermionic): the involution census in D_N on the refined 2N-point seam circle (N=16: 17 involutions, EXACTLY ONE free = the antipodal shift by 8; all 16 reflections have exactly 2 circle fixed points — sites for even axis, edge midpoints for odd; robustness N=8,12; odd control N=9: ZERO free — an even carrier count is needed, consistent with 16=2^{\gcar-1}); the census is COMPLETE because \Aut(C_{16})=D_{16} exactly (brute force over all 8!=40320 permutations at N=8, propagation at N=16); quotient topology: circle/antipode = CIRCLE (\chi=0, closed), circle/reflection = INTERVAL (\chi=1, two boundary corners) — a fixed-point deck would break the 6\pi=3\times2\pi closed-circle count, so “deck free on the seam circle” is exactly the established “the seam is a closed circle”, not a new premise; the complete Fock census: ALL 17 involutions implemented in exact \mathrm{Cl}(16) — NONSPLIT \iff FREE with no exception, root dichotomy 2 vs 0; the chain: free \Rightarrow shift 8 \Rightarrow crossing (all 28 invariant 4-sets) \Rightarrow U^2=(-1)^F\Rightarrow central class \Rightarrow the edge/silver arrangement is EXCLUDED; Klein control (covering decks are free regardless of the gluing); branch control (the pillowcase involution has exactly 4 fixed points = the marks and restricts to a REFLECTION: the silver arrangement mistakes the mark-creating branch \Z_2 for the seam deck). Part B (Möbius/RP): the cylinder-double-cover deck is free on all 120 model points and restricts to the antipode on the \Z_2 seam fibre; one-sidedness TYPES the \Z_2 as the free antipode (quotient \mathbb{RP}^2 = the |\Z_2| of \cthree=1/(|\Z_2|2\pi\chi)); the deck-commuting real structures are EXACTLY two families (\mu\bar z: fix line through the poles, NOT free; \mu/\bar z, \mu>0: fix circle |z|=\sqrt\mu, FREE; \mu<0: empty); the per-configuration table (\mu_4 equatorial/free, silver real-axis/not free, generic-unit free without harmonicity, hexagonal RP-incompatible); harmonic + free solves EXACTLY to b=\pm i (on free circles the silver family “harmonic AND edge” is EMPTY — harmonicity becomes sufficient); honest counterwitnesses: the discrete \{0,1,8,9\} and the continuous \{\pm1,\pm(3{+}4i)/5\} (free, central, j=148176/25, no clock) — the Fock dichotomy measures the CLASS, not the modulus. Premise typing: (P-i) deck = Möbius covering deck \Rightarrow free [standard topology; the one-sidedness strand origin_theory/v456]; (P-ii) marks on the \Theta-fix seam circle = existing REALIZE [C]content (QGEO.REALIZE.02 hypotheses, v264) — NO new input. Audit finding documented: “circle-freeness” was nowhere typed before (only mark-freeness, v216); v507 F1.1 explicitly admitted the reflection reading of the edge class — v510 closes that door (a reflection is not a covering deck). The [C]rest, verbatim: the alignment bit reduces from “harmonic AND central” to the square-modulus datum \tau=i alone; the position half is topology (edge class excluded), the modulus half stays the one carrier input. Discovery provenance: experiments/tfpt-discovery/qgeo_realize_deck_freedom_dihedral_probe.py (14/14) and qgeo_realize_rp_fixcircle_moebius_probe.py (16/16), same day.
• Ledger + papers + Wolfram + website. Ledger: new row SEAM.BIT.FREEDOM.01 ([E]dihedral theorem + complete Fock census + real-structure classification + harmonic-free collapse + counterwitnesses; [C]the square-modulus datum); TEXT precisions (markers unchanged) in SEAM.CLOCK.RIGIDITY.01, SEAM.BIT.ORIGIN.01, QGEO.REALIZE.01 (bit \to square-modulus datum; edge class topologically excluded) and P2.TYPING.01 (R1 halved). Papers: tfpt_3 celestial/QGEO passage extended (“the position half is topology”), tfpt_research_contracts QGEO kill-test + QGEO.MARKS.03 + celestial-contract passages extended, one-sentence precisions in introduction, tfpt_1, tfpt_2, origin_theory where v506/v507 are cited. Wolfram: six mirrors (dihedral census + quotient topology; Aut completeness incl. the 8! brute force; the \mathrm{Cl}(16) dichotomy census over all 17 involutions on 256\times256 Jordan–Wigner gammas; the real-structure table; harmonic + free \Rightarrow\pm i; the counterwitnesses) verified with the active engine the same day (457\to463, ALL WOLFRAM EXTENSION CHECKS PASSED; base file re-verified 116/116). Registry + run_all (504 modules, suite green: ALL CHECKS PASSED). No gate closes, no marker moves.

Details

• v509_celestial_wp5e_eps2_level_from_flux.py (CELEST.WP5E.EPS2.01, 28 checks, <1 s; sympy/Fraction exact, no floats; deterministic). Target: derive the boundary level k=1 from flux quantisation on the lockstep spheres, following the Costello–Paquette–Sharma Burns-holography mechanism (PRL 130, 061602; arXiv:2306.00940). CPS skeleton replicated exactly: the S^3 period of the brane back-reaction 3-form is (2\pi i)^2N (their eq. 3.33; large-gauge invariance quantises N), the exceptional-sphere Kähler flux is 2\pi N exactly (K=|u|^2{+}N\log|u|^2, the flat part dies at t\to0), and the boundary level magnitude is 2N via the symplectic-boson double contraction (N=1,2,3; their eq. 9.4), Dynkin normalisation T(\text{so}_8\ \text{vector})=1, T(\text{adj})=6. Pairing matrix pinned: the v505 \mathrm{ch}_2 ledger (f(m)=-(C^{-1})_{mm}/2=(-\tfrac38,-\tfrac12,-\tfrac38)) + integrality + unimodularity + effectivity force P_{mi}=c_1(T_m).[\Sigma_i]=\delta_{mi} by COMPLETE enumeration (row candidates 8/6/8; 48 unimodular solutions of PC^{-1}P^T=C^{-1}; with P\ge0 exactly 2: identity + A_3 diagram flip); fluxes F_i=(64,60,64)=\dim\mathfrak g_i, total 188=248-60, flip-invariant; flux per charged root current uniformly ONE quantum through exactly one sphere. The level formula, honest: “level = total open-string flux” is KILLED by the lockstep test itself (k_i=(64,60,64) unequal — F_i is the FLAVOUR multiplicity, the brane matter content); what works: k= flux per adjoint current =F_i/\dim\mathfrak g_i=(1,1,1), anchored non-circularly by the lattice current count (240,0,0,0) for k=1..4 (v505 S4.8) and the embedding index j(D_5{\subset}E_8)=j(A_3{\subset}E_8)=1 (60=2h^\vee). Ord-4-vs-level-1 resolved: the clock order counts FRACTIONAL flux sectors, not the level (\mu=\det(D_5)\det(A_3)=16=\mathrm{ord}^2; the \mu_4 glue diagonal isotropic + Lagrangian; KLM 16\to16/4^2=1, zero residual fractional flux); NEW DIAL: \#\mathrm{primaries}((E_8)_k)=(1,3,5,10,15,27) for k=1..6 from the affine marks — exactly ONE sector iff k=1 (independent of the CPS transplantation). The lockstep theorem (the lifting of v505 S4.10): clock invariance P(iZ)=P(Z) forces a_3=a_2=a_1=0 sharply, the four branch points are ONE \mu_4 orbit, |\Pi_j|=\sqrt2\,t EQUAL on all three spheres, the monodromy is the pure phase i and \det(A{-}1)=-4\neq0 (no invariant flux vector) — k_1=k_2=k_3 IS A THEOREM of clock invariance; NEGATIVE PROBE (new falsifier): the clock-forbidden family Z^4{-}Z (smooth, disc -27, branch points \{0,1,\omega,\omega^2\} not a \mu_4 orbit): over ALL 24 orderings ZERO chains with three equal period moduli (1 vs \sqrt3); HONEST LIMIT: uniform flux doubling keeps the lockstep — the theorem proves EQUALITY, not the value (1 from the counters and the CPS minimal quantum N=1). Negative controls: k=2 (current count 0, residual 360, prefactor -31/48, \#prim 3 — while the lockstep dial honestly does NOT fire); SO(16) (fluxes (0,60,0): spheres 1/3 FLUX-DARK, defect -30\neq-78, condensation stuck at 4=\det\mathrm{Cartan}(D_8)); A_2 (det 3\neq4, glue diagonal 23/24\notin\Z, \mu=12 no perfect square: NO Lagrangian subgroup, Coxeter 3\neq4). Typed honestly: the CPS dictionary itself on \mathbb{PT}/\Z_4 stays [C](their geometry is the \mathbb C^2 blow-up with one (-1)-sphere and \mathfrak{so}(8) at negative level — not the A_3 ALE); the type-I B-model back-reaction stays [O](the same fence as v502/v505).
• Ledger + papers + Wolfram + website. Ledger: new row CELEST.WP5E.EPS2.01 ([E]CPS skeleton/pairing/lockstep/sector arithmetic + [C]CPS dictionary + [O]back-reaction); CELEST.SEAM.01 updated (“\varepsilon_2 executed via v509: the one-level statement is now a THEOREM (lockstep from clock invariance, falsifier Z^4{-}Z), level =1 carries the new sector-counter dial \#\mathrm{prim}((E_8)_k)=1 \Leftrightarrow k=1; the \mathbb{PT}/\Z_4 back-reaction stays [O]”); the v505 “one scale \Rightarrow one level” sentence LIFTED to theorem status in ledger and prose (text move, no marker move). Papers: tfpt_research_contracts WP5e section extended by the \varepsilon_2 block (done, verdict B; roadmap now \delta_1/\delta_2/\varepsilon_1 + back-reaction); tfpt_3 celestial section becomes a THIRTEEN-step story with Step 13 “one level is a theorem; the sector counter pins k=1” and the “What remains” paragraph updated. Wolfram: seven mirrors (the CPS period integrals; the 2N contraction + Dynkin normalisation; the pairing enumeration 48\to2; the fluxes (64,60,64); the \#prim table (1,3,5,10,15,27); the lockstep periods + the 0/24 negative probe; the SO(16)/A_2 controls) verified with the active engine the same day (450\to457, ALL WOLFRAM EXTENSION CHECKS PASSED). VerificationDag: v509 on the QFT/celestial node; registry + run_all (503 modules, suite green). No gate closes, no marker moves — WP5e proper stays [O].

Details

• experiments/e8-ladder-bed v2 (2DCS): amendment preregistered, run, and killed by its own controls. Trigger: arXiv:2512.16829 (Watanabe, Trebst, Hickey) — a THEORY-only paper (four-kink/ED localized-limit spectra at B_y=0–3 T, not the E8 point; no measurements); as of 2026-07-21 no experimental 2DCS measurement of CoNb_2O_6 exists anywhere (web-checked). amendment_v2 was frozen in hypotheses/e8_ladder_bed_v1.yaml BEFORE any 2D recovery run (detector, catalog F1/F2/F3, gates, null model, verdict criteria); amendment_v2_1 discloses a null-region correction (add n_matched to the false-alarm region) found on synthetic diagnostics, timing disclosed. Runs (results/results2d.json, deterministic): known-answer KA2D 8/8 rungs with m_7/m_8 recovered ONLY via F3 difference coordinates (\chi^2/\mathrm{dof}=1.30, MC p=2.0\times10^{-4}, 0/5000) — the mechanism that pulls m_7/m_8 inside a low-pass window works; the NEG2D localized-limit map (the paper's own non-E8 closed forms) is correctly rejected (p=0.029); offset and wrong-ladder controls rejected. BUT scramble2d fires at 54/200=27\% and jitter2d shows no dose-response decay (29/21/17/18\%; 18\% at the destructive 30\% level) \Rightarrow preregistered verdict kill_2d: permuting the detection axis preserves the pump-axis marginal (which carries the full 1D ladder) and the 120-entry catalog is dense enough that rung coverage follows from the marginal alone — the v2 detector does not test the 2D pairing; failure kills the detector, not TFPT. The CoNb_2O_6 leg stays 6/8 (p=0.0038) per the preregistered m7_m8_upgrade_rule; a future v3 would need coincidence-based coverage and a sparser catalog, preregistered BEFORE real data exists.
• Surfaces synced (scorecard row updated; counters unchanged, 120 rows). Scorecard row “E8 mass-ladder blind recovery bed”: data_value extended (“CoNb2O6 6/8 unchanged; 2DCS extension killed at detector level (scramble 27%, jitter 18% @30%); KA 8/8 incl. m7/m8 via F3, p=2e-4; no experimental 2DCS data exists (arXiv:2512.16829 theory-only)”) and status_note extended (amendment v2/v2.1 preregistered+disclosed; real-data leg blocked; v3 criteria) via build_evidence_scorecard.py; status/stage UNCHANGED (consistent/not_applicable, analog_positive_control basket — the v1 detector validation stands). Experiment README v2 section, hypotheses YAML amendments, experiments/next.txt note; data-request plan on record (Armitage lab/JHU, Wang lab/Ames; 2D peak table at B\parallel b, 4.7–5.5 T, sub-Kelvin). No claim; firewall intact.

Details

• v508_celestial_wp5e_gamma_pairing.py (CELEST.WP5E.GAMMA.01, 27 checks, \sim4 s; sympy/Fraction exact, no floats; deterministic). Question: can the three twisted sphere axions \eta_m (Coxeter eigencharacters \{i,-1,-i\} on H^2(\mathrm{ALE}), v505 S5.8) cancel the rigid 32\,T_3 residual of the Atiyah–Bott ledger (A_{\mathrm{fix}}=Q^{(1)}/2{+}Q^{(2)}/4{+}Q^{(3)}/2=9P_1{-}30P_2{-}15P_3{+}32\,T_3, v505 S3.7) by Green–Schwarz-type EXCHANGE with sector-compatible quadratic vertices? No, with a certificate. Vertex-space theorem: exact Weyl nullspace arithmetic over all bidegrees gives quadratics dim 2 (bidegrees (1,0,1)) and quartics dim 5 (bidegrees (2,0,1,0,2)) — the vertex basis is COMPLETE; product theorem (the master kill): (a s_5{+}b s_3)(c s_5{+}d s_3)=ac\,P_1{+}(ad{+}bc)P_2{+}bd\,P_3 identically — every exchange image (any axion count, any rule, any couplings) is annihilated by \Phi_{T_3}, and \Phi_{T_3}(A_{\mathrm{fix}})=32\neq0. Selection enumeration: the strict two-index reading collapses (C_{j,j'}\neq0 only for j'=-j, forcing m=0: 16 raw triples, 4 nonzero, all m=0); twist insertion: \eta_1{\leftrightarrow}K^{(3)}, \eta_2{\leftrightarrow}K^{(2)}, \eta_3{\leftrightarrow}K^{(1)}, \eta_0{\leftrightarrow}K^{(0)}; channels E_{13}=(16,-96,144,0,0), E_{22}=(16,32,16,0,0), E_{00}=225\,E_{22}; SIDE DISCOVERY: K^{(0)}=-15\,K^{(2)} (the even-sector quadratics are PARALLEL — this collapse drives the \Phi_P obstruction). Unsolvability with certificate (the core): M is 5{\times}2, rank 2, kernel 0; \operatorname{rank}([M\,|\,A_{\mathrm{fix}}])=3; annihilator certificate (\Phi_{T_5},\Phi_{T_3},\Phi_P)(A_{\mathrm{fix}})=(0,32,72) — two independent obstructions (32\,T_3 structural, rule-independent; \Phi_P=72 selection-rule-driven); \ker M=0 forces the unique non-disturbing coupling c=0 (the sphere axions are RIGIDLY SILENT); \Phi_P(Q^{(0)})=0 (the sector-0 Okubo mechanism undisturbed); relaxed control: dropping charge conservation reaches the P-block (\det=-64) but T_5=T_3=0 persists. Naturalness resolution: ch_2-natural couplings (9/64,1/4) give (25/4,-11/2,97/4,0,0) and AB-weight couplings (1/2,1/4) give (12,-40,76,0,0) — BOTH with certificate (0,0) vs required (32,72): the failure is scale- and coupling-INDEPENDENT (no “unnatural factor”). Negative controls: invariant-only = v505 S3.9 replicated + sharpened; flipped rule m{+}j{-}j': \Phi_{T_3} kill flip-invariant, \Phi_P rule-dependent; SO(16): odd classes EMPTY and AB sum (15,-18,-15,-4,40) — T_5 does not even cancel (E_8 doubly special); D_8: 16\,T_8{+}12\,s_8^2, no T_3 structure — the question cannot even be posed. Number fences [C]only: 32=(h,h){+}(h',h')=20{+}12, 72=2\tilde\lambda^2_{e_8}.
• Ledger + papers + Wolfram + website. Ledger: new row CELEST.WP5E.GAMMA.01 ([E]vertex space/selection/certificate + [C]fences + [O] contact terms); CELEST.SEAM.01 stays Open with the WP5e targets UPDATED (“the exchange sub-branch is closed by v508; the named remaining targets are the twisted BCOV contact terms (\delta_1, binary test) and the full-tensor ledger (\delta_2)”). Papers: tfpt_research_contracts WP5e section extended by the \gamma block (\gamma done — negative result with certificate; the \delta_1/\delta_2/\varepsilon_1/\varepsilon_2 roadmap with success/kill criteria); tfpt_3 celestial section becomes a TWELVE-step story with Step 12 “the exchange no-go and the contact-term criterion” and the “What remains” paragraph updated. Wolfram: six mirrors (the vertex-space dims 2/5 by generator-wise nullspace arithmetic; the twisted quadratics with K^{(0)}=-15\,K^{(2)}; the product theorem, symbolic; the rank-3 certificate (0,32,72) with channels and naturalness images; the SO(16) AB sum (15,-18,-15,-4,40); the D_8 quartic 16\,T_8{+}12\,s_8^2) verified with the active engine the same day (444\to450, ALL WOLFRAM EXTENSION CHECKS PASSED). VerificationDag: v508 on the QFT/celestial node; registry + run_all (502 modules, suite green). No gate closes, no marker moves — the O-fence is only sharpened.

Details

• v507_seam_bit_origin.py (SEAM.BIT.ORIGIN.01, 26 checks, \sim2.6 s; Part A sympy-exact torus/Möbius arithmetic + Part B exact \mathrm{Cl}(16) Fractions; deterministic). Conjecture under attack: marks and deck both descend from the SAME torus structure (the seam double T^2_\tau\to S^2, v214/v260), so the common origin would force the central alignment automatically and the v506 bit would be a tautology. Origin theorem (die Herkunft): the 4 marks are the fixed points of w\mapsto-w on \mathbb C/(\mathbb Z{+}\tau\mathbb Z) = the half-periods \Lambda/2\Lambda (a \tfrac13-coefficient never qualifies); t_c descends iff 2c\in\Lambda — exactly the three half-period translations (control c=\tfrac13 fails); the descended maps are the EXPLICIT rational involutions \sigma_i(x)=e_i+(e_i{-}e_j)(e_i{-}e_k)/(x{-}e_i) with the exact curve-lift identity \prod(\sigma_i{-}e_l)=\psi_i^2\prod(x{-}e_l), \psi_i RATIONAL (the \wp-addition formula as pure algebra), closing to Klein V_4; CONVERSELY the freely-acting involutions in ALL three stabiliser types (generic V_4 (4,3,3), harmonic D_4 (8,5,3), hexagonal A_4 (12,3,3)) are EXACTLY the three \sigma_c \Rightarrow every mark-free collar deck (v216) IS a half-period translation. Core solve (symbolic \lambda): \sigma_c admits a marks-preserving \mathrm{PGL}_2 root \iff\lambda=-1 (\sigma_1), 2 (\sigma_2), \tfrac12 (\sigma_3) — per half-period exactly ONE solution, intrinsically “the deck pairs separate harmonically” (pair cross-ratio -1); generic modulus: V_4 abelian, ALL \sigma_c central yet clockless (“central alone is empty” — the bit is the CONJUNCTION D_4 \wedge central); at \tau=i the stabiliser jumps to D_4 with \sigma_1 central (2 roots = the clock pair), \sigma_2/\sigma_3 edge-type (0 roots); the v506 seam deck lands on the central \sigma_1=-1/x under all 8 mark bijections; CM dictionary: mult-by-i fixes exactly c^*=(1{+}\tau)/2 (mult-by-\rho 3-cycles all three), the clock lift needs the CM unit i (\psi=2i/(x{+}1)^2, t^2{+}1 irreducible over \mathbb Q) while half-period decks lift rationally. Equivalence tetrad + circularity audit: clock exists \iff (\tau{=}i AND deck = the CM-fixed half-period) \iff (Stab =D_4 AND deck central) \iff deck pairs harmonic \iff NS Fock lift nonsplit; the v493 converse independently re-verified — two independently computed converses of ONE equivalence, no circle. Fermion dichotomy (Part B): central arrangement (shift by 8): \tilde U^2=256\,\gamma_1\cdots\gamma_{16}\Rightarrow U^2=(-1)^F, nonsplit \Z_4, 2 roots in the 32-element dihedral lift (the \Z_8 tower); edge arrangement (reflection j\mapsto4{-}j, fixed sites = non-marks = the silver \{0,\infty\} between neighbour marks): the implementer EXISTS but \tilde U_e^2=+2^7\cdot\mathbf1\Rightarrow U^2=+1, the extension SPLITS, 0 roots, no \Z_8; one-particle (\lambda^2{+}1)^8 vs (\lambda^2{-}1)^8; census: all 28 shift-invariant 4-sets cross (central type), all 21 mark-free reflection-invariant sets do not; robust at 8 sites; R-sector edge control splits too — the bit IS a physical extension class (Fidkowski–Kitaev-type): distinguished, but not derived (WHICH class nature realises stays the input). Silver placement: \{\pm1,\pm(3{+}2\sqrt2)\} = “\tau=i with deck = a non-CM-fixed half-period translation” (all 8 bijections \to\sigma_2/\sigma_3, never \sigma_1) — the in-family witness: TAUTOLOGY REFUTED. The one [C]rest (verbatim): the alignment bit remains a carrier datum; “no tautology — the origin forces the CLASS of the deck (half-period translation), not its POSITION”. No overclaim.
• Ledger + papers + Wolfram + website. Ledger: new row SEAM.BIT.ORIGIN.01 ([E]origin theorem/core solve/tetrad/fermion dichotomy/silver placement + [C]the bit); TEXT precisions at unchanged markers: SEAM.CLOCK.RIGIDITY.01 (“the bit is now fully characterised as a tetrad … and carries a physical face: the non-split NS Fock extension class — v507”), QGEO.REALIZE.01 (“the class of the deck IS derived (half-period translation, v507); only its position stays the input”, stays [C]/[O]), P2.TYPING.01 (origin cross-reference on the common R1 bit). Papers: tfpt_3 celestial Step-1/2 environment + QGEO paragraph, contracts marks/kill-test paragraph, introduction/tfpt_1/tfpt_2/% origin_theory one-sentence updates where v506 is cited. Wolfram: six mirrors (half-period descent 2c\in\Lambda with the rational \sigma_i lift identity; the core solve \lambda\in\{-1,2,\tfrac12\}; the stabiliser counts (4,3,3)/(8,5,3)/(12,3,3) via the complete 24-triple enumeration; the CM fixed point c^* + the lemniscatic \psi=2i/(x{+}1)^2; the \mathrm{Cl}(16) dichotomy 256\,\gamma-monomial vs scalar 2^7 on explicit 256{\times}256 Jordan–Wigner matrices + the 2-vs-0 root census in the 32-element dihedral lift; the silver bijection census) verified with the active engine the same day (438\to444, ALL WOLFRAM EXTENSION CHECKS PASSED); the v493 audit direction and the 28/21 arrangement census stay Python-side (flagged). VerificationDag: v507 on the P2/carrier and QGEO nodes; registry + run_all (501 modules, suite green). No gate closes, no marker moves.

Details

• v506_seam_clock_rigidity.py (SEAM.CLOCK.RIGIDITY.01, 34 checks, \sim3.5 s; Part A sympy-exact Möbius/divisor arithmetic + Part B exact \mathrm{Cl}(16) Fractions; deterministic). v503 localised the one remaining identification input (“why does the raw seam carry the order-4 clock”, QGEO-R1 \equiv P2-R1); v506 reduces it. Theorem A (Möbius side, complete): Cayley–Hamilton + the complete case analysis give g^2= deck \Rightarrow \operatorname{tr}g\neq0 forces g diagonal with (a/d)^2=-1, and \operatorname{tr}g=0 forces \det g=0 (impossible) — exactly two roots z\mapsto\pm iz, both projective order 4, mutually inverse (\operatorname{Aut}(\Z_4)), conjugation-covariant; marks filter: a deck-invariant deck-free 4-set \{u,-u,w,-w\} admits a marks-preserving root \iff w=\pm iu \iff marks = one \mu_4 orbit \Rightarrow cross-ratio class \{2,\tfrac12,-1\}, j=1728, \tau=i, free 4-cycle, deck = clock^2, only the scale free; group form: |\mathrm{Stab}(\mu_4)|=8=2|\mu_4| with order profile \{1{:}1,2{:}5,4{:}2\}=D_4, center =\{\mathrm{id},\text{deck}\}, exactly 2 order-4 elements = the clocks — “deck central in the mark-D_4” \iff “clock exists”; the counterexample \{\pm1,\pm(3{+}2\sqrt2)\} is deck-free AND harmonic (j=1728, cross-ratio \tfrac12) with full D_4, but the deck sits non-centrally (the central involution crosses the \pm pairs) \Rightarrow 0 roots; the RP filter passes BOTH roots (the \operatorname{Aut}(\Z_4) ambiguity is irreducible, v492 NC3). B3/B4 (n-arithmetic + controls): free orbit \Rightarrow n\mid4; transitivity \Rightarrow n=4; faithfulness kills n\ge5 (kernel argument through S_4); n=3 fixes a mark; n=6 dies independently at the cusp degree 3 (v117); the deck is modulus-blind (v503 B5 exact); n=8 impossible on 4 marks; hexagonal stabiliser A_4 (0 order-4 elements, squares only orders \{1,3\}); generic stabiliser V_4 (exponent 2); stabiliser trichotomy V_4/D_4/A_4 with order-4 counts (0,2,0). Theorem B (Fock side, the core result): S^2 has b_1=0 \Rightarrow unique spin structure \Rightarrow NS/bounding on every circle (established P1 datum); one-particle: D^2=-1 (charpoly (\lambda^2{+}1)^8), quarter-shift C^2=D, C^8=1 (charpoly (\lambda^4{+}1)^4, \zeta_8 spectrum); Fock (exact \mathrm{Cl}(16)): \tilde U=\prod_j(1-\gamma_j\gamma_{j+8}) implements the deck on all 16 generators and \tilde U^2=256\,\gamma_1\cdots\gamma_{16} \Rightarrow U^2=(-1)^F EXACTLY; (cU)^2=c^2(-1)^F is never 1 (center of \mathrm{Cl}(16) = scalars): the \Z_4 lift is FORCED, 1\to\Z_2^{(-1)^F}\to\Z_4\to\Z_2^{\mathrm{deck}}\to1 nonsplit; the quarter-shift lift V (unique even block implementer) has V^2\propto\tilde U, V^4\propto(-1)^F, projective order 8=2|\mu_4| — the v492 \Z_8 spin bridge realised on the 16-Majorana Fock space; R control U_R^2=+1 (split — the forcing rests on NS = established geometry); 16-fold way \theta_v=1 at \nu=16; robustness: the forcing already holds at 8 sites (16 = the carrier count 2^{\gcar-1}, not a tuning). The one [C]rest (verbatim): “the collar deck is the central involution of the mark-D_4” (\iff square-root existence on the mark data) — ONE alignment bit; the fermionic \Z_4 forces the ORDER, not the spatial alignment. No overclaim.
• Ledger + papers + Wolfram + website. Ledger: new row SEAM.CLOCK.RIGIDITY.01 ([E]both theorems + [C]the alignment bit); TEXT precisions at unchanged markers: QGEO.REALIZE.01 (“the rest is now ONE bit: the central position of the deck in the mark-D_4; order/uniqueness/orbit/\tau{=}i follow — v506”, stays [C]/[O]), P2.TYPING.01 (R1 mapped onto the same bit, fermionic halving), QGEO.EMERGE.LIGHT.01 context sharpened. Papers: tfpt_3 celestial Step 1 (fermionic U^2=(-1)^F sentence) + Step 2 (the one-bit reduction), contracts QGEO kill-test paragraph + marks paragraph + CELEST hypothesis note, introduction/tfpt_1/tfpt_2/origin_theory one-sentence updates at the v499/v503 citation sites. Wolfram: four mirrors (the complete square-root solve; D_4 center + order-4 counts + the silver counterexample; the \mathrm{Cl}(16) identity \tilde U^2=256\,\gamma_1\cdots\gamma_{16} on explicit 256{\times}256 Jordan–Wigner matrices + the split R control; the n-arithmetic + generic kill) verified with the active engine the same day (434\to438, ALL WOLFRAM EXTENSION CHECKS PASSED); the 24-triple stabiliser enumeration and the A_4 profile stay Python-side (flagged). VerificationDag: v506 on the P2/carrier node; registry + run_all (500 modules, suite green). No gate closes, no marker moves.

Details

• v505_celestial_wp5e_beta_equivariant_ledger.py (CELEST.WP5E.BETA.01, 47 checks, \sim1.3 s; sympy/Fraction exact, no floats; deterministic). The twistor side of WP5e pushed as far as exact arithmetic reaches: the Atiyah–Bott/Lefschetz skeleton of the one-loop box anomaly (Costello; Bittleston–Sharma–Skinner) on \mathbb C^2/\Z_4 with the E_8 \mu_4-glue monodromy. AB ledger: \det(1-g_j^{-1})=(2,4,2); \sum_j1/\det_j=\tfrac54=(|\Z_4|^2{-}1)/12; \operatorname{tr}_{g_j}(\mathrm{Ad}\,E_8)=(248,0,-8,0) two routes; invariant average 60= carrier; Frobenius 61568; \operatorname{tr}_{g_2}=-8=120-128. Okubo per sector (honest sharpening): the invariant sector is exactly Okubo-quadratic (Q^{(0)}=36\langle x,x\rangle^2, 36=\tilde\lambda^2_{e_8} — v495 re-derived through the glue grading); the twisted sectors do NOT reduce (T_5: (-8,32,-8), T_3: (48,-64,48)); the AB sum cancels the D_5 quartic exactly and leaves the rigid 32\,T_3 (the only quartic-free weighting is the \Z_4 projector; the graded GS exchange is rank-obstructed in sectors 1–3); heterotic identities (\mathrm{Tr}_{120}, \mathrm{Tr}_{128}, \mathrm{Tr}_{E_8}=9(\mathrm{tr}^2)^2) exact. Index bridge: f(m)=\tfrac14\sum_j(i^{jm}{-}1)/\det_j=(0,-\tfrac38,-\tfrac12,-\tfrac38) =\mathrm{ch}_2(T_m)=-(C^{-1})_{mm}/2 EXACT (E1 = E2 arithmetic); glue \mathrm{ch}_2 defect -78 both routes (classes (-24,-30,-24)); Killing sums (320,320,240,320), total 1200=2h^\vee(h,h). k=1 geometric: lattice current count 240 at k{=}1, exactly 0 at k=2,3,4; embedding residual 47k^2{+}219k{-}266=(0,360,814,1362); integrality alone fixes nothing (honest, as v502); one scale \Rightarrow one level (v493 periods). E3 refutation (prominent): a_0\in O(8) fills the GRAVITON slot O(2), NOT the BSS axion slot O(-2) (X=-2\neq+2=Y; mismatch 4=|\mu_4| as a [C]coincidence) — “the theory brings its GS axion as a_0” is REFUTED; instead the three H^2(\mathrm{ALE}) classes carry exactly the Coxeter eigencharacters \{i,-1,-i\} = the three twisted slots (bijection); slot count 1{+}3=4 =|\mu_4| soft. Negative controls: \operatorname{diag}(i,i) breaks the ledger four ways (\sum1/\det=\tfrac14, \omega-weight -1, 5 generators); SO(16) glue: characters (120,0,+120,0), defect -30\neq-78, bulk Okubo fails (independent 4th Casimir); k=2: current count 0, c-prefactor -31/48\neq-\tfrac13. [C]dictionaries (\mathrm{CS}(\rho)\bmod1 = discriminant weight; 32=(h,h){+}(h',h'); -30=-h^\vee; \Omega-contraction). [O]the global BCOV/KS quantisation, non-fixed-point contributions, the O(-2) bulk axion, CPS level-from-flux, the pairing 32\,T_3\leftrightarrow sphere axions. VERDICT B; the kill test does not fire on the equivariant skeleton.
• Ledger + papers + Wolfram + website. Ledger: new row CELEST.WP5E.BETA.01; CELEST.SEAM.01 updated to “WP5e: \alpha (CFT side, v502) + \beta (equivariant twistor skeleton, v505) executed; the global BCOV quantisation stays [O]— with the rigid 32\,T_3 twisted residual and the sphere-axion pairing as the named remaining targets” at unchanged Open status. Papers: contracts WP5e bullet \to “\alpha + \beta done” with the a_0 refutation as an honest finding (kill-test and honest-success paragraphs consistent); tfpt_3 celestial section becomes the eleven-step story (new Step 11 “the equivariant anomaly ledger on twistor space”, “what remains” = the global BCOV quantisation with named targets). Wolfram: five mirrors (AB denominators + characters two routes; the index bridge + -78 defect; the per-sector Okubo sums over the 240 glue roots with the 32\,T_3 residual; the lattice current counts + embedding residuals; the axion-slot weight arithmetic + twisted-slot bijection) verified with the active engine the same day (429\to434, ALL WOLFRAM EXTENSION CHECKS PASSED). VerificationDag qft node gains v505; registry + run_all (499 modules at this step; 500 with v506, suite green). No gate closes, no marker moves — SEAM.EQUIV.01 stays [O].

Details

• v504_celestial_wp5d_klm_completeness.py (CELEST.WP5DB.01, 37 checks, \sim12 s; numpy/scipy Gaussian machinery ED-validated + exact GF2/integer algebra; deterministic). Complete rationality (Kawahigashi–Longo–Müger) = split + strong additivity + finite two-interval \mu-index; v501 (WP5d-\alpha) witnessed the index, v504 witnesses the other two legs on the same critical Majorana seam edge, with the U(1)/Dirac number-conserving subnet as the preregistered negative control (the current net, where strong additivity famously fails in the continuum — Buchholz–Schulz–Mirbach). Strong additivity, exact algebra: “touching = shared boundary Majorana” gives \mathrm{Even}(A)\vee\mathrm{Even}(B)=\mathrm{Even}(A\cup B) EXACTLY (GF2 spans full 64/64, 256/256, 512/512, 1024/1024; literal matrix rank 32/32); disjoint intervals give exactly HALF (rank 16/32, index 2 — the missing sector \mathrm{odd}\otimes\mathrm{odd} is the v501 \ln2 bit, localised at the split point); the neutral U(1) algebras do NOT generate the union even with the shared site (gaps 2/10/52, growing). Entropic witness: the fermionic touching MI carries the pure UV point-gap law ((c/3)\ln2 increments, 0.115525, dev 1.2\times10^{-6} — present for every net, no discriminator); the honest discriminator is bounded vs divergent: the \Z_2/orbifold deficit rises 0.4045\to0.5899 (N=64\to4096), stays <\ln2, and (\ln2-\Delta_2)\sim N^{-p} with p=0.2444 vs 2\Delta_\mu=\tfrac14 (Ising disorder exponent); split-point scan reflection-symmetric to 3.9\times10^{-14}; ED vN same direction; gap continuity to the v501 x\to1 endpoint; honest note: “defect \to0” would be false at the sharp lattice split — bounded \Leftrightarrow finite index (Longo–Xu) is the correct statement; the U(1) control bursts \ln2 from L=128 and grows as \tfrac12\ln\operatorname{Var}Q_A (Klich–Levitov slope 0.10134 vs 1/\pi^2=0.10132). Split/nuclearity: coupling-block \sigma_k fall exponentially at the elliptic-nome rate \pi K(1-x)/K(x) to 1.3–2.0\% (rates 2.29/3.24/4.82/5.81 at d=16/64/256/512); \sigma_1\sim x^{0.5044} (vs \tfrac12); trace norm summable; dimerised control exponential. Orbifold inheritance exact: P_A flips C\to-C (\sigma_k identical, 8.3\times10^{-17}) and the even-bilinear coupling Gram is the second compound \Lambda^2C (singular values \{\sigma_a\sigma_b\}, 2.8\times10^{-14}) — the same ladder at doubled rate (Longo heredity, [C]). Pimsner–Popa, exact: E(a)-a/2=PaP/2 identically; attainment in integers (all 16384 (even, odd) monomial pairs at N=8: 8192 extremal with M^2=4, \operatorname{Tr}M=0, PMP=-M, 0 violations; 2048/2048 at N=12; pencil \min\lambda=0.5000000000); \lambda_{E_4}=\tfrac14=1/\mu; index consistency e^{\Delta_\infty}=2.000000000= 1/\lambda_{\mathrm{PP}} and e^{2\Delta_\infty}=4=\mu; U(1): \lambda=1/(m{+}1)\to0. Controls + honesty: dimerised phase trivial (drift 3.6\times10^{-15}); wrong prescription \{1,P_{AB}\} deficit exactly 0; \Z_2 finite-size honesty (14.9\% below \ln2 at N=4096 — witness, not proof). VERDICT: with v501 all THREE KLM ingredients of complete rationality are witnessed on the lattice; the continuum theorem for finite-group orbifolds is Xu's (cited, not claimed), the concrete seam quotient net and the condensed (E_8)_1 net stay WP5e/Costello–Li.
• Ledger + papers + Wolfram + website. Ledger: new row CELEST.WP5DB.01; CELEST.SEAM.01 updated to “WP5d: \alpha (\mu-index) + \beta (split + strong additivity) executed via v501/v504 — all three KLM ingredients witnessed on the lattice; continuum uplift stays [O](Xu's theorem cited)” at unchanged Open status. Papers: contracts WP5d bullet \to “both stages done” with the \beta-stage numbers + Xu fence (kill-test and honest-success paragraphs consistent); tfpt_3 celestial section becomes the ten-step story (new Step 10 “the KLM completion on the lattice”, “what remains” = WP5e twistor inflow only); tfpt_4 keystone note updated. Wolfram: five mirrors (GF2 span theorem; Pimsner–Popa identity + attainment on explicit gamma matrices; two-interval group \lambda_{E_4}=\tfrac14 + index consistency; the \Lambda^2/Minors compound with Cauchy–Binet and \mathrm{Minors}[-C,2]=\mathrm{Minors}[C,2]; the U(1) dephasing \lambda=1/(m{+}1) family) verified with the active engine the same day (424\to429, ALL WOLFRAM EXTENSION CHECKS PASSED); the lattice curves stay Python-only (flagged). VerificationDag qft node gains v504; registry + run_all (498 modules, suite green). No gate closes, no marker moves — SEAM.EQUIV.01 stays [O].

Details

• v503_qgeo_emergence_light.py (QGEO.EMERGE.LIGHT.01, 17 checks; mpmath 40 digits + v280-style lattice DtN + sympy exact). The v215 lever 7 (“the decisive kill is deferred: compute the raw seam-collar DtN without inputting the marks and check whether four square marks emerge”) executed in its first honest light version. No dynamical selection: \log F=\log(\tau_2|\eta|^4) (Osgood–Phillips–Sarnak) is exactly S-invariant (dev <10^{-35}), so t{=}1 is symmetry-forced critical on the rectangular family \tau=it; there g''(1)=-1.298<0 (strict maximum) but the \tau_1-direction has h''(0)=+0.298>0: \tau{=}i is a saddle of the full moduli space; the only critical points are the CM points (E_2^*(i)=E_2^*(\rho)=0 at 40 digits, generic 0.163\neq0); the global \det{}' winner at fixed area is the hexagonal torus (\log F(\rho)=-1.03352>\log F(i)=-1.05469, diff 0.021169); criticality at the square is Palais symmetric criticality (the independent heat trace is equally critical). Sharp rigidity: the DtN mod-4 off-character indicator (the v215-K3 lever as a modulus function) vanishes to 8\times10^{-15} exactly at \tau{=}i and rises strictly monotonically both ways (min 8.4\times10^{-2} away) — an isolated zero; R^2=\Z_2 exact as permutation matrices (the v492 \Z_8 bridge on the lattice); j(it)\ge1728 with equality only at t{=}1. The \Z_2 chain is modulus-blind: 4 marks + [\Z_2,\Delta_t]=0 on every torus (v216 confirmed without mark input); pillowcase \det{}'=\tfrac12 torus \det{}' exactly (\pm-pair structure) — only the order-4 clock distinguishes \tau{=}i; \mathrm{Aut}(E_\rho)=\Z_6 has no order-4 element (the dynamical winner cannot carry the clock); given the clock the modulus freezes automatically (j=1728 identically, v493 mini). Residual convergence (the core finding): the order-4 element on H^1 is a Coxeter element of W(A_3) (char poly \lambda^3{+}\lambda^2{+}\lambda{+}1, order 4=|\mu_4|=\Nfam{+}1) = the P2 cusp-class carrier datum of v499 R1 \Rightarrow QGEO-R1 \equiv P2-R1, one common residual; chain: marks derived (v216), square derived given the clock (v493), clock = carrier input [[C]], rest = its dynamical realisation [O]; route classification (v347 analogue): B-rigidity in the sharpened form “RP seam data \Rightarrow order-4 clock” is the live route. v215 lever 7 thereby moves from “deferred, not executable” to “light version executed: rigidity, not dynamical selection” — documented in v503, v215 itself unchanged.
• Ledger + papers + Wolfram + website. Ledger: new row QGEO.EMERGE.LIGHT.01; QGEO.REALIZE.01 text precised (rest = the order-4 clock, identical with the P2 typing rest R1; square + mark count derived) at unchanged marker [C]/[O]; cross-reference added in P2.TYPING.01 (convergence theorem). Papers: contracts kill-test paragraph (lever-7 light result + the residual-convergence sentence, both with v503), tfpt_3 Step 2 (light-converse sentence), origin_theory necessity paragraph, introduction/tfpt_1/tfpt_2 one-residual cross-references. Wolfram: four mirrors (CM-point identities E_2(i)=3/\pi, E_2(\rho)=2\sqrt3/\pi \Rightarrow E_2^*=0 exact, j/DedekindEta special values; the no-dynamical-selection witnesses; the pillowcase halving as an exact algebraic identity; clock \Rightarrow square + Coxeter) verified with the active engine the same day (420\to424, ALL WOLFRAM EXTENSION CHECKS PASSED); the lattice DtN curves stay Python-only (flagged). VerificationDag masses node gains v503; registry + run_all (497 modules, suite green). No gate closes, no marker moves.

Details

• v502_celestial_wp5e_prefactor_level.py (CELEST.WP5E.ALPHA.01, 33 checks, exact sympy/Fraction, no floats). The two exactly computable consistency anchors of the WP5e twistor uplift established. Part A — the prefactor: Hurwitz-\zeta exact (\zeta(-1,\theta)=-B_2(\theta)/2; E_b(\theta)=-\tfrac1{24}+\theta(1-\theta)/4; Ising shift \tfrac1{16} per Majorana); the clock is inner (h=(2,2,2,2,2;0^4) and h'=(0^5;1,1,1,-3) both read the class mod 4 on all 240 roots; (h,h)=20, (h',h')=12, sum 32) \Rightarrow twist \theta=0^8 in all four sectors — a shift orbifold, NOT a rotation orbifold — so E_{\mathrm{osc}}=-\tfrac13=-c/24 is sector-independent (c=8= rank = Sugawara 248/31; one common \eta^8); sector ground exponents -\tfrac13+(0,1,1,1); discriminant form = Casimir energy two ways (spectral flow j^2(h,h)/32 mod 1; free fermions exact: D_5=10 Majorana \to\tfrac58, A_3=D_3=6\to\tfrac38, glue diagonal =16=2^{\gcar-1} = the WP5d seam carrier \to1); the rotation reading fails (deck \tfrac3{16}\neq\tfrac38 — “clock \neq deck” at the Casimir level); the four sector characters sum exactly to E_4/\eta^8=q^{-1/3}(1,248,4124,34752). Part B — the level: k=1 forced three independent ways — (i) the current condition h(J)=k=1 (adjoint closure 60{+}64{+}60{+}64=248 at spin 1); (ii) the conformal embedding (47k^2{+}219k{-}266=47(k{-}1)(k{+}266/47)=0; D_8: 128k(1{-}k)=0); (iii) c=8\Leftrightarrow240(k{-}1)=0 (= the prefactor); plus the v498 singular-vector dial (\|(E^\theta_{-1})^m0\|^2=m!\,k(k{-}1)\cdots(k{-}m{+}1), singular vector at grade k{+}1; 31124-27000=4124 at k{=}1 only, \langle s|s\rangle=+4 at k{=}2). Honest sharpening (mandatory): glue-h integrality h(J^j;k)=k\,(0,1,1,1) is integer for ALL k=1..8 — the originally conjectured integrality route fixes nothing; the three other dials carry. Negative controls: D_8/SO(16)_1 (same c=8, SAME q^{-1/3} prefactor, but h=(0,\tfrac12,1,1): the discriminator is glue-h integrality, not the prefactor; \Theta_{D_8}+\Theta_s=E_4); \Z_2 rotation (E=+\tfrac16, h=\tfrac12); \mu_4 rotation (sector-dependent (-\tfrac13,\tfrac1{24},\tfrac16,\tfrac1{24}) — the common prefactor breaks); wrong k\in\{2,3,4\} fails all five dials coherently (residuals 360/814/1362; \langle s|s\rangle=4/12/24; 31124\neq4124). [[C]] Kac generation for k\ge3 (k=1,2 machine-anchored by WP5b/WP5c). [O]the BCOV/Costello–Li twistor inflow itself (Costello's one-loop axion \lambda_g/(8\pi\sqrt3) on \mathbb{PT}, the Costello–Paquette–Sharma level-from-flux mechanism) — v502 anchors the CFT side of WP5e only; the KILL branch (“inflow demands a level \neq1”) is not triggered on the CFT side, its twistor side is untested.
• Contract + ledger + Wolfram + website. tfpt_research_contracts: the WP5e bullet subdivided — “WP5e-\alpha (done) [E]/[C]: the CFT-side prefactor + level pinning” with the core numbers and v502_celestial_wp5e_prefactor_level.py; “WP5e proper [O]: the twistor inflow”; summit header, honest-success paragraph and status roster updated (“WP5a–WP5d plus WP5e-\alpha landed; WP5e proper the remaining open piece”). tfpt_3 celestial section: new Step 9 (prefactor + level, one paragraph); tfpt_4 keystone note updated. Ledger: new row CELEST.WP5E.ALPHA.01; CELEST.SEAM.01 status \to “WP5e: alpha stage (CFT-side prefactor + level pinning) executed via v502; the BCOV/twistor inflow itself stays [O]”. Wolfram: four exact mirrors (Hurwitz-\zeta vacuum energies; discriminant = Casimir with the deck failure; the k-fixing equations incl. the “fixes nothing” integrality table; the character sums + D_8 control) verified with the active engine the same day (416\to420, ALL WOLFRAM EXTENSION CHECKS PASSED). wolfram/README.md, GATE.WOLFRAM.02, tfpt_safeguards Layer-6 count and the website counters (116/116+424/424 after (XIII)) updated; VerificationDag qft node gains v502; registry + run_all (497 modules, suite green). SEAM.EQUIV.01 untouched; no marker moves.

Details

• v501_celestial_wp5d_two_interval_index.py (CELEST.WP5D.01, 39 checks, numpy/scipy Gaussian machinery + exact Fractions). The WP5d roadmap discriminator — (E_8)_1\Rightarrow\mu{=}1 (Haag-dual two-interval net, one sector) vs SO(16)_1\Rightarrow\mu{=}4 — executed at the \alpha (lattice-index) stage on the v367 seam edge (16 critical Majorana layers, c_-=8, v450-style ring; Peschel-exact, every Gaussian formula ED-validated to \le1.5\times10^{-15}; the superselection identity [\rho,P_{AB}]=0 machine-exact). The \mu=1 reference: the fermionic (fixed-spin-structure) two-interval mutual information is extensive — c fit 0.5000, \max|F(x)|=5.97\times10^{-5} at N{=}512, falling monotonically to 1.3\times10^{-6} at N{=}1024 (Casini–Huerta extensivity as a lattice witness). The \mu-offset: Rényi-2 exactly \Delta_2=-\ln[(1{+}R)/2]; the 16-layer seam carrier saturates \ln2 at machine precision (|\Delta_2-\ln2|=1.1\times10^{-15} at N{=}512; R\sim N^{-0.500}); index reading [F:F_{\mathrm{even}}]=2.000000, \mu_{\mathrm{gauged}}=4.000000 = the v490 parity census (two independent lattice witnesses); per-layer gauging costs 16\ln2 (the offset measures \ln[\text{index}], not a blob). Haag-duality indicator: fermionic S(E)=S(E') to 8.2\times10^{-12}; the orbifold breaks complementarity (D(E)-D(E')=0.427 at a{=}64); the complementary-pair sum is exchange-symmetric and \le\ln4= \ln\mu(SO(16)_1) with max 1.2637 at the self-dual x=\tfrac12 — a double-limit statement (the first naive fixed-x check was wrong and was replaced — honestly typed in the module). Condensation arithmetic anchored at both measured ends: \det\mathrm{Cartan}(D_5)\cdot\det \mathrm{Cartan}(A_3)=16; KLM/Longo–Rehren 16/4^2=1 and 16/2^2=4\to4/2^2=1 (both routes exact); \Sigma d_i^2=(4,4,1); \theta_v=1 exactly at \nu=2c_-=16, rivals \neq1 — so \mu=1 after condensation ([[C]] the condensed net is interacting, the v490 fence): KILL not triggered. Negative controls: \nu{=}1 single Majorana (offset non-removable, \theta_v=e^{i\pi/8}\neq1 — the discriminator has teeth); trivial phase (all \le10^{-7}); wrong sector sum \{1,P_{AB}\} (offset exactly 0). [O]WP5d-\beta (continuum Haag duality / local normality / split of the quotient net; runs as exploration) and WP5e (Costello–Li) stay open.
• Contract + ledger + Wolfram + website. tfpt_research_contracts: the WP5d bullet \to “\alpha stage done” with the core numbers and v501_celestial_wp5d_two_interval_index.py; status line and honest-success paragraph now “four of five milestones landed; WP5e (twistor uplift) is the remaining open piece”. tfpt_3 celestial section: new Step 8 (one paragraph, conservative); tfpt_4 keystone note updated to WP1–WP5d. Ledger: new row CELEST.WP5D.01 (fine types Formal + Lattice/Numerical + [[C]] dictionary); CELEST.SEAM.01 status \to “WP1–WP5d executed; WP5e [O](single remaining milestone)”. Wolfram: four exact mirrors (Cartan determinants; KLM/Longo–Rehren quotients; \Sigma d^2 + index-reading arithmetic; 16-fold-way \theta_v) verified with the active engine the same day (412\to416, ALL WOLFRAM EXTENSION CHECKS PASSED); the lattice curves stay Python-only (flagged). wolfram/README.md, GATE.WOLFRAM.02, tfpt_safeguards Layer-6 count and the website counters (116/116+416/416) updated; VerificationDag qft node gains v501; registry + run_all (495 modules, suite green). SEAM.EQUIV.01 untouched; no marker moves.

Details

• v500_celestial_wp5c_gns_limit_state.py (CELEST.WP5C.01, 35 checks, exact integer/Fraction, no floats, deterministic). WP5a (v497) made the boundary limit precise, WP5b (v498) built the deleting operator; WP5c constructs the state. \omega_w is quasi-free: loop sector = the affine k{=}1 vacuum n-point functions via the machine-determined compact anti-involution \theta(e_\alpha)={-}e_{-\alpha} (the unique sign that is an anti-automorphism on all 61504 basis pairs; contravariance exact); radial sector = oscillator pairings x^{wr} (the exact Gibbs regulator, x-adic). Convergence sharp: \omega_w=\omega_\infty\bmod x^{N+1} for all w\ge N{+}1 (N{=}8), not yet at w=N — the WP5a threshold w=n{+}1 holds identically at the state level; positivity of every \omega_w checked. The kernel (core, fully machine): the complete exact level-2 Gram — 31124 monomials in 9361 weight blocks (profile 164{:}1, 37{:}240, 7{:}2160, 1{:}6960) — has level-1 rank 248 positive definite (Cartan block = the E_8 Cartan matrix, \det1), level-2 rank 4124 exactly (the preregistered target check), kernel 27000, every block PSD; rank per Weyl orbit (0,0,1,8,44); kernel =V(2\theta) weight by weight (Freudenthal cross-check on all 9361 weights); highest-weight norms (3844,49,2,0) at (0,\omega,\theta,2\theta); direct ideal checks F_i|s\rangle, F_jF_i|s\rangle norm 0. \mu_4-equivariance: \omega_w\circ\text{clock}=\omega_w for all w; the clock descends to GNS; level-2 rank split by clock class (1036,1024,1040,1024)=\Theta_{C_j}/\eta^8 at q^2 — the two-routes identity at the state level. Resolution of WP5b S4.10: |s\rangle is the zero vector of GNS(\omega_\infty) — the twisted two-C_1-mode floor (6 quarters =q^{3/2}) concerns only pre-limit objects (GNS has no state there, 6\bmod4\neq0). Negative controls: k{=}2 (\langle s|s\rangle={+}4, dominant blocks full rank — no ideal in the kernel); k{=}0 (level-1 Gram \equiv0 — no 248 layer without the central extension); D_8 (rank 2076=7380-5304, kernel =V(2\theta_{D_8}) PSD, but h=(0,\tfrac12,1,1): one block of four — one-block completeness stays E_8/\mu_4-specific); wrong family E'_w=mw{+}r (the 248 layer erased, 710955\neq248); no damping (897266\neq248); CCR obstruction: at fixed radial pairing c>0 positivity forces \omega(bb^*)\ge c — no w-uniform damping state exists, the family formulation is necessary and the family exists: the preregistered KILL is probed and not triggered. [C]quasi-free extension to the full *-algebra; Kac positivity beyond level 2. [O]local normality, net structure/Haag duality (WP5d), Costello–Li (WP5e).
• Contract + ledger + Wolfram + website. tfpt_research_contracts: the WP5c bullet \to done with the core numbers (state-level threshold w=N{+}1, the full 9361-block Gram rank 4124/kernel 27000 weight by weight, the clock-class split, the CCR necessity of the family), citing v500_celestial_wp5c_gns_limit_state.py; the WP5b handover marked answered. tfpt_3 celestial section: new Step 7 (one paragraph, conservative). Ledger: new row CELEST.WP5C.01 (fine type Formal/Identity [E]+ [C] identifications + [O]net questions). Wolfram: four exact mirrors (level-2 block census from the root data; rank/kernel arithmetic with \dim V(\omega)=3875 by Weyl; the glue-frame clock-class split (1036,1024,1040,1024); threshold + level-dial + D_8 census) verified with the active engine the same day (408\to412); the exact Gram/PSD computation stays Python-side (flagged). VerificationDag qft node gains v500; registry + run_all (494 modules at this step, suite green). SEAM.EQUIV.01 untouched; no marker moves.

Details

• v499_p2_weights_deligne_bg.py (P2.TYPING.01, 23 checks, sympy exact, no floats). v491 reduced P2 to “3 positive integers summing to 4”; its residual was the typing (why integers, why positive, why sum 4). Now typed: given the [E]anchors (4 marks, v216; rank 3=\#\text{marks}-1, v137/v228; cusp-class local monodromy \lambda^3-1 and exact irreducibility |\langle U,M_0\rangle|=24=|S_4|, \sum_g|\chi|^2=24, v117; unitarity (U)), the Deligne canonical extension E of the flavor connection satisfies: \deg E=-4 as a theorem (residue exponents \{0,\tfrac13,\tfrac23\} the unique lift to [0,1), trace 1 per mark; par-deg 0; \infty apparent, monodromy product =\mathbf1 exactly); h^0(E)=0 as a theorem (Mehta–Seshadri stability; all 324 section-saturation cases; BG dictionary h^0{=}0\Leftrightarrow a_i\ge1); integrality = Birkhoff–Grothendieck. Hence \{a\}=\{1,1,2\} unique and \gcar=e_2=5. Selector typed: the Schur–Horn permutohedron has integer points exactly \{(2,2,0),(2,1,1)\} (the two options of the tfpt_2 “Global computation” theorem) and h^0{=}0 kills \{0,2,2\} — the “balanced/minimal-variance” heuristic is now stability/positivity. The h^1 discrepancy, real and resolved: h^1(\mathcal O(-2){\oplus}\mathcal O(-1)^2)=1\neq3, so “generations =H^1 of the rank-3 bundle” is FALSE (negative control: forcing it demands \deg-6 and loses uniqueness, e_2\in\{9,11,12\} never 5); correct two-object typing: L=\mathcal O(-D) carries the generations (h^1(L)=3=\Nfam, the v137 basis =H^0(\Omega^1(\log D))), E carries them as its fiber. Biswas explicit (independent witness): the weight denominator 3=\Nfam forces the \Z_3 cusp cover y^3=(z{-}1)(z{-}i)(z{+}1)^2(z{+}i)^2 (genus 2; NOT the pillowcase double, genus 1, j=1728 — the deck side); p_*\mathcal O_Y=\mathcal O\oplus\mathcal O(-2)^2; the regular-representation model lands on \{0,2,2\} = the unstable Schur–Horn companion (h^0=1): integrality and the sum rule come out of the correspondence, stability is the selector. Negative controls: no positivity \to4/31 solutions; \deg-3/-5 breaks; n-mark scan e_2=(n{-}2)(n{+}1)/2=5 ONLY at n=4 (the value, not the uniqueness, is the n{=}4 content); MS-rational e_2=\tfrac{16}3, \tfrac{44}9. Honest residuals (P2 stays an axiom): R1 module identification [C] (QGEO.SYM territory), R2 (U) [C](v33/v40), R3 analytic monodromy (Numerical, v117 check 4); non-circular (neither v23's splitting input nor v115's \operatorname{diag}A_0=a/4 input used); the v491 direction-reversal note carries over. Ledger: new row P2.TYPING.01; the v491/P2.PARTITION.01 row records “typing residual sharpened by v499” (v491 stays active, no supersede). Registry + run_all (493 modules, suite green).
• Wolfram second path + papers + website. Five exact mirrors added to the extension file tfpt_readouts_extension.wl (partition enumeration under the full typing + naive-typing control + h^1 discrepancy; Deligne residue-trace degree with the A_0^\star witness char poly; Schur–Horn lattice points; Biswas \Z_3-cover degree arithmetic incl. deck double j=1728; the n-mark formula), verified with the active engine the same day (403\to408, ALL WOLFRAM EXTENSION CHECKS PASSED); wolfram/README.md + GATE.WOLFRAM.02 + tfpt_safeguards Layer-6 count + website counters (116/116+408/408). Papers (one sentence each, conservative): introduction anchor paragraph, tfpt_1 (5,3) keybox, tfpt_2 H2 winbox and the “Global computation” theorem (the balanced selection typed as stability), origin_theory (5,3) keybox — all citing v499_p2_weights_deligne_bg.py; no status card changes. Website: VerificationDag P2 node gains v499; generated indices via build; no marker moved anywhere.

Details

• tfpt_3_e8_audit_bootstrap: new section “The celestial route: from the \mu_4 clock to the (E_8)_1 boundary shadow”. The celestial-contract material that had accumulated as one long paragraph inside “The Möbius origin of the irreducible \pi” is consolidated into a dedicated section (placed after the bootstrap audit, before “Honest status”), telling the executed WP1–WP5b story in six typed steps: (1) setup — the E_8 \mu_4-glue as a flat \Z_4 monodromy on the A_3 ALE space, clock^2= deck (spin \Z_8, 8=2|\mu_4|) [E](v492); (2) deformation — XY=Z^4{+}a_0 pure scale, j=1728 frozen, clock = Picard–Lefschetz/Coxeter monodromy [E] (v493); (3) consistency honestly demoted — (\kappa/\cthree)^2=12 exact but the alignment format passes 8/8: alignment survives, selectivity does not [E]/[C](v495); (4) type mismatch + boundary limit — the (E_8)_1 character is not a jet-grading conformal block (growth n^{2/3} vs n^{1/2}) but the \mu_4 period sum 248 holds exactly: a boundary-limit shadow of SEAM.EQUIV.01/MMST shape [E](v496); (5) the null ideal quantitative — stabilisation threshold w=n{+}1, 27000=(h^\vee)^3 derived, quotient = sector sum =4124, SO(16)_1 contrast [E](v497); (6) the deleting operator explicit — |s\rangle=(E^\theta_{-1})^2|0\rangle, J^a_1|s\rangle=0 on all 248 (tally 190/57/1), level dial 2(1{-}k), one-block closure E_8/\mu_4-specific [E](v498); plus the WP5c–e roadmap and the honest status line (the route does not close SEAM.EQUIV.01; the GNS/limit question stays [O]). All pre-existing script citations (v492–v498) are carried into the new section; the Möbius section now points to it with one sentence. tfpt_research_contracts: a reader's note at the head of the CELEST.SEAM.01 contract cross-references the new Paper-3 section (the contract stays the full technical reference). tfpt_4_frontier: one sentence in the metric-sector keybox records the celestial route as a third, quantitative approach to the keystone (not closing it).
• Website mirror. papers.ts: the Paper-3 section list gains the celestial-route section (six-step summary in the existing style) and a “Celestial route” highlight; the contracts mirror gains the same cross-reference sentence. glossary.ts: three new entries (celestial route, ALE space, null ideal) in the established technical style. Generated indices via build; no status component changes (no marker moved anywhere).

Details

• v498_celestial_wp5b_singular_vector.py (CELEST.WP5B.01, 53 checks, exact integer/Fraction, deterministic, no floats). Success on the preregistered WP5b criterion: the deleting object exists and is explicit. Construction: |s\rangle=(E^\theta_{-1})^2|0\rangle (weight 2\theta, level 2=8 quarter units =q^2, an integer level) in a machine-built Chevalley/Frenkel–Kac basis — bimultiplicative 2-cocycle with asymmetry verified on all 57600 ordered root pairs, the [e_\alpha,e_{-\alpha}] sign forced by the Jacobi identity (\mathrm{SGN}=-1, not chosen), the invariant form \kappa derived (\kappa(\theta^\vee,\theta^\vee)=2), Jacobi exact on all 57600 \mathfrak{sl}_2-slice triples + 6000 seeded + all 496 \theta-adjacent, all N_{\alpha\beta}=\pm1. Singularity (machine): J^a_1|s\rangle=0 for all 248 basis elements (0 violations), with the case classification 190 (first bracket) / 57 (second) / 1 (a=F^\theta via the central-term cancellation — the only case that sees the level k); plus J^a_2|s\rangle=0, E^a_0|s\rangle=0, exact weight, Shapovalov norm 0; the affine PBW engine unit-tested on all 61504 basis pairs. Level dial: the F^\theta_1 coefficient is 2(1{-}k) — 2/0/{-2} at k=0/1/2: without the central extension the deletion operator does not exist. \mu_4 data: glue class j(\theta)=1 (machine-built h-adapted chamber, \langle\theta,h\rangle=5, height 29=h^\vee{-}1), clock phase i^{2j}=-1, \operatorname{class}(2\theta)=2 sheet-even; the deleting \theta-\mathfrak{sl}_2 crosses the sheet-odd classes (1,3) while its square is sheet-even; cross-table \theta-grading \times glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)); 8 quarters =q^2 via the per-period dictionary. Level-2 match: weight 2\theta has multiplicity 1 in the level-2 Fock, so U(\mathfrak g)|s\rangle is the 27000=(h^\vee)^3; the direct \mathfrak g_0-orbit BFS matches the Freudenthal multiplicities of V(2\theta) exactly through depth 4 ([C]Weyl complete reducibility beyond); quotient 31124-27000=4124; lattice reading (2\theta,2\theta)/2=4>2: |s\rangle is the kernel generator of Fock \to E_4/\eta^8. Negative controls (honest separation): the level-1 current state E^\theta_{-1}|0\rangle (one witness, \to-k|0\rangle — the 248-current state that must remain) and 4 generic level-2 states (14–33 witnesses) are not singular; at k{=}2 the cube is (generic Kac (E^\theta)^{k+1} mechanics, honestly typed); SO(16)_1 has the same singular vector but keeps three extra level-1 primaries (h=\tfrac12 breaks one-block fusion; 5304\neq14^3) vs E_8 (comarks all \ge2, glue weights (0,1,1,1) integer) — the one-block closure is the E_8/\mu_4-specific part. Honest [O]: in the twisted quarter-slot moding two sector-C_1 modes never sum to 8 quarters (minimum 6=q^{3/2}) — the per-period dictionary (v496), not the per-slot identification, carries |s\rangle to q^2: exactly the WP5c GNS/limit-state question; WP5c–e stay open; SEAM.EQUIV.01 untouched.
• Contract + surfaces synced. tfpt_research_contracts: the WP5b bullet moved to done with the key numbers (case tally 190/57/1; level dial 2(1{-}k); j(\theta)=1, clock phase -1; orbit BFS through depth 4; the D_8 contrast) and the twisted-slot tension as the explicit WP5c handover; honest status line and K4 paragraph updated (“WP1–WP4 + WP5a + WP5b executed; WP5c–e [O]”). tfpt_3_e8_audit_bootstrap: the celestial-contract paragraph extended by the WP5b sentence. Ledger: CELEST.WP5B.01 (new) + CELEST.SEAM.01 (“WP1–WP4 + WP5a + WP5b executed; WP5c–e [O](roadmap)”). Wolfram: 5 exact v498 mirrors added to tfpt_readouts_extension.wl and verified with the active engine the same day (398\to403, ALL WOLFRAM EXTENSION CHECKS PASSED); wolfram/README.md + GATE.WOLFRAM.02 counters advanced; tfpt_safeguards Layer-6 count + website mirrors 116/116+403/403. Website: papers.ts contract mirror (“WP1–WP4 + WP5a + WP5b done, WP5c–e roadmap”), DAG qft node + v498, generated indices via build. Suite now 492 scripts, all green; SEAM.EQUIV.01 stays [O].

Details

• v497_celestial_wp5a_null_ideal.py (CELEST.WP5A.01, 34 checks, exact integer/Fraction, no floats). Three results make the WP4 limit reading precise. Stabilisation theorem: the one-parameter family \chi_w of graded Fock characters on the chiral jet generators (loop energy m, radial number r; E_w=m+wr in quarter units u=q^{1/4}) contains the chiral jet grading as its w{=}2 member (generator counts 64,120,128,180,192,240,256,300 per degree), and its u^n coefficient equals the quarter-moded loop Fock for all w\ge n{+}1, strictly larger for every w\le n (verified n\le8, w\le10) — the v496 “boundary-limit shadow” is a precise coefficientwise limit \varepsilon=1/w\to0 with explicit threshold w=n{+}1, not a slice (convention clarified: 897266= the quarter-moded loop Fock at u^4= integer level 1). The null ideal derived, not cited: Freudenthal multiplicities + the Weyl dimension formula + generic character peeling give \operatorname{Sym}^2(248)=V(2\theta)\oplus V(\omega)\oplus V(0)=27000+3875+1 with residual exactly zero (independent Weyl-orbit cross-check); 27000=30^3=(h^\vee)^3 exactly; C_2=(124,96), Dynkin indices 6750/750; level-1 integrability (\langle2\theta,\theta^\vee\rangle=4, \langle\omega,\theta^\vee\rangle=2) leaves the vacuum as the only level-1 primary (Kac, used only through its counting consequence); level-2 quotient 31124-27000=4124=1+248+3875 exactly. Two routes, one number: the \mu_4 theta-split sector sum at q^2=(1036,1024,1040,1024) gives the same 4124 — the untwisted null-ideal quotient and the boundary/twisted lattice presentation count one space (q^3: 34752=\chi_3). Negative controls: SO(16)_1 through the same pipeline gives four components (5304+1820+135+1), 5304\neq14^3 (the h^{\vee\,3} identity is not generic), quotient 2076=\Theta_{D_8}/\eta^8 at q^2 (the recipe validated on a second algebra), block weights (0,\tfrac12,1,1) break one-block fusion (vs E_8's integer (0,1,1,1)); only P=4=|\mu_4| periodisation reproduces the 248 layer; swapped glue breaks the zero-mode anchor (64\neq60). Honest [O]: the limit does not generate the truncation — the singular vector as an operator in the limit net (WP5b), the GNS limit state (WP5c) and local normality/Haag duality (WP5d) stay open; the celestial route now has the same two-step shape as MMST (limit + maximal ideal), with the ideal quantitatively fixed; SEAM.EQUIV.01 untouched.
• Contract + surfaces synced. tfpt_research_contracts: WP5 subdivided — WP5a moved to done with the key numbers (stabilisation threshold w=n{+}1; 27000=(h^\vee)^3 derived; quotient = sector sum =4124; SO(16)_1 contrast), WP5b–e added as roadmap bullets with success/kill criteria (WP5b singular-vector operator in the quarter-moded current Fock; WP5c GNS limit state, kernel \supseteq ideal; WP5d local net/Haag duality via the KLM route; WP5e BCOV/Costello–Li twistor uplift, partition function =E_4/\eta^8 incl. the q^{-1/3} prefactor); K4 paragraph and the honest-success status line updated. Ledger: CELEST.WP5A.01 (new) + CELEST.SEAM.01 (“WP1–WP4 + WP5a executed; WP5b–e [O](roadmap)”). Wolfram: 4 exact v497 mirrors added to tfpt_readouts_extension.wl and verified with the active engine the same day (394\to398, ALL WOLFRAM EXTENSION CHECKS PASSED); wolfram/README.md + GATE.WOLFRAM.02 counters advanced; tfpt_safeguards Layer-6 count + website mirrors 116/116+398/398. Website: papers.ts contract mirror (“WP5a done, WP5b–e roadmap”), DAG qft node + v497, generated indices via build. Suite now 491 scripts, all green; SEAM.EQUIV.01 stays [O].

Details

• experiments/comb-meta-limit channel update (methodology untouched). The Vela X-ray channel (channels.py) now ingests the PG.06b FULL reduction (pg06b_full_vela.json in pulsar-glitch-recovery/results: 665 obs \to 491 usable \to 119 segment-coherent \nu(t) points) instead of the superseded single-obs pg06b_vela.json; the stale code note quoting the noise-peak “detection” F_0=11.19275 Hz is corrected on record (true Vela F_0=11.1861692 Hz), and the channel key is fixed from the mislabel PG.07 to PG.06b. Censoring handled exactly by the existing Tier-B rule (nothing invented): the comb observable is a linear \delta\nu(\tau) residual (\muHz, not a fractional \ln-flux modulation), both legs sit below the 2.8-period gate (2019: 2.17, 2021: 1.38), and the surrogate-calibrated injection never reaches \varepsilon_{90} below \varepsilon=1 (\varepsilon_{50}=0.55 in 2019 only) \Rightarrow no absolute per-channel \varepsilon-UL exists for the DL/HKSJ combination; the channel stays enumerated-only (“no constraint”), carrying the normalised context amplitude \sqrt{2g}=0.40 (leg gains 0.046/0.114, min p=0.014 — the 2019 audit-level chance feature: Bonferroni \times10=0.14, shuffle p=0.40).
• Full re-run (seed 0, deterministic): both limits numerically unchanged. Boundary/horizon-scoped: data_limited (still no horizon channel with an absolute \varepsilon); all-channel universal-DSI 95% UL \varepsilon<0.11954 (\approx12.0\%; HKSJ t_2=2.92, \rho=0.3-inflated 0.137; 6.91\times the predicted \varepsilon=0.0173), still driven by the surface log-flux channels A1/A4/A5 (A4 GRB the anchor, per-channel UL 0.108); now 10 channels enumerated / 3 usable. Injection self-consistency unchanged (inject 5% \to recover 4.8%, UL 5.5%; coverage 0.92/150 trials). Surfaces synced: experiment README (tier table, ingest list, PG.06b FULL bullet), results/results.json, scorecard row data_value/source via build_evidence_scorecard.py (status stays data_limited; counters unchanged, 120 rows), experiments/README.md catalog row. No claim; firewall intact.

Details

• Verified engine run 2026-07-21 (Wolfram Engine 14.3, license reactivated). wolframscript -file verification/wolfram/tfpt_readouts.wl \to 116 passed, 0 failed, ALL WOLFRAM CHECKS PASSED; tfpt_readouts_extension.wl \to 394 passed, 0 failed, ALL WOLFRAM EXTENSION CHECKS PASSED. The 16 exact mirrors added while the engine was unactivated all enter the counted total (378\to394): v479 Kronheimer bridge (2), v491 partition corollary (3), v493 celestial WP2 clock deformation (4), v495 celestial WP3 Green–Schwarz alignment (3), v496 celestial WP4 character shadow (4).
• Counter surfaces synced. verification/wolfram/README.md (current-state block + the five deferral notes resolved; v484/v485/v486/v487/v488/v492 stay mirrorable-but-unmirrored, Python-exact, so they are not counted); ledger GATE.WOLFRAM.01 (re-verified 2026-07-21) + GATE.WOLFRAM.02 (394/394, UPDATE note); tfpt_safeguards Layer-6 count + website papers.ts mirrors (116/116 + 394/394); v479 registry/run_all deferral note resolved. No numeric claim changes; the second computational path is simply verified-current again.

Details

• v495_celestial_wp3_gs_alignment.py (CELEST.WP3.01, 25 checks, exact Fraction/sympy). The pre-registered WP3 “alignment test only”: the Okubo coefficient \mathrm{Tr}_{\mathrm{adj}}X^4/(\mathrm{Tr}_{\mathrm{adj}}X^2)^2=5/(2(\dim\mathfrak g{+}2)) is derived as an exact polynomial identity for all 8 algebras on Costello's list (sl_2, sl_3, g_2, so_8, f_4, e_6, e_7, e_8; e_6/e_7 carved out of the v492 E_8 root set; negative control: sl_4 fails, 5/32 vs 3/32); the closed form \tilde\lambda^2=10h^{\vee2}/(\dim{+}2)=h^\vee{+}6 holds exactly across the Deligne series; E_8: \tilde\lambda=6 (unit-trace) resp. \lambda_{\mathrm{fund}}=1/10 (adjoint-trace) exact, so (\kappa/c_3)^2=12=|\mu_4|N_{\mathrm{fam}} resp. 1/300 — with \kappa/c_3 itself irrational (2\sqrt3). Byproduct: the printed \lambda^2_{\mathrm{so}_8}=3/2 in Costello's appendix A is a factor-2 slip (his own weight content gives 3). Look-elsewhere (the hard fence): the alignment format passes 8/8 across the whole list (zero selectivity), \tilde\lambda-integrality 2/8 (shared with sl_3), only g_{\mathrm{car}}{=}5\mid h^\vee is e_8-selective (1/8), the isolating conjunction is post hoc. Kill test K3 does not fire but is scope-demoted: alignment survives; selectivity does not — compatibility, not e_8-selective evidence. Verdict B.
• v496_celestial_wp4_character_shadow.py (CELEST.WP4.01, 25 checks, exact integer/Fraction/sympy). WP4, the type mismatch head-on: the (E_8)_1 character E_4/\eta^8=(1,248,4124,34752,213126,\dots) is not a conformal block of the glue-equivariant E_8[\mathbb C^2] S-algebra in its own jet grading — localised three ways: (a) spin 1-d/2 unbounded below and 248 never a tower dimension (d\le100); (b) growth n^{2/3} vs n^{1/2} (quadratic cumulative count 31s^2{+}92s{+}60, 31=k{+}h^\vee; f_n/\chi_n strictly increasing n=1..12); (c) the level-2 null ideal deletes exactly 27000=30^3=(h^\vee)^3 out of \mathrm{Sym}^2(248)=1{+}3875{+}27000 — no jet analogue. But the boundary/period reading holds exactly at the current stratum (zero modes =60=C_0 currents, jet slice cycles (60,64,60,64), one \mu_4 period sums to 248, glue weights (0,1,1,1) integer; free loop Fock 897266\gg248, so the truncation must come from a limit) — the same MMST scaling-limit shape as SEAM.EQUIV.01 (v336/v449). K4 fires only against the exact-block reading; the constructive limit question passes to WP5. Verdict B(ii).
• Contract + surfaces synced. tfpt_research_contracts WP3/WP4 bullets moved to done with the look-elsewhere demotion in the text, K3/K4 marked evaluated, status line now “WP1–WP4 executed (verdicts B, B, B, B(ii)), WP5 [O]”; one conservative sentence added to the tfpt_3 celestial paragraph; ledger rows CELEST.WP3.01 + CELEST.WP4.01 (new) and CELEST.SEAM.01 updated; v495/v496 Wolfram mirrors added to tfpt_readouts_extension.wl (3 + 4 exact checks; count with the next verified engine run — engine still unactivated); website papers.ts contract highlight “WP1–WP4 done” + DAG qft node extended. Suite now 490 scripts, all green; SEAM.EQUIV.01 stays [O].

Details

• experiments/pulsar-glitch-recovery PG.06b FULL executed (tfpt-pulsar vela-full). The full NICER Vela-pulsar (PSR B0833-45) L2 archive is reduced, with the manifest frozen before the bulk download (manifest_full_reduction.json: all 665 ObsIDs, preregistered 2019/2021/2024 glitch windows, 20-GB abort gate): 665/665 obs downloaded (2.6 GB, raw gitignored) \to 491 usable \to 240 per-obs detections \to 119 segment-coherent \nu(t) points (108 fully coherent), median \sigma_\nu\approx0.6\,\muHz; barycentring validated against PINT to 3.4\,\mus — the first X-ray \nu(t) probe of the search program. Sanity gate PASS: \Delta\nu/\nu 2019 =2.39\times10^{-6} vs 2.5012\times10^{-6} published, 2021 =1.272\times10^{-6} vs 1.26\times10^{-6}; the 2024 glitch is not reducible (only \sim150-s snapshots, 9.7 ks — documented).
• Frozen-kernel comb (\omega=2.5827, \varepsilon_{\mathrm{pred}}=0.0173): verdict data_limited (range- and amplitude-limited). Reach 2019: 2.17 comb periods (< the 2.8-period gate; beats the PG.08 public Vela reach 1.83), 2021: 1.38; the 2019 raw p=0.014 has the kernel as smallest-p battery member BUT Bonferroni \times10=0.14 and shuffle p=0.395 \Rightarrow chance-level by the preregistered escalation rule (audit flag, NOT a candidate); 2021 p=0.731; \ln\tau stack p=0.049/shuffle 0.307. Surrogate-calibrated injection at the real sampling and noise: \varepsilon_{50}\approx0.55 (2019 only), \varepsilon_{90} never reached below \varepsilon=1, 0% power at the predicted \varepsilon=0.0173 — a weaker amplitude probe than the PG.08 radio residuals, but it independently reproduces the PG.06 range-blindness on real data. Second off-kernel flag: \lambda=e in the 2021 leg (raw p=0.002, Bonferroni 0.018; the same non-kernel periodicity class as PG.08 J0742-2822 — flagged, not claimed). The sub-day early window that would clear the gate is absent from the archive (first usable obs 3.4 d / 7 d post-glitch); next decider: <3-d post-glitch X-ray observations of a future Vela glitch (eXTP class).
• Correction on record. The old PG.06b single-obs “pipeline proven” detection at F_0=11.19275 Hz (H=18.4) was a noise peak; the true Vela frequency at that epoch is F_0=11.1861692 Hz (H=99.7, consistent with the phase-connected PuMA ephemeris). The pipeline remains proven — by the corrected fold. Scorecard row updated to “recovery comb on the full NICER Vela X-ray nu(t) (PG.06b FULL)” (status stays data_limited; counters unchanged, 120 rows: 49 consistent / 9 tension / 26 null / 33 data-limited / 3 parked); experiments/README.md catalog + 1 prose, pulsar README, experiments/next.txt synced. No claim; firewall intact.

Details

• v494_cosmo_running_mu_record.py (23 checks, all green). The \alpha_s-running and \mu-distortion scorecard rows of 2026-07-20 (IV) get their load-bearing module so the new paper/website statements are machine-backed. [E]leading-order identities (sympy exact): \alpha_s=\mathrm dn_s/\mathrm d\ln k=-2/N_\star^2=-r/6, \beta_s=-4/N_\star^3 (NOT -8/N^3); LO band N_\star\in[50,60]: [-8.0,-5.6]\times10^{-4}. [E]exact slow roll on the Einstein-frame Starobinsky potential (third-order Lyth–Riotto + finite-difference cross-check <1\%): \alpha_s(51.4)=-7.09\times10^{-4}, \beta_s(51.4)=-2.70\times10^{-5}. [C]data: Planck 2018 (-0.0045\pm0.0067) +0.57\sigma; record leg P-ACT-LB (+0.0062\pm0.0052) -1.33\sigma (watch channel: positive central value; kill: robust \alpha_s<-5\times10^{-3} or >+3\times10^{-3} at \ge5\sigma; any robust POSITIVE running kills the plateau branch). [C]\mu-distortion (Chluba window, +21\% calibrated on the \LambdaCDM anchor): sharp branch N_\star{=}51.4 \mu=1.6\times10^{-8}, profiled record N_\star{=}56.1 \mu=1.94\times10^{-8}, band 1.5–2.3\times10^{-8}; branch split \Delta\mu=3.0\times10^{-9} \Rightarrow 3\sigma A_s/N_\star branch decision at \sigma(\mu)=10^{-9} (Voyage-2050 class; PIXIE baseline 9\times10^{-8} does not reach; kill: robust \mu<0.9\times10^{-8} or >4\times10^{-8}). [E]comb blindness (honest S15): the frozen log-comb (\omega=2.5827, \varepsilon=0.0173) is structurally invisible in the \mu window (|F(\omega)|=0.033, \delta\mu=7.4\times10^{-12}=0.007\sigma even at Voyage-2050). Fences: N_\star stays an external reheating input (GATE.NSTAR.01); the frozen band [50,60] stays the surface of record (v84/v86). Ledger rows COSMO.RUNNING.01 and COSMO.MUDIST.01 added.
• Prediction surfaces synced (the missing mirror of 2026-07-20 (IV)). predictions.txt gains the two rows (\alpha_s [C]with the -1.33\sigma P-ACT-LB pull and the positive-running kill; \mu-distortion [C]data-limited with the Voyage-2050 branch split); website lib/predictions.ts gains the matching cards alpha-s-running and mu-distortion (confrontation + experiment blocks pointing at experiments/tfpt-discovery); the cosmology DAG node lists v494 and states the running/\mu readouts.
• Paper bodies. tfpt_1 (Inflation interface) gains the running consistency-relation paragraph (\alpha_s=-2/N_\star^2=-r/6, exact slow roll, pulls, positive-running kill) with \veri{v494_cosmo_running_mu_record.py}; tfpt_2 (inflation-pivot keybox) gains the \mu-distortion branch-decider sentence (sharp 1.6\times10^{-8} vs profiled 2.0\times10^{-8}, 3\sigma at Voyage-2050) with the same \veri. No marker of any existing claim changes. (v494 is numerical/quadrature, Python-only by the suite convention, flagged in the docstring; GATE.WOLFRAM.01/02 untouched.)