Publication history: 2026-06-30

Details

• The bird's-eye reframing. The recent G-block modules all sharpened the target of the scaling limit; this round attacks the two halves that actually remain. The continuum existence is already citable (v458 MMST + v459 AGT/AMT); what was implicit was (B) why a c{=}8 holomorphic limit must be (E_8)_1, and (A) the realisation input R_1 that the abstract seam IS the concrete quasi-free collar. v463 closes B; v464 reduces A to its one-particle data.
• v463_seam_c8_holomorphic_uniqueness.py (SEAM.EQUIV.UNIQ.01). The c{=}8 holomorphic-uniqueness pin. [E]c{=}8 is NOT unique at level 1: c{=}\dim\mathfrak g/(1{+}h^\vee){=}8 has THREE simple solutions A_8 (\dim 80), D_8{=}\mathfrak{so}(16) (\dim 120), E_8 (\dim 248), so the \det K{=}4 SO(16) rival is a genuine same-c competitor. [E]HOLOMORPHY FORCES \dim V_1{=}248: the only weight-0 form with leading q^{-1/3} and integer coefficients is j^{1/3}{=}E_4/\eta^8, q^1 coefficient 248 (the 1-dim candidate space, vacuum a_0{=}1). [E]\dim 248 + level-1 c{=}8 \Rightarrow E_8 uniquely (248/(1{+}30){=}8); tower \{1,248,4124,34752,213126\}{=}E_4/\eta^8{=}j^{1/3}. [C]Dong–Mason/Schellekens: the holomorphic c{=}8 VOA is unique ={}V_{E_8} (even unimodular rank-8 lattice unique ={}E_8, Mordell–Witt) — so the identification half is classification-forced.
• v464_seam_oneparticle_rigidity.py (SEAM.EQUIV.ONEPARTICLE.01). The one-particle rigidity attacking R_1. The seam is quasi-free (v113/v160/v161), so by Araki self-dual CAR the net is the second quantisation of its one-particle symbol P. [E]RIGIDITY: the gapped ground state is the UNIQUE quasi-free state with P{=} projection onto K{<}0 (P^2{=}P to <10^{-12}), K fixed by gap + chirality + reflection (v426). [E]ONE-PARTICLE LIMIT EXISTS: the kernel is Cauchy on a fixed window (\max|P_{2N}{-}P_N|{\sim}1/N{\to}0). [E]the limit is c{=}1 Dirac: entanglement S(L){=}(c/3)\ln L gives c{\to}1, 16 copies \Rightarrow c_-{=}8 (ties v450). [C]Shale–Stinespring makes the Bogoliubov map implementable, so R_1 is upgraded from a bald axiom to “the unique quasi-free realisation, modulo the named CAR functor”. Python-only (numpy).
• Lean (SeamResidualAxiom.lean). Adds the v463 kernel facts c8_three_candidates (A_8/D_8/E_8 each \dim{=}8(1{+}h^\vee)) and holomorphy_selects_e8 (forced \dim V_1{=}248{=}E_8 \neq 80,120) to residual_arithmetic (still [propext]-only); v464 is numerical. #print axioms seamResidualClosed still lists exactly FOUR axioms.
• Wolfram + integration. v463 mirrored (the three candidates, the forced 248, the E_4/\eta^8 tower); the extension count is now 365/365. Integrated across the contracts paper (attacks (xxx)/(xxxi)), README, zenodo, ledger and website. Net effect: across (v)–(xxxi) the SEAM.EQUIV.01 residual is now entirely certification — a named, hypothesis-audited package (MMST, AGT/AMT, Dong–Mason, Araki, Shale–Stinespring) with no open internal mechanism — yet SEAM.EQUIV.01 stays [O]because it still rests on cited continuum-existence theorems we do not re-prove.

Details

• v461_seam_strict_locality.py (SEAM.S3.LOCALITY.01). Sharpens v460's verdict from a cited statement to an exhibited topological obstruction on the same v367 p{+}ip collar. [E]CHERN INTEGER: the FHS Chern is C{=}{+}1 (chiral M{=}1), 0 (trivial M{=}3), -1 (opposite mass). [E]WILSON-LOOP / WANNIER-CENTRE WINDING ={} CHERN: the hybrid Wannier centre \theta(k_y) winds by |C|{=}1 (chiral) and 0 (trivial) over the BZ; exponentially-localised Wannier exist IFF the winding is 0 (Brouder–Panati et al., PRL 98 046402; Thouless), so the chiral collar has NONE. [E]OBSTRUCTION INTEGERS TIED: winding ={}|C|{=}1{\neq}0 and c_-{=}8{=}\gcar{+}\Nfam{=}\operatorname{rank}E_8{\neq}0 (the same 8 as c_3's one-sided count, v456). [E]NEG CONTROL: BOTH phases are gapped, yet only the trivial (winding 0) admits exp-localised Wannier and a strict finite-range commuting projector — it is the CHIRALITY, not the gap. [C][O] Kapustin–Fidkowski (arXiv:1810.07756): a strictly finite-range commuting projector would force winding 0, C{=}0; since C{=}1{\neq}0 NONE exists, so v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net (strict locality topologically forbidden, not a missing premise). SEAM.EQUIV.01 stays [O]. Python-only (numpy); the winding/Chern integers Wolfram- and Lean-mirrored.
• v462_seam_spinor_continuum.py (SEAM.EQUIV.SPINOR.01). The constructive companion of the v458 (MMST audit) / v459 (lattice-VOA) residual: the 128-spinor extension SO(16)_1{\to}(E_8)_1 exhibited three ways. [E](E_8)_1 TOWER FROM SO(16)_1 SECTORS: the exact Jacobi/E_8 identity \theta_2^8{+}\theta_3^8{+}\theta_4^8{=}2E_4 gives \chi_{(E_8)_1}{=}E_4/\eta^8{=}\chi_o^{SO16}{+}\chi_s^{SO16} — the holomorphic (E_8)_1 character IS the SO(16)_1 vacuum + spinor sector sum. [E]THE 128 SPINOR WEIGHT-1 STATES FILL 120{\to}248: \chi_o gives 120, \chi_s gives 128, summing to the exact tower \{1,248,4124,34752\} (248{=}120{+}128). [E]FINITE 16-MAJORANA FOCK: C(16,2){=}120 NS bilinear currents and 2^{16/2}{=}256 Ramond ground states ={}128 spinor + 128 cospinor. [E]FINITE-L\to CONTINUUM CONVERGENCE: the critical free-fermion ring Casimir gives c_-{\to}8 and the lowest scaled NS gap x_1{\to}1 with |x_1{-}1|{\sim}1/L^2 (L{=}64..1024). [C][O]AGT/AMT (v459) supply the rigorous extension; SEAM.EQUIV.01 existence (v336) stays [O]. Mixed exact (\theta identity + tower + 120/128/256, Wolfram-mirrored) + numerical (finite-L, Python-only).
• Lean (TfptCarrier/SeamResidualAxiom.lean, FORM.SEAM.RESIDUAL.01). The two former cited axioms mmst_existence \circ agt_lattice_extension are MERGED into ONE combined external axiom seam_realisation_theorem (dropping the intermediate ChiralCFT_c8), so #print axioms seamResidualClosed now lists exactly FOUR axioms (down from six). Two more kernel facts are added axiom-free: strict_locality_forbidden (v461: winding ={}|C|{=}1{\neq}0, c_-{\neq}0) and fock_sectors (v462: C(16,2){=}120, 2^8{=}256, spinor 128, 120{+}128{=}248); residual_arithmetic (now including both) depends only on [propext]. lake build OK. So the residual is reduced to ONE realisation axiom + ONE combined cited theorem.
• Wolfram. tfpt_readouts_extension.wl adds 3 exact mirrors (total 363/363): the v461 obstruction integer-logic and the v462 \theta-identity + 120{+}128{=}248 sector / Fock counts.

Details

• v460_seam_s3_lto.py (SEAM.S3.LTO.01). NPW26 (arXiv:2605.10693) prove the intrinsic Bisognano–Wichmann lemma for commuting-projector (local-topological-order) models with a \Z_2 reflection; v424 reduced the keystone BW residual to that theorem modulo two open sub-steps, of which (ii) was settled by v426 and the continuum lift of BW by the uniform-in-N tower v455. This module attacks the commuting-projector side of sub-step (i) on the same explicit v367 p{+}ip collar. [E]FLAT-BAND PARENT: the flattened Bloch Q{=}h/|h| has Q^2{=}I (flat \pm1 bands) and P{=}(I{-}Q)/2 is an exact projector EQUAL to v367's occupied ground-state projector — a frustration-free commuting-projector parent with the same gapped ground state (machine precision over a k-grid). [E]GAP-PROTECTED QUASI-LOCALITY: the gapped projector kernel |P(n,0)| decays rapidly (\xi{\approx}1.14 at M{=}1, exp R^2{=}0.995, |P(20,0)|{<}10^{-6}) and \xi grows as the gap closes (2.7 at gap 0.02); at the gapless point it decays as a clean power law |R|^{-2.1} (R^2{=}0.999) — locality needs the gap. [E]\Z_2 REFLECTION + \mu_4 INHERITED: the flattened modular Hamiltonian K_{\mathrm{flat}}{=}\operatorname{sign}(K) inherits \Theta K_{\mathrm{flat}}\Theta{=}{-}K_{\mathrm{flat}} (u_\Theta{=}J) and [\rho,K_{\mathrm{flat}}]{=}0 EXACTLY (\operatorname{sign} is odd; \rho commutes with any f(K)), with side-even and off-\mu_4 neg controls firing. [C][O]the chiral c_-{=}8{\neq}0 (FHS Chern |C|{=}1) obstructs STRICT finite-range commuting-projector locality (Kapustin–Fidkowski, arXiv:1810.07756; C{\neq}0 forbids exponentially-localised Wannier), so the realisation is the quasi-local LTO net NPW26 use — the form sub-step (i) asks for. Advances v424 sub-step (i); SEAM.EQUIV.01 stays [O](the continuum theorem MMST v336 is the backbone). Python-only (numpy); registered in run_all.py, the registry, the ledger and cited in tfpt_research_contracts (route (xxvii) of the convergent attacks).

Details

• v454_seam_edge_virasoro.py (SEAM.EQUIV.EDGE.VIRASORO.01). Tests the cited MMST “Koo–Saleur lattice generators \to continuum Virasoro with the correct c” fact on our own free-fermion edge (G2). [E]the Affleck–Cardy Casimir energy fits c_{\mathrm{Dirac}}{=}1.000 (\tfrac12 per Majorana, L{=}64..1024); [E]the lattice U(1) current closes its affine level exactly (\langle0|J_nJ_{-n}|0\rangle{=}n, n{=}1..8; a gapped control vanishes); [E]the level-1 Sugawara c{=}\dim G/(1{+}h^\vee){=}8 for both SO(16)_1 (120/15) and (E_8)_1 (248/31). [C]16{\cdot}\tfrac12{=}8{=}\gcar{+}\Nfam is the level-1 current-algebra c_-; SEAM.EQUIV.01 stays [O]. Sugawara Wolfram-mirrored.
• v455_seam_bw_uniform.py (SEAM.EQUIV.BW.UNIFORM.01). A uniform-in-N Tomita–Takesaki tower for the intrinsic Bisognano–Wichmann condition u_\Theta{=}J — the inductive-limit companion to v426, advancing v424 sub-step (i) (G1). [E]\|\Theta K\Theta{+}K\|{=}0 exactly at every N (N{=}8..128); [E]an N-independent gap 6\log\tfrac32; [E][\rho,K]{=}0 exactly at every N; [E]an N-independent neg-control margin 2\|K\|. [C][O]the inductive limit inherits u_\Theta{=}J; SEAM.EQUIV.01 stays [O]. Python-only.
• v456_seam_chirality_from_c3.py (SEAM.S3.FROM-P1.01). The edge chirality (c_-{\neq}0, the S3 phase) is forced by the one-sidedness that defines c_3{=}1/(8\pi) (P1) (G6). [E] c_3's “8” is the one-sided count 8{=}|\Z_2|\cdot(\!\int\!K/\pi){=}2\cdot4; [E]a reflection sends the Chern integer C{\to}{-}C (+1\!\to\!-1 on the p{+}ip model); [E]a two-sided boundary forces C{=}{-}C{=}0. [C]the one-sided seam (no reflection) forces C{\neq}0, magnitude c_-{=}8{=}\operatorname{rank}E_8 — S3 is a consequence of P_1; SEAM.EQUIV.01 existence stays [O]. The arithmetic is Wolfram-mirrored.
• v457_seam_character_killtest.py (SEAM.EQUIV.KILLTEST.01). An explicit character/sector kill test of (E_8)_1 against SO(16)_1 (G7). [E]the (E_8)_1 character \chi{=}E_4/\eta^8 with tower degeneracies \{1,248,4124,34752,\dots\}; [E]the weight-1 count 248{=}120{+}128 (not the free 120); [E]the sector count |\!\det\mathrm{Cartan}\,E_8|{=}1 (not |\!\det\mathrm{Cartan}\,D_8|{=}4). [X]a measured 120, four sectors, or a wrong coefficient would falsify (E_8)_1; the data give 248/1/\{1,248,4124\} (passed). SEAM.EQUIV.01 stays [O]. Wolfram-mirrored.
• v458_seam_mmst_citation_audit.py (SEAM.EQUIV.MMST.AUDIT.01). The exact hypothesis-by-hypothesis citation audit of MMST (arXiv:2107.13834) against the collar (G3, the “exakter Zitat-Abgleich”). [C]H1 CAR class, H2 massless limit, [E]H3 per-copy c{=}\tfrac12 (Thm 4.11), [C]H4 decoupled additivity to c{=}8, H5 the WZW range 8{\le}8{\le}16 with the 120 bilinear currents all match; [O]the single residual H6 is the 128-spinor extension SO(16)_1{\to}(E_8)_1 (\det K\,4{\to}1), outside MMST's bilinear method. SEAM.EQUIV.01 stays [O], its open part narrowed to this one holomorphic extension.
• v459_seam_lattice_voa_route.py (SEAM.EQUIV.LATTICEVOA.01). The second, independent route to (E_8)_1 supplying exactly the residual of v458 (G5). [E]E_8 even unimodular (\det{=}1); [E] the weight-1 space 248{=}8{+}240; [E]the 240 roots split 112 integer + 128 half-integer (the spinor, 248{=}120{+}128). [C]the lattice net A_{E_8} (Adamo–Giorgetti–Tanimoto, arXiv:2506.01008; OS arXiv:2407.18222) constructs the holomorphic extension; [O]the two routes leave one shared realisation input (collar ={}A_{E_8}, tied to P_1 by v456). SEAM.EQUIV.01 stays [O]. Wolfram-mirrored.
• Lean (FORM.SEAM.RESIDUAL.01, TfptCarrier/SeamResidualAxiom.lean). Kernel-checks the whole G-block arithmetic (the one-sided 8{=}|\Z_2|\!\int\!K/\pi, the reflection C{\to}{-}C, c_-{=}8, the MMST range 8{\le}8{\le}16, decoupled additivity, 248{=}8{+}240{=}120{+}128, 240{=}112{+}128, \det K\,1 vs 4) by decide, reducing the SEAM.EQUIV.01 residual to ONE named TFPT-internal realisation axiom (CollarRealizedAsLatticePhase) plus two cited external theorems (MMST v458, AGT v459); lake build OK.
• Wolfram + integration. Six exact mirrors added to tfpt_readouts_extension.wl (Sugawara c{=}8 for 120/15 and 248/31; the one-sided 8{=}2\cdot4 and reflection C{=}{-}C\Rightarrow0; the (E_8)_1 tower \{1,248,4124,34752\} with 248{=}120{+}128 and |\!\det\mathrm{Cartan}| 1 vs 4; the root split 112{+}128 and 248{=}8{+}240) — extension total now 360/360. Registry, ledger, master index, website mirror and manifests updated in the same change.

Details

• v449_seam_mmst_uniform.py (SEAM.EQUIV.MMST.UNIFORM.01). Strengthens the cited MMST hypothesis from convergence at a single cross-ratio (v444/v448) to the uniformity the theorem actually asserts, on the same v367/v368 collar. [E]cross-ratio A Q(1,2,4,7)/12\to2{+}\sqrt3, the deviation shrinking monotonically; [E]an independent cross-ratio B Q(1,3,5,9)/12\to its own value 2, so convergence is not a tuned point; [E]a uniform 1/N rate |Q(N){-}Q_\infty|\cdot N\le4 for both observables at every N_x\in\{36,48,60\} (one N-independent constant); [E]the bulk control is off-scale. [C]the MMST uniform-convergence hypothesis is empirically satisfied; SEAM.EQUIV.01 stays [O].
• v450_seam_edge_entanglement.py (SEAM.EQUIV.EDGE.ENTANGLEMENT.01). A third independent reading of the edge central charge (after the correlator v444 and the topology v447). [E]the Calabrese–Cardy entanglement law gives c_{\mathrm{Dirac}}{=}1.000 for the critical free-fermion chain; [E]so c per Majorana =1/2; [E]a gapped control saturates (c{=}0). [C]bulk–edge: c_-{=}N_{\mathrm{Maj}}\cdot\tfrac12{=}16\cdot\tfrac12{=}8{=}\gcar{+}\Nfam. SEAM.EQUIV.01 stays [O].
• v451_seam_edge_cardy_tower.py (SEAM.EQUIV.EDGE.CARDY.01). Names the edge CFT by its operator content. [E]the Cardy finite-size spectrum gives the spin dimension h_\sigma{=}1/16 (the periodic–antiperiodic ground-energy split) and [E]the energy dimension h_\epsilon{=}1/2 (the lowest NS pair excitation), both [E]L-independent (conformal 1/L scaling). [C]\{c,h_\sigma,h_\epsilon\}{=}\{\tfrac12, \tfrac1{16},\tfrac12\} is the chiral Ising (free-Majorana) minimal model M(3,4); 16 copies give c_-{=}8. SEAM.EQUIV.01 stays [O].
• v452_seam_e8_modular.py (SEAM.EQUIV.E8.MODULAR.01). The (E_8)_1 torus modular signature, extending v156's q-expansion to the transformation data. [E]the character \chi{=}E_4/\eta^8 is S-invariant \chi(-1/\tau){=}\chi(\tau) (one primary, S{=}[1]); [E]T-phase \chi(\tau{+}1)/\chi(\tau){=}e^{-2\pi i/3}{=}e^{-2\pi i c/24} (c{=}8); [E]leading power \chi\,q^{1/3}\to1 so q^{-c/24}{=}q^{-1/3}\Rightarrow c{=}8; [E]holomorphic c{=}8\equiv0\bmod8 with T-phase order 24/\gcd(8,24){=}3. [C]the (E_8)_1 identity is now pinned on the torus and the edge. SEAM.EQUIV.01 stays [O].
• v453_seam_mu4_from_marks.py (SEAM.RIGIDITY.MU4FROMMARKS.01). Folds the last structural premise of the rigidity chain (v446's residual QGEO.SYM.01) into the existing realisation premise. [E]the four Gauss–Bonnet marks are \mu_4{=} roots of z^4{-}1; [E]the order-4 clock z\!\to\!iz is a 4-cycle permuting them (the v445 clock); [E]it preserves the cross-ratio \lambda{=}2 (in the D_4 stabiliser, v168); [E]the form basis \omega_k{=}z^{k-1}dz/(z^4{-}1)\to i^k\omega_k is \mu_4-graded (weights (1,2,3){=}A_3 exponents). [C]so QGEO.SYM.01 follows from marks=\mu_4 + the existing QGEO.REALIZE.01 — no new premise; the chain (v453\tov446\tov445) closes modulo realisation. SEAM.EQUIV.01 stays [O].
• Lean: TfptCarrier/SeamEdgeChern.lean (FORM.SEAM.EDGE.CHERN.01). Kernel-hardens the integer backbone the four edge reads converge on: the FHS Chern integers (+1/0/-1), the bulk–edge channel count |C|{=}1, N_{\mathrm{Maj}}{=}2^{\gcar-1}{=}16, the assembly 2c_-{=}N_{\mathrm{Maj}}\cdot|C| (so c_-{=}8), c_-{=}8{=}\gcar{+}\Nfam{=}\mathrm{rank}\,E_8, holomorphic c\equiv0\bmod8 and the order-3 modular T-phase — all by kernel decide (edge_kernel_facts, axioms propext only). lake build OK, #print axioms clean.
• Wolfram mirror. tfpt_readouts_extension.wl gains nine exact checks (v452 modular data \times4, v450 entanglement assembly, v451 Ising Kac weights, v453 marks=\mu_4 \times3): 345\to354 checks, all passing (GATE.WOLFRAM.02). E_4 is evaluated via its q-series since EisensteinE stays symbolic for complex \tau.
• Papers. The adversarial audit (tfpt_5_redteam, Target A) and the research-contracts note record the edge CFT as named end-to-end — central charge c_-{=}8 from three independent observables (correlator v444, topology v447, entanglement v450), the Ising operator tower (v451), the (E_8)_1 torus modular data (v452), and the \mu_4 premise derived from the marks (v453) — while the only residual that stays [O]remains the cited continuum-existence theorem (v336).

Details

• v446_seam_clock_invariance.py (SEAM.RIGIDITY.CLOCKINV.01). Closes v445's single residual one notch further: the operator condition [\rho,\Lambda_\Sigma]{=}0 (clock-invariance) is shown to be a consequence of a structural symmetry, not an independent assumption. [E]the OS canonical transfer \Lambda{=}C(1)C(0)^{-1}{=}e^{-h} (fixed by RP+reflection, v54) of a \mu_4-invariant RP covariance satisfies [\rho,\Lambda]{=}0 (swept N{=}4..32); [E]a \mu_4-breaking covariance gives [\rho,\Lambda]\neq0 (iff-sharp neg control); [E]\Lambda is a positive contraction (spectrum in (0,1]); [E]the free modular Hamiltonian K{=}\log((1{-}C)C^{-1}) (v258) of the invariant state has [\rho,K]{=}0 (Tomita–Takesaki), broken control \neq0. [C]so with v445 (commute \Leftrightarrow block-diagonal), RP-definability plus the four marks force the block-diagonal \Lambda_\Sigma; the residual reduces from an operator commutation to “the seam RP data is \mu_4-symmetric” (QGEO.SYM.01). SEAM.EQUIV.01 stays [O].
• v447_seam_edge_chern.py (SEAM.EQUIV.EDGE.CHERN.01). An independent (topological) reading of the edge central charge, corroborating v444's correlator route from a different observable. [E]the M{=}1 collar has integer Chern number C{=}{+}1 (Fukui–Hatsugai–Suzuki link-variable method, no fit); [E]C{=}0 in the trivial phase and C{=}{-}1 at M{=}{-}1 (opposite chirality); [E]the open strip carries exactly |C|{=}1 chiral edge branch per boundary; [E]the Li–Haldane entanglement spectrum has in-gap modes at 1/2 only in the topological phase. [C]bulk–edge: c_-{=}N_{\mathrm{Maj}}\cdot\tfrac12\cdot|C| {=}16\cdot\tfrac12{=}8{=}\gcar{+}\Nfam — the same c_-{=}8 as v444. SEAM.EQUIV.01 stays [O].
• v448_seam_edge_fourpoint.py (SEAM.EQUIV.EDGE.FOURPOINT.01). Pins the one external MMST fact (continuum existence of the chiral scaling limit) empirically at the next order. [E]the ground-state correlation matrix is an exact projector G^2{=}G (\sim10^{-13}), so the state is quasi-free (the MMST CAR hypothesis holds); [E]the edge four-point cross-ratio Q{=}[G_{01}G_{23}]/[G_{02}G_{13}] at commensurate positions is Cauchy-convergent under refinement; [E]Q converges to the free-fermion conformal cross-ratio \sin(5\pi/12)/\sin(\pi/12){=}2{+}\sqrt3\approx3.732; [E]the bulk control is off-scale (exponential). [C]so the one external fact is supported at both two-point (v444) and four-point order. SEAM.EQUIV.01 stays [O].
• Lean: TfptCarrier/SeamRigidityForcing.lean (FORM.SEAM.RIGIDITY.FORCING.01). Kernel-hardens v445 (and v442's uniformity): the entrywise forcing iff \mathrm{clock4}\,a{=}\mathrm{clock4}\,b\Leftrightarrow a{=}b (i.e. i^a{=}i^b\Leftrightarrow a\equiv b\bmod4, residue-only hence uniform in N), the four distinct eigenphases, the exact commutant dimension \sum n_s^2{=}64{<}256 and the order discriminator (128{>}64) are all proved by kernel decide (forcing_kernel_facts, no domain axioms); only the operator-level lift to the full L^2 is the named premise. lake build OK, #print axioms clean.
• Wolfram mirror. tfpt_readouts_extension.wl gains three exact v445 checks (entrywise forcing for all a,b\in0..31; commutant dimension 64{<}256; order-2 commutant 128{>}64 swept N{=}4..64): 342\to345 checks, all passing (GATE.WOLFRAM.02).
• Papers. The adversarial audit (tfpt_5_redteam, Target A) records the five-direction narrowing of the SEAM.EQUIV.01 keystone (edge scaling v444, forcing converse v445 + Lean, derived clock-invariance v446, independent topological c_-{=}8 v447, four-point pinning v448) explicitly as “not promoted”: the only residual that stays [O]is still the cited continuum-existence theorem (v336). The research-contracts note lists the same as “eleven convergent attacks”.

Details

• v445_seam_rigidity_forcing.py (SEAM.RIGIDITY.FORCING.01). The rigidity route (v398/v442) showed a \mu_4-block-diagonal K commutes with the order-4 clock \rho{=}\operatorname{diag}(i^n). This module proves the converse and the exact structure, isolating the single remaining physical premise. [E]FORCING (entrywise iff): [\rho,K]{=}0 iff K_{ij}{=}0 whenever i{\not\equiv}j\bmod4, verified entrywise and swept N{=}4..128 — commuting forces \mu_4-block-diagonality, it does not merely permit it. [E]EXACT COMMUTANT DIMENSION: \dim\{K:[\rho,K]{=}0\}{=}\sum_s n_s^2 (the \mathrm{ad}_\rho nullspace), a proper subspace of the full N^2 algebra (N{=}16: 64{<}256) — a genuine quantitative restriction. [E]OFF-BLOCK \Leftrightarrow NON-COMMUTING: every off-block matrix unit has \|[\rho,E_{ij}]\|{\ge}\sqrt2{>}0 (zero slack). [E]THE ORDER IS THE DISCRIMINATOR: the order-4 clock gives 4 sectors; the order-2 clock leaves a strictly larger commutant (N{=}16: 64 vs 128) — the four Gauss–Bonnet marks (order 4{=}|\mu_4|{=}h(A_3)) force strictly more structure than two, and only the index-4 extension is (E_8)_1 (\det{=}1; v281/v154/v351). [E]NONDEGENERACY: exactly 4 distinct eigenvalues \{1,i,-1,-i\} for all N{\ge}4. [C]SYNTHESIS: commuting with the order-4 \mu_4 clock is equivalent to \mu_4-block-diagonality, so RP-definability (v54/v398) + clock-invariance force the block-diagonal \Lambda_\Sigma — v442's “permit” upgraded to “force”. [O]the single residual is now precisely that the physical transfer lies in that commutant (clock-invariance = QGEO.SYM.01, v323/v335); SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O]. Python-only (numpy; det facts Wolfram-mirrored via v89/v281/v422). Registered in run_all.py, script_registry.csv, the ledger, and cited in tfpt_research_contracts (route (xii) of the convergent attacks, and the sharpened residual in the Seam State Rigidity contract).

Details

• Context. Two of the seven convergent attacks on the keystone residual are turned from numerical witnesses into kernel-checked facts, in the audit-contract style of SeamScalingLimit.lean / BWKeystone.lean: the load-bearing algebra/bookkeeping is proved by the Lean kernel, while the cited theorems and the one open continuum input remain named axioms. lake build OK; #print axioms clean. Tracked as two new FORM.* ledger rows; \veri-context in tfpt_research_contracts; Lean manifest refreshed.
• FORM.SEAM.BW.HSMI.01 (TfptCarrier/SeamStandardPair.lean) — standard-pair relations, kernel-checked (hardens v438). The four Borchers/Wiesbrock relations on the centred 5-mode ladder over \mathbb{Z} are proved by kernel decide (axioms propext / Classical.choice / Quot.sound only, no domain axiom): [K,P]{=}P, J^2{=}1, JKJ{=}{-}K, JPJ{=}P_+ (and \operatorname{tr}K{=}0). The cited Borchers/Wiesbrock theorem and the one continuum input (the continuum standard pair) are named axioms; the composition pins face (ii) of SEAM.EQUIV.01 to exactly those two — intrinsic Bisognano–Wichmann, no rotation-covariance presupposed.
• FORM.SEAM.APPLICABILITY.01 (TfptCarrier/SeamApplicabilityLedger.lean) — the audit count, kernel-checked (hardens v441). The headline bookkeeping is itself machine-verified: of the 12 cited-theorem hypotheses (MMST/NPW26/Adamo, mirrored in order) exactly 11 are internal and 1 external (total_count=12, internal_count=11, external_count=1), plus mmst_range (8\le8\le16) and npw_outside_bucket (\det K{=}1\neq4) — ALL with NO axioms (pure kernel decide). The single external fact (continuum scaling-limit existence) is the named axiom; the keystone reduction pins its external dependence to exactly that one fact.
• Net. Both modules are imported by the root TfptCarrier.lean, axiom-audited in AxiomCheck.lean, and signature-locked in AuditContract.lean; full lake build succeeds. No new mathematics — the same facts v438/v441 verify numerically are now kernel-proved; SEAM.EQUIV.01 stays [O].

Details

• Context. v439 grounded the bulk half of the cited MMST existence (uniform gap, exponential clustering, finite light cone) and explicitly narrowed the one open analytic step to the chiral massless edge scaling limit. v444 attacks exactly that residual on the SAME explicit v367/v368 p{+}ip collar. vN-registered, runs in run_all.py (suite ends ALL CHECKS PASSED), \veri-cited in the body of tfpt_research_contracts (the G_{\mathrm{net}} contract). Python-only (numpy; no new exact identity to Wolfram-mirror).
• v444_seam_edge_scaling.py (SEAM.EQUIV.CONTINUUM.EDGE.01) — chiral-edge motor. The lattice edge carries the kinematic and conformal data of a chiral c{=}1/2 Majorana CFT: [E]a linear, chiral in-gap dispersion (velocity v{\approx}0.98, R^2{=}0.9999, with opposite per-edge velocities — one chiral Majorana per edge); [E]an algebraic equal-time edge two-point \sim r^{-\eta} with \eta{\approx}1, versus the exponential bulk row and the exponential trivial-phase (M{=}3, no edge) control; [E]a Cauchy-convergent scaling-collapse under refinement (\eta{=}1.001{\pm}0.007, successive profile spread 0.028{\to}0.013 across N_x{=}36..60) — the numerical hallmark of an existing scaling limit. [C]\eta{=}1\Leftrightarrow one chiral Majorana (c{=}1/2) per edge, so the 16-copy carrier has a chiral c_-{=}8 (E_8)_1 edge CFT (c_-{=}\gcar{+}\Nfam). [O]the rigorous uniform-convergence theorem (cited MMST, v336) is not supplied.
• Net. With v439 (bulk) and v444 (edge), both halves of the one continuum residual are numerically grounded on our explicit model and pinned to a single cited existence theorem; SEAM.EQUIV.01 stays [O]. Ledger row, script_registry.csv index and the generated surfaces (verification.tex, ScriptIndex.tsx) updated in the same change; manifest refreshed last.

Details

• Context. The keystone SEAM.EQUIV.01 is “closed modulo a cited theorem”: its one open input is the continuum existence of the seam's chiral massless scaling limit. Six new modules attack that residual from independent directions; each is honest about not closing it. All six are vN-registered, run in run_all.py (suite ends ALL CHECKS PASSED), and are \veri-cited in the body of tfpt_research_contracts (the G_{\mathrm{net}} contract). Python-only (no new exact identity to Wolfram-mirror).
• v438_seam_hsmi_borchers.py (SEAM.EQUIV.BW.HSMI.01) — intrinsic BW. The seam supplies a Longo standard pair / +half-sided modular inclusion (positive lightlike translation P\ge0, boost K with [K,P]{=}P, reflection J with JKJ{=}{-}K, JPJ{=}P_-), so by Borchers' theorem the modular flow is geometric intrinsically — without presupposing rotation-covariance of the vacuum (complementary to Adamo–Giorgetti–Tanimoto). [E]ladder algebra exact, sl(2,\mathbb{R}) positivity (min eig >0); [C]synthesis; [O]the continuum standard pair.
• v439_seam_lieb_robinson.py (SEAM.EQUIV.CONTINUUM.LR.01) — continuum motor. The explicit gapped p{+}ip collar (v367) has a uniform-in-N gap, exponential clustering with N-independent \xi, and a finite Lieb–Robinson light cone, so by Nachtergaele–Sims its bulk thermodynamic limit exists — grounding the bulk half of the cited MMST existence in our model and narrowing the open step to the chiral edge scaling limit. [E]/[C], [O]the edge CFT.
• v440_seam_lto_rp_beta.py (SEAM.EQUIV.LTORP.BETA.01) — \beta-homotopy. The (LTO-RP)/BW data (\Theta K\Theta{=}{-}K, [\rho,K]{=}0) are \beta-independent, and along the whole path C_\beta{=}1/(1{+}e^{\beta K}) from the NPW26-proven trace state (\beta{=}0) to the seam KMS state (\beta{=}1) every covariance is reflection-related, \mu_4-invariant and valid — a norm-continuous, obstruction-free homotopy, upgrading the single point of v426 to a path argument for sub-step (ii) of v424.
• v441_seam_applicability_ledger.py (SEAM.EQUIV.APPLICABILITY.01) — audit. A hypothesis-by-hypothesis ledger of the three cited theorems (MMST / NPW26 / Adamo–Giorgetti–Tanimoto) finds 11 of 12 hypotheses are TFPT-internal computed facts and exactly one is external — the continuum existence of the chiral scaling limit; covariance is an Adamo consequence (defuses the v215 circularity), not an input. The keystone's external dependence is pinned to that one analytic fact.
• v442_seam_rigidity_uniform.py (SEAM.RIGIDITY.UNIFORM.01) — uniform rigidity. The band-limited \mu_4-character rigidity of v398 is uniform in N: exact 4-sector commutation to N{=}256 (not merely the N\le64 sweep), a uniform gap, and an N-independent clock-breaking lower bound \|[\rho,V]\|{=}2 — so it is not a small-N artifact (the full-L^2 forcing stays [O]).
• v443_seam_e8_discriminator.py (SEAM.E8.DISCRIMINATOR.01) — kill tests. Mode D of the closure-mode classification (v347) as concrete falsifiable discriminators against the nearest c{=}8 rival SO(16)_1: torus ground-state degeneracy 1 vs 4, level-1 currents 248 vs 120, lattice theta 240 vs 112 norm-2 roots; c{=}8 alone cannot decide (v277), only \det K{=}1 (the genus-1 statement, v90) selects E_8. A seam with GSD{>}1 would falsify (E_8)_1. [X]/[C].
• Status discipline. SEAM.EQUIV.01 stays [O]throughout; the six modules reduce, audit and falsify the residual but do not supply the one cited continuum-existence theorem. Ledger rows, the script_registry.csv index, and the generated surfaces (verification.tex, ScriptIndex.tsx) are updated in the same change; manifest refreshed last.

Details

• Context (reviewer finding, Alessandro). On the exported tfpt5_shadow.zip the independent surfaces reproduced (build.sh notes () 11 ok; Wolfram base/extension; Red Team; Lean transcript), but the package layer did not: make_manifest.py --check reported a stale verification/website_map.csv and build.sh audit crashed because it unconditionally required website/lib/release.ts, which the shadow subset deliberately excludes.
• make_docs_map.py is now shadow-aware. website_map.csv mirrors the website/ sources; on a tree without website/ the generator now skips it instead of regenerating it down to a single README.md row. This removes the silent corruption that made build.sh notes/gen leave website_map.csv stale against the manifest on the export (it now matches the established skips in make_script_index.py and make_changelog_web.py).
• audit_sync.py is now shadow-aware. When website/ is absent it skips the website mirror (section D), the website version stamps (E) and the website Wolfram-count card (F), and the website generated-surface freshness in (A), printing exactly which sections were skipped. build.sh audit therefore runs to a clean AUDIT OK on the shadow subset instead of raising FileNotFoundError.
• Export coherence gate. scripts/export-shadow.sh now re-runs make_manifest.py --check and the shadow-aware audit_sync.py on the filtered tree before the mirror is published. The manifest check catches the earlier v235--v298 symptom at the source: a module referenced by run_all.py/script_registry.csv/manifest.sha256 but not committed (and thus dropped by git ls-files) now fails the export loudly instead of shipping a package that says ALL CHECKS PASSED while run_all.py stops at the first missing script.
• Manifest refreshed. manifest.sha256 was regenerated as the last release step so the shipped digests match the committed changelog.tex and website_map.csv; the full Python suite reproduces end-to-end on the regenerated export. No theory content, status marker or vN module changed — this is an export/infrastructure fix only.