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Original ledger classification: Formal [F] (Lean kernel): the G-block arithmetic (one-sided 8=|Z2|*intK/pi, reflection C->-C, c_-=8, MMST range 8<=8<=16, decoupled additivity, 248=8+240=120+128, 240=112+128, det K 1 vs 4, + v461 winding=|C|=1!=0 obstruction + v462 Fock counts 120/128/256 + v463 holomorphy selector forced dimV1=248=E8) all kernel-proved by decide; the two cited theorems MERGED into ONE combined axiom seam_realisation_theorem so seamResidualClosed depends on FOUR axioms (down from six); v464 one-particle rigidity is numerical; residual reduced to ONE realization axiom + ONE combined cited theorem; PLUS the v469 parallel peer-reviewed route seamResidualClosed' (invariant-level collar_invariants + crossedproduct_route_theorem; new kernel facts lr_locality_integer, glue_locality_integers, klm_holomorphic, sixteenfold_pin, all [propext only]); SEAM.EQUIV.01 stays [O]

Recorded statement: LEAN 4 FORMALISATION reducing the SEAM.EQUIV.01 residual to a SINGLE named realization axiom (2026-06-30, G4 of the post-F next steps) -- new module TfptCarrier/SeamResidualAxiom.lean, lake build OK. It kernel-checks the arithmetic…

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Paper context: tfpt research contracts

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FORM.SEAM.RESIDUAL.01

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LEAN 4 FORMALISATION reducing the SEAM.EQUIV.01 residual to a SINGLE named realization axiom (2026-06-30, G4 of the post-F next steps) -- new module TfptCarrier/SeamResidualAxiom.lean, lake build OK. It kernel-checks the arithmetic backbone of the whole G-block over Nat/Int: [F] c3_eight_one_sided (the one-sided count |Z2|*(intK/pi)=2*4=8, v456), [F] two_sided_nonchiral (a reflection sends C->-C so a two-sided boundary forces C=0), [F] cMinus_eq_eight (c_-=8=g_car+N_fam=rank E8), [F] mmst_range (rank<=c<=D, 8<=8<=16), [F] decoupled_additivity (16*1/2=8), [F] currents_fermionic (120 so(16) bilinears) + currents_lattice (248=8+240), [F] roots_split (240=112+128, 248=120+128), [F] holomorphic_discriminator (det K_E8=1 vs det K_SO16=4) -- ALL by kernel decide. UPDATE (2026-06-30, v461/v462 round): the two cited existence theorems are MERGED into ONE combined external axiom seam_realisation_theorem (dropping the intermediate ChiralCFT_c8), so #print axioms seamResidualClosed now lists exactly FOUR axioms (CollarRealizedAsLatticePhase, SeamIsE8, collar_realizes, seam_realisation_theorem), down from six; and two more kernel facts are added -- [F] strict_locality_forbidden (v461: the Wannier winding=|C|=1!=0 and c_-=8!=0, so a strict finite-range commuting projector is topologically forbidden) and [F] fock_sectors (v462: C(16,2)=120 currents, 2^8=256 Ramond states, spinor 128, 120+128=248). residual_arithmetic (now incl. v461/v462) depends only on [propext]. So the residual is reduced to ONE TFPT-internal realization axiom + ONE combined cited theorem. UPDATE (2026-06-30, v463/v464 round): two more kernel facts -- [F] c8_three_candidates (A8/D8/E8 each dim=8*(1+h^v), so c=8 is not unique at level 1) and [F] holomorphy_selects_e8 (the forced dim V_1=248=dim E8 != 80(A8) != 120(D8), so holomorphy promotes 'c=8' to 'E8' uniquely, v463); v464 (the one-particle realization rigidity) is NUMERICAL (symbol idempotency + Cauchy convergence + entanglement c-fit), so it sharpens the meaning of collar_realizes ('the unique quasi-free realization, modulo Araki/Shale-Stinespring') but is not a kernel fact. seamResidualClosed stays exactly FOUR axioms; residual_arithmetic (now incl. v463) depends only on [propext]. UPDATE (2026-07-02, v469 round): a PARALLEL peer-reviewed derivation seamResidualClosed' is added -- the invariant-level axiom collar_invariants (R1': quasi-free [C] + gap [E] + class D + c_-=8 [E from P1], replacing the model-level fiat) plus ONE combined 1995-2001 package crossedproduct_route_theorem (Kitaev/Kapustin-Spodyneiko phase classification o MMST+Boeckenhauer scaling limit o Longo-Rehren/BE local Z2 crossed product + KLM o Dong-Mason uniqueness); its joints are NEW kernel facts with NO axioms -- [F] lr_locality_integer (h_s = 16/16 in Z), [F] glue_locality_integers (mu4 h(J^k) = {1,1,1} in Z), [F] klm_holomorphic (4/2^2 = 1 and 16/4^2 = 1), [F] sixteenfold_pin (nu = 16 = 0 mod 16, c_- = nu/2 = 8), bundled as crossedproduct_arithmetic [propext only]. #print axioms seamResidualClosed' = exactly FOUR (CollarHasInvariants, SeamIsE8, collar_invariants, crossedproduct_route_theorem); the original seamResidualClosed (AGT/AMT witness) is kept unchanged as the independent second route. lake build OK (3285 jobs).

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Formal

Dependencies as recorded

SEAM.EQUIV.01; SEAM.S3.FROM-P1.01; SEAM.S3.LOCALITY.01; SEAM.EQUIV.KILLTEST.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.SPINOR.01; SEAM.EQUIV.UNIQ.01; SEAM.EQUIV.ONEPARTICLE.01; SEAM.EQUIV.CROSSEDPRODUCT.01; v456; v457; v458; v459; v461; v462; v463; v464; v469

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Research contracts
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Research Contract 1 — [formula in source]tfpt research contracts
2026-06-30 ( v454– v459 + Lean + Wolfram — the post-F ``G-block'': the lattice Sugawara current algebra, a uniform-in- [formula in source] Tomita–Takesaki tower, chirality forced by [formula in source] , an [formula in source] character kill test, the exact MMST citation audit and the complementary lattice-VOA route)Research changelog; TFPT — Introduction and reading guide
2026-06-30 ( v461/ v462 + Lean axiom merge — the strict-locality obstruction made explicit, the 128-spinor extension exhibited at character level, and the SEAM.EQUIV.01 residual reduced to ONE realisation axiom [formula in source] ONE combined cited theorem)Research changelog; TFPT — Introduction and reading guide
Front matter / supportResearch contracts
Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
2026-07-02 ( v469/ v470 — the closure-route round on the two named targets: the SEAM.EQUIV.01 extension leg re-founded on 1995–2001 peer-reviewed crossed-product theorems [formula in source] the realisation axiom reduced to invariant level; the [formula in source] level [formula in source] the computed bulk Chern invariant [formula in source] the seam [formula in source] -normalisation [formula in source] the embedding index [formula in source] ; both targets stay )Research changelog; TFPT — Introduction and reading guide
Research Contract 1 — [formula in source]tfpt research contracts

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