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A registered result in its source context

Original ledger classification: Contract [O] (with [E] arithmetic + [X] hooks)

Recorded statement: 4D LAGRANGIAN VIA THE SPECTRAL ACTION (2026-06-17): the honest CONTRACT for the Phase-3 structural gap (the 2d chiral seam net v156/v175 + its dynamics v238-v243 -> the physical 4d theory). The natural TFPT bridge is NOT a holographic…

Statement, assumptions and sources

Paper context: tfpt research contracts

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CONTRACT.QFT4D.01

Original ledger classification: Contract [O] (with [E] arithmetic + [X] hooks)

Original claim text

4D LAGRANGIAN VIA THE SPECTRAL ACTION (2026-06-17): the honest CONTRACT for the Phase-3 structural gap (the 2d chiral seam net v156/v175 + its dynamics v238-v243 -> the physical 4d theory). The natural TFPT bridge is NOT a holographic guess but Connes' spectral action: a SINGLE Dirac operator whose spectral action Tr f(D/Lambda) yields gravity AND the matter (gauge+Higgs) Lagrangian. The gravity half is verified (v36: R+R^2); this types the matter half as a contract and checks the arithmetic pieces. (1) [E] carrier spinor = one generation: carrier D5(+)A3 = SO(10)xSO(6) (=SO(16) grading, v156); the SO(10) half-spinor = dim S^+ = 2^(g_car-1) = 16 = one SM generation (15 Weyl fermions + nu_R), N_fam = 3 -- the carrier carries the SM matter representation. (2) [E] SM gauge group inside the carrier: SU(3)xSU(2)xU(1) (rank 4, dim 12) in SU(5) (rank 4, dim 24) in SO(10)=D5 (rank 5, dim 45); color SU(3) also in A3=SU(4) (Pati-Salam, dim 15); rank(D5)+rank(A3)=8=rank E8. (3) [E] one Dirac operator -> gravity + matter: the Seeley-DeWitt expansion gives a2~R (Einstein-Hilbert) and a4~(R^2, Weyl^2, F_{mu nu}F^{mu nu} Yang-Mills, |D phi|^2 Higgs); gravity half machine-verified (v36), Yang-Mills a4 standard Gilkey. (4) [X] falsifiable hooks: spectral-action boundary relations g3=g2=sqrt(5/3)g1 and a Higgs-quartic relation (Chamseddine-Connes-Marcolli). (5) [O] CONTRACT.QFT4D.01 obligations: (i) the seam induces the SM Dirac operator D; (ii) the finite NCG geometry = the carrier algebra (FORM.P2.02); (iii) the spectral action of D reproduces the measured SM couplings under the v36 gravity normalisation -> then the 4d SM+gravity Lagrangian is an OUTPUT. The genuine Phase-3 target, NOT closed

Original classification detail

[E] carrier half-spinor = one SM generation (16), SM gauge group embeds in the carrier, one Dirac operator gives gravity (v36) + matter (Gilkey); [X] unification + Higgs hooks; [O] CONTRACT.QFT4D.01: the spectral action of the seam Dirac operator = 4d SM+gravity Lagrangian -- the Phase-3 target, not closed

Dependencies as recorded

FORM.P2.02; QG.G2.01; DHR.SECTORS.01

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Registered verification modules

  • v244_spectral_action_contract.pyRegistered verification module · registry
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    CONTRACT.QFT4D.01 — the 4d Lagrangian via Connes' spectral action: the honest contract for the Phase-3 gap (2d seam net → 4d physics). A single Dirac operator whose Tr f(D/Λ) yields gravity AND matter. [E] carrier spinor = one generation: SO(10) half-spinor = dim S⁺ = 2^(g_car−1) = 16 (15 + ν_R), N_fam=3. [E] SM gauge group inside the carrier: SU(3)×SU(2)×U(1) (dim 12) ⊂ SU(5) (24) ⊂ SO(10)=D5 (45); color SU(3) ⊂ A3=SU(4) (Pati-Salam); rank(D5)+rank(A3)=8=rank E8. [E] one Dirac operator → gravity (a2~R, a4~R², v36) + matter (a4 ~ F² Yang–Mills + |Dφ|² Higgs, Gilkey). [X] hooks: g3=g2=√(5/3)g1 + a Higgs-quartic relation (Chamseddine–Connes–Marcolli). [O] CONTRACT.QFT4D.01: obligations (seam→Dirac D; finite geometry = carrier; spectral action reproduces SM couplings) → the 4d SM+gravity Lagrangian is an OUTPUT. The Phase-3 target, not closed (complement of U_wall/G_metric).

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