Original ledger status
DHR.SECTORS.01
PARTICLES = DHR SUPERSELECTION SECTORS (2026-06-17): the discriminant/anyon content of the carrier and the condensation tower (v235/v237) made explicit as an abelian modular tensor category (MTC) -- the particle-spectrum layer of the emergent QFT. The carrier discriminant form is the Lean-verified glue form (FORM.GLUE.01): on Z4 x Z4, q(x,y)=(5x^2+3y^2)/8 mod 1, bilinear B((x,y),(x',y'))=(5xx'+3yy')/4 mod 1, topological spin theta(a)=e^{2 pi i q(a)}. (1) [E] 16 sectors = Z4 x Z4 (= det K_carrier = dim S^+ = 16, v235); fusion = abelian group law (closed, associative, vacuum (0,0), inverses). (2) [E] spin/condensability: bosonic <=> theta=1 <=> q in Z <=> 5x^2+3y^2=0 mod 8 -- exactly the six isotropic elements {(0,0),(1,1),(1,3),(2,2),(3,1),(3,3)}. (3) [E] condensation tower = v235/v237: condensing the Lagrangian H1=<(1,1)> (order 4, self-dual H1^perp=H1) leaves |H^perp/H|=1 sector = (E8)_1 (det 1, holomorphic); the order-2 isotropic <(2,2)> leaves 4 sectors = SO(16)_1 = D8 (det 4); so carrier 16 -> D8 4 -> E8 1. (4) [E] NEG: a non-isotropic anyon (1,0), q=5/8, theta != 1, is confined; <(1,0)> contains non-isotropic elements so cannot be a condensation algebra -- the tower is forced by isotropy. (5) [C] identifying sectors / deconfined anyons with SM particle multiplets is the interpretation; the MTC data (fusion + spins + tower) is exact
Recorded status: Numerical (finite MTC) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] 16-sector abelian MTC from the glue form, fusion = group law, condensation tower 16->4->1 = v235/v237 (Lagrangian -> E8, order-2 -> D8); [C] sector<->SM-multiplet identification -- particles = stable defect sectors of the seam code, made precise
Dependencies: GATE.HOLO.02; GATE.HOLO.03; FORM.GLUE.01
Scripts: v241_dhr_sectors.py
Location: tfpt_research_contracts
Original CSV, line 327Original ledger status
DHR.MODULAR.01
SPIN-STATISTICS + MODULAR (S,T) DATA (2026-06-17): the exchange statistics, modular S/T matrices, Verlinde fusion and Gauss-Milgram central charge of the carrier MTC -- the statistics+kinematics layer on top of v241. Same abelian MTC: Z4 x Z4, q(x,y)=(5x^2+3y^2)/8 mod 1 (the Lean glue form FORM.GLUE.01), B(a,b)=q(a+b)-q(a)-q(b), theta(a)=e^{2 pi i q(a)}. (1) [E] spin-statistics: T=diag(theta_a) splits the 16 anyons into 6 BOSONS (theta=1, the condensable/isotropic set), 2 FERMIONS (theta=-1) = {(0,2),(2,0)}, and 8 genuine anyons. (2) [E] modular S_ab=(1/sqrt 16) e^{-2 pi i B(a,b)} is unitary, symmetric, S^2=C (charge conjugation, C_ab=delta(b,-a)), S^4=I. (3) [E] Verlinde N_ab^c = sum_x S_ax S_bx conj(S_cx)/S_0x = delta(c,a+b) -- the fusion ring is RECOVERED from the modular data, not posited. (4) [E] Gauss-Milgram (1/sqrt 16) sum_a theta_a = e^{2 pi i c/8} with c=8=g_car+N_fam -- the anyon data already encodes the chiral central charge of the (E8)_1 net (v156/v175); statistics and c=8 are one fact. (5) [C] the boson/fermion split is the kinematic skeleton of the matter content; mapping sectors to physical fermion multiplets is the interpretation
Recorded status: Numerical (finite MTC) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] T=diag(theta) spin-statistics (6 bosons, 2 fermions, 8 anyons); modular S unitary/symmetric, S^2=C, S^4=I; Verlinde reproduces the group-law fusion; Gauss-Milgram gives c=8=g_car+N_fam. [C] the boson/fermion-to-field map is the interpretation
Dependencies: DHR.SECTORS.01; GATE.HOLO.02
Scripts: v242_spin_statistics_modular.py
Location: tfpt_research_contracts
Original CSV, line 328Original ledger status
DHR.SMATRIX.01
SCATTERING: HAAG-RUELLE SKELETON + BRAIDING S-MATRIX (2026-06-17): the S-matrix layer of the emergent QFT. The seam net has a mass gap (v64/v238), so Haag-Ruelle scattering theory constructs asymptotic in/out states; and for the abelian anyon MTC (v241/v242) the 2-particle S-matrix is the monodromy (double braiding), a pure phase. Same MTC: Z4 x Z4, q=(5x^2+3y^2)/8, theta=e^{2 pi i q}, M(a,b)=theta(a+b)/(theta(a)theta(b))=e^{2 pi i B(a,b)}. (1) [E] mass gap Delta=6 log(3/2)>0 (v64/v238) is the verified input; HR then constructs asymptotic states (standard gapped-net theorem). (2) [E] 2-particle S = monodromy: |M|=1 (unitary), diagonal in the conserved anyon labels -> NO particle production, factorised (integrable) scattering. (3) [E] Yang-Baxter/factorisation: M is a bicharacter (M(a+a',b)=M(a,b)M(a',b)), checked on all triples -> multi-particle factorisation. (4) [E] crossing+symmetry: M(a,b)=M(b,a) and M(a,-b)=conj(M(a,b)). (5) [E] holomorphic point: on the condensed (E8)_1 (single deconfined sector, v241/v235/v237) M=1, trivial braiding -- a closed anomaly-free bulk (det 1/SRE); scattering non-trivial only in the un-condensed matter sectors. (6) [O] residual: the 4d S-matrix and physical cross-sections are the open holographic uplift (Phase 3 / the spectral-action route) -- this exhibits the EXACT 2d topological scattering data (braiding = S-matrix) with the mass gap from v238
Recorded status: Numerical (finite MTC + cited HR) [E]/[O]
Dependencies, status detail and source
Canonical status: [E] mass gap -> HR asymptotic states; 2-particle S = monodromy (unitary, diagonal, factorised); Yang-Baxter bicharacter; crossing; trivial S on condensed E8. [O] the 4d S-matrix / cross-sections = the open holographic uplift
Dependencies: DHR.MODULAR.01; DYN.SEMIGROUP.01; v64
Scripts: v243_haag_ruelle_braiding.py
Location: tfpt_research_contracts
Original CSV, line 329Original ledger status
CONTRACT.QFT4D.01
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
Recorded status: Contract [O] (with [E] arithmetic + [X] hooks)
Dependencies, status detail and source
Canonical status: [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: FORM.P2.02; QG.G2.01; DHR.SECTORS.01
Scripts: v244_spectral_action_contract.py
Location: tfpt_research_contracts
Original CSV, line 330Original ledger status
QFT4D.MATTER.01
SPECTRAL-ACTION MATTER CONTENT (2026-06-17): the computable core of CONTRACT.QFT4D.01 (v244) -- the NCG matter representation, discharging the representation-theory + tree-relation core to [E] and leaving only the seam->Dirac realisation and the measured-coupling reproduction as the open residual. The carrier half-spinor is the SO(10) 16 (dim S^+ = 2^(g_car-1) = 16). One generation = SO(10) 16 = SU(5)(10+5bar+1) = SM: Q=(3,2)_{1/6}, u^c=(3bar,1)_{-2/3}, e^c=(1,1)_{1}, L=(1,2)_{-1/2}, d^c=(3bar,1)_{1/3}, nu^c=(1,1)_{0}. (1) [E] 16 = one generation: SM multiplet dims sum to 6+3+1+2+3+1 = 16 = dim S^+, N_fam=3 (15 SM Weyl fermions + a right-handed neutrino). (2) [E] electric charges Q=T3+Y are exactly the SM values (quarks +2/3,-1/3; anti-quarks -2/3,+1/3; leptons 0,-1; e^c +1; nu^c 0) -- the hypercharges are forced by the SO(10) embedding. (3) [E] anomaly-free: all four gauge anomalies cancel exactly over the 16: sum Y = 0, sum Y^3 = 0, [SU(2)]^2 U(1) = 0, [SU(3)]^2 U(1) = 0. (4) [E] spectral-action unification: sin^2 th_W = (sum T3^2)/(sum Q^2) = 3/8 at the spectral scale, i.e. g3=g2=sqrt(5/3)g1 (Chamseddine-Connes-Marcolli). (5) [E] Higgs = the (1,2)_{1/2} off-diagonal part of the finite Dirac operator (inner fluctuation), one SM doublet. (6) [O] residual: (i) the seam induces the Dirac operator D and (iii) the spectral action reproduces the MEASURED couplings after RG running -- still a contract, now with its rep-theory core verified
Recorded status: Numerical (exact rep theory) [E]/[O]
Dependencies, status detail and source
Canonical status: [E] SO(10) 16 = one anomaly-free SM generation with correct hypercharges/charges; sin^2 th_W = 3/8 (g3=g2=sqrt(5/3)g1); Higgs = (1,2)_{1/2} inner fluctuation. [O] the seam->Dirac realisation + measured-coupling reproduction stay the open contract
Dependencies: CONTRACT.QFT4D.01; FORM.P2.02
Scripts: v245_spectral_matter_content.py
Location: tfpt_research_contracts
Original CSV, line 331Original ledger status
QFT4D.RGTEST.01
UNIFICATION DATA CROSS-CHECK (2026-06-17): the honest data-confrontation kill-test for the spectral-action prediction (CONTRACT.QFT4D.01, v244/v245). Does the boundary condition g1=g2=g3, sin^2 th_W=3/8 at the unification scale survive contact with the MEASURED Standard-Model couplings? Answer: NOT in the plain SM. Inputs (PDG, GUT-normalised, at M_Z; consistent with the PyR@TE cross-check v159): alpha_i^-1 = (59.01, 29.59, 8.47), SM 1-loop betas b=(41/10,-19/6,-7), sin^2 th_W(M_Z)=0.23122. (1) [E] the prediction: at Lambda, alpha1=alpha2=alpha3 and sin^2 th_W=3/8. (2) [X] tension: running the measured couplings up 1-loop, the three do NOT meet at one scale -- pairwise crossings a1=a2 ~1.0e13, a1=a3 ~2.4e14, a2=a3 ~9.9e16 GeV (~4 decades spread), residual spread at 2e16 GeV Delta(alpha^-1) ~ 8.8. (3) [X] sin^2 th_W: predicted 3/8 vs measured 0.23122 (the known few-percent non-SUSY shortfall). (4) [X] two-loop also misses: an RK4 2-loop run gives minimal spread Delta(alpha^-1) ~ 3.2 at mu ~ 1.7e14 GeV (never a common point), ~8.7 at 2e16. (5) [O] NO rescue in the current theory: TFPT has SM gauge content and no new states (tfpt_3 'Dark matter = no new state'), and the E8 cascade is an arithmetic spine (D_n=60-2n, v5), NOT a threshold tower -- there is no admissible gauge-threshold source; closing the spread would require EITHER new gauge-charged matter (a new postulate, contradicting 'no new state') OR a non-GUT/boundary reading of the spectral-action cutoff. The spectral-action 4d-GUT route is in genuine tension -- the SAME status as the Connes-Chamseddine NCG Standard Model -- likely falsified unless the theory is extended (the AQFT skeleton + DHR + SM rep-core survive as an emergent boundary QFT). Typed [X]/[O], NEVER [E]-as-success; the tension IS the result
Recorded status: Data confrontation (kill-test) [X]/[O]
Dependencies, status detail and source
Canonical status: [X] plain-SM run misses unification at 1- and 2-loop (1-loop spread ~9; 2-loop min spread ~3.2 at 1.7e14) and sin^2 th_W 3/8 vs 0.231; [O] no admissible threshold source (SM content, no new state; the E8 spine is arithmetic, not a tower) -- spectral-action 4d-GUT route in tension like NCG-SM, likely falsified unless extended; not a confirmation
Dependencies: QFT4D.MATTER.01; CONTRACT.QFT4D.01
Scripts: v246_unification_data_crosscheck.py
Location: tfpt_research_contracts
Original CSV, line 332Original ledger status
PS.E8BRANCH.01
E8 HULL FORBIDS THE 126 (2026-06-17): the E8 audit hull supplies exactly {1,10,16,45} for SO(10) -- turning the scalaron-match '16-Higgs over 126-Higgs' selection into an algebraic statement. The carrier D5(+)A3 = SO(10)xSU(4) is a rank-8 subalgebra of E8 via SO(16); the adjoint 248 branches as (45,1)+(1,15)+(10,6)+(16,4)+(16bar,4bar) = 45+15+60+64+64 = 248 (SO(16) split: 120 adjoint + 128 spinor = spinor_SO10 (x) spinor_SO6). (1) [E] ranks 5+3=8=rank E8. (2) [E] dims sum to 248. (3) [E] the SO(10) reps in the hull are exactly {1,10,16,45}; the 126 (dim 126) is ABSENT -> the renormalisable 126_H is algebraically forbidden if fields are E8-hull reps, and 10+16+45 is E8-supplied, not fitted. (4) [E] MULTIPLICITY CAVEAT: only ONE (45,1) and ONE (1,15) in the 248 -> the minimal E8-allowed content (one 45) is proton-MARGINAL, a sharp falsifiable constraint. (5) [O] the field-content principle ('TFPT fields are E8-hull reps') + the NCG inner-fluctuation complement (v250).
Recorded status: Lattice/rep theory [E]
Dependencies, status detail and source
Canonical status: [E] E8->SO(10)xSU(4) branching: SO(10) content {1,10,16,45}, 126 absent (126_H forbidden), only one 45/15 (proton-marginal); [O] the field-content principle
Dependencies: E8.GLU.01; QFT4D.MATTER.01
Scripts: v247_e8_branching_no126.py
Location: tfpt_research_contracts
Original CSV, line 333Original ledger status
PS.ALGEBRA.01
PATI-SALAM NCG ALGEBRA + EXACT SM MATCHING (2026-06-17): the finite algebra A_F = H_L (+) H_R (+) M_4(C) and its consequences (Reviewer-A Theorem 2). (1) [E] unimodular gauge group SU(2)_L x SU(2)_R x SU(4)_c (ranks 1+1+3=5=rank SO(10), dims 3+3+15=21). (2) [E] one generation 16 = (4,2,1)+(4bar,1,2) = 8+8. (3) [E] Y = T_3R + (B-L)/2 reproduces the EXACT SM hypercharges for all six multiplets (Q,u^c,d^c,L,e^c,nu^c), and Q = T_3L + Y the exact electric charges {2/3,-1/3,0,-1,1,0}; B-L is the SU(4)_c generator (lepton = 4th colour). (4) [E] coupling matching a1^-1=(3/5)a2R^-1+(2/5)a4^-1 (coeffs sum to 1, so a1=a_GUT at the SO(10) point), sin^2 thetaW=3/8 (=v245). (5) [O] that A_F is the UNIQUE real even algebra generated by seam+sheet-Z2+mu4 (so 'carrier gauged' is a theorem, not a fork). (6) [O] CAVEAT: the PS colour SU(4)_c sits inside SO(10)=D5; it is NOT the carrier factor A3=SU(4) (= family structure, N_fam=3=rank A3) -- a precision the reviews blur.
Recorded status: Numerical (exact rep theory) [E]/[O]
Dependencies, status detail and source
Canonical status: [E] A_F -> SU(2)xSU(2)xSU(4) PS group, 16=(4,2,1)+(4bar,1,2), exact hypercharges/charges, matching a1^-1=(3/5)a2R^-1+(2/5)a4^-1, sin^2=3/8; [O] seam-reconstruction + the colour/family distinction
Dependencies: QFT4D.MATTER.01; PS.E8BRANCH.01
Scripts: v248_ps_algebra.py
Location: tfpt_research_contracts
Original CSV, line 334Original ledger status
PS.RGTEST.01
CARRIER-NATIVE PATI-SALAM/SO(10) UNIFICATION KILL-TEST + NEGATIVE CONTROLS (2026-06-17): the suite claim-check of experiments/gauge-unification. Two-step M_Z->(SM)->M_PS->(PS)->M_GUT (SO(10): a4=a2L=a2R), matching a4=a3, a2L=a2, a2R^-1=(5/3)a1^-1-(2/3)a3^-1. (1) [E] matching holds and a valid 1-loop unification solution exists (M_PS < M_GUT < M_Pl). (2) [X] NEG: pure SM running does NOT unify (residual spread ~8.8 at 2e16 GeV, the universal non-SUSY gap). (3) [C] SCALE COINCIDENCE: the unifying M_PS lands on the scalaron scale M_s=c3^(7/2)Mbar -- M_PS/M_s = 1.0-1.7 at 1-loop (experiment 2-loop shrinks to ~1.0-1.2); two independent TFPT scales (gauge unification + gravitational scalaron) coincide. (4) [X] NEG: 126-type (10,1,3) content (large SU(2)_R, b2R>0) yields NO scalaron-matched solution (1-loop unphysical / 2-loop ~33x) -- match lost; 126 also E8-forbidden (v247). (5) [X] NEG: proton decay selects content -- minimal 16-Higgs M_GUT~2e15 -> tau_p < Super-K (2.4e34 yr) EXCLUDED; +(15,1,1) [E8-allowed 45] raises M_GUT proton-safer. (6) [O] residuals: the gauging fork, the exact kappa in M_PS=kappa M_s (now ~1.0-1.2), the one-45 E8 marginality (v247).
Recorded status: Data confrontation (kill-test) [X]/[O]
Dependencies, status detail and source
Canonical status: [E] matching+unification-exists; [X] SM-only fails, 126 breaks the match, proton decay selects content; [C] M_PS=scalaron coincidence; [O] gauging fork + exact kappa + one-45 marginality
Dependencies: PS.ALGEBRA.01; QFT4D.RGTEST.01
Scripts: v249_ps_unification.py
Location: tfpt_research_contracts
Original CSV, line 335Original ledger status
CONTRACT.QFT4D.DIRAC.01
THE SPECTRAL-TRIPLE CONTRACT (2026-06-17): the object that would turn 'carrier is gauged' from a fork into a theorem (Reviewer-A Theorem 3 + the sigma=scalaron identification). Target: a real even almost-commutative triple (A,H,D,J,gamma) with A_F = PS algebra (v248), H_F = 3x16, D_F = D_M (x) 1 + gamma_5 (x) D_F(R,L,phi0,Majorana). (1) [E] H_F = N_fam x dim S^+ = 3x16 = 48 (three generations of the SO(10) spinor); A_F acts on it. (2) [E] the charged-fermion Yukawas are ALREADY fixed (FLAV.QUARK.01/v18), so the open test is NCG CONSISTENCY of D_F, not 'nice Yukawas' (Reviewer A). (3) [O] the 7 NCG obligations: reality J, grading gamma, KO-dimension (6 mod 8), the FIRST-ORDER CONDITION (the historic SO(10)-NCG obstruction), orientability, Poincare duality, no-junk forms. (4) [O] sigma=SCALARON HYPOTHESIS: the Chamseddine-Connes-Mukhanov singlet sigma (rescues first-order + gives nu_R Majorana) is conjectured to BE the R+R^2 scalaron (supported by M_PS=kappa M_s, v249); CAVEAT: the scalaron<->nu_R nu_R coupling is NOT automatic and must be proven. (5) [O] inner fluctuations Omega^1_D -> Higgs content 10+16+45 not 126 (NCG complement of v247). (6) [C] verdict: reduces the '4d QFT' residual to ONE constructive object (the triple) + the finite NCG checks; gauge group (v248), matter rep (v245), Yukawas (v18) and unification corridor (v249) already in place -- what is missing is D_F + its axioms.
Recorded status: Contract [O] (with [E] finite facts)
Dependencies, status detail and source
Canonical status: [E] H_F=3x16=48, Yukawas already fixed (test is NCG consistency); [O] the 7 NCG axioms (first-order condition the hard one), sigma=scalaron, inner-fluctuation Higgs content -- reduces 4d QFT to one spectral triple
Dependencies: PS.ALGEBRA.01; CONTRACT.QFT4D.01
Scripts: v250_dirac_triple_contract.py
Location: tfpt_research_contracts
Original CSV, line 336Original ledger status
PS.DIRAC.01
FIRST-ORDER CONDITION VIOLATED BY THE MAJORANA/sigma (2026-06-17): the Chamseddine-Connes-van Suijlekom 'beyond first order' mechanism exhibited on a faithful minimal model -- advances CONTRACT.QFT4D.DIRAC.01 (v250) from a contract to an exhibited mechanism. HONEST CORRECTION to the external review: the sigma field does NOT 'rescue' the first-order condition; the opposite is true (CCvS 2013) -- one DROPS the first-order condition, and the inner fluctuations of D THEN generate the fields that promote the SM gauge group to Pati-Salam SU(2)xSU(2)xSU(4) plus a singlet sigma (which gives the nu_R Majorana mass and drives the breaking). Model: H = H_F (+) H_F^c, H_F=C^3 (nu_L,nu_R,e_R); pi(a)=diag(rho(a),0) on particles; J = swap.conj so J pi(b) J^-1 = diag(0,conj rho(b)); D=[[D_p,M],[M^dag,conj D_p]] with D_p the Yukawa block and M the Majorana (nu_R nu_R^c = sigma) block. (1) [E] order-zero [pi(a),J pi(b) J^-1]=0. (2) [E] real-even: J^2=+1, gamma^2=1, D gamma=-gamma D (KO-dim 6 signs; physical SM triple is KO-6). (3) [E] first order HOLDS for the Yukawa block (M=0): [[D,pi(a)],J pi(b) J^-1]=0. (4) [E] first order VIOLATED with M!=0, supported EXACTLY on the off-diagonal (nu_R nu_R^c) Majorana blocks -- the sigma is the unique first-order violator; the controlled violation IS the Pati-Salam mechanism. (5) [C] sigma=scalaron (M_PS=kappa M_s, v249); physical identification [O] (scalaron<->nu_R nu_R coupling to be proven). (6) [O] the full 96-dim SM/PS finite triple + spectral-action expansion remain the open contract (v250).
Recorded status: Numerical (NCG mechanism) [E]/[C]/[O]
Dependencies, status detail and source
Canonical status: [E] order-zero + real-even (J^2=1,KO-6 signs,D odd) + first order holds(Yukawa)/violated(Majorana, localized to sigma); the CCvS beyond-first-order Pati-Salam mechanism exhibited; [C] sigma=scalaron; [O] full 96-dim triple + spectral action
Dependencies: CONTRACT.QFT4D.DIRAC.01; PS.ALGEBRA.01
Scripts: v251_first_order_sigma.py
Location: tfpt_research_contracts
Original CSV, line 337Original ledger status
PS.DIRAC.02
FULL 96-DIM FINITE SPECTRAL TRIPLE BUILT + NCG AXIOMS VERIFIED (2026-06-17): advances CONTRACT.QFT4D.DIRAC.01 (v250) from a contract listing 7 open obligations to a concrete object on which 5 are discharged to [E]. H_F = H_particle (+) H_antiparticle = particle/antiparticle doubling of three SO(10) 16-plets, dim 96 = 2 x Omega_adm. One generation = nu_L,nu_R,e_L,e_R (colour singlets) + (u,d)x(L,R)x3 colours = 16. D_F = [[S,T],[T^dag,conj S]] with S the Dirac/Yukawa block and T the Majorana (nu_R nu_R^c = sigma). J = (particle<->antiparticle swap).conj; gamma = chirality (flipped on antiparticles). A_F = C(+)H(+)M_3(C) as a bimodule: particles via (lambda,q) colour-trivial, antiparticles via (lambda,m) weak-trivial. (1) [E] dim 96. (2) [E] D_F=D_F^dag and D_F gamma=-gamma D_F. (3) [E] reality J^2=+1, JD=DJ. (4) [E] gamma^2=1, [gamma,pi(a)]=0, J gamma=-gamma J. (5) [E] KO-dimension 6 mod 8 (Chamseddine-Connes-Marcolli). (6) [E] order-zero [pi(a),J pi(b) J^-1]=0 for the SM algebra. (7) [E] first-order condition HOLDS for the SM algebra C(+)H(+)M_3 INCLUDING the full Majorana term (van Suijlekom). (8) [E] first-order VIOLATED for the Pati-Salam algebra (an SU(2)_R pairing nu_R with e_R) EXACTLY by the Majorana, with the violation supported exactly on the right-handed lepton doublet {nu_R,e_R} -- the Chamseddine-Connes-van Suijlekom beyond-first-order mechanism (SM -> Pati-Salam + sigma, cf v251) now on the full 96-dim triple. (9) [E] eigenvalues of D_F reproduce the fermion spectrum: charged Dirac eigenvalues = +/- m_f (all 9 charged masses) and the neutrino block gives the type-I seesaw m_light = m_D^2/M_R. (10) [C] Poincare duality: the intersection (multiplicity) form is non-degenerate (det!=0). (11) [O] orientability (Hochschild cycle), no-junk 1-forms and the spectral-action heat-kernel expansion remain open.
Recorded status: Numerical (NCG axioms) [E]/[C]/[O]
Dependencies, status detail and source
Canonical status: [E] dim 96 + Hermitian/odd + reality J + grading + KO-6 + order-zero + first-order holds(SM)/violated(PS, localized to nu_R,e_R) + spectrum=masses+seesaw; [C] Poincare duality; [O] orientability + no-junk + spectral action; discharges 5 of the 7 v250 obligations to [E]
Dependencies: CONTRACT.QFT4D.DIRAC.01; PS.DIRAC.01; PS.ALGEBRA.01
Scripts: v252_full_finite_triple.py
Location: tfpt_research_contracts
Original CSV, line 338Original ledger status
PS.KAPPA.01
kappa IN M_PS = kappa M_s PINNED TO O(1) ~ 1 (2026-06-17): discharges the open residual 6(b) of v249 ('exact kappa, now ~1.0-1.2, not pinned') as far as is honest. The scalaron scale M_s = c3^{7/2} Mbar ~ 3.06e13 GeV is exact from {c3=1/(8pi), Mbar}. kappa := M_PS/M_s with M_PS the RG output of the carrier-native PS two-step (v249). HEAT-KERNEL ORIGIN: in the spectral action S = Tr f(D/Lambda) the gauge kinetic terms (=> unification scale), the Einstein-Hilbert term and the R^2 term (=> scalaron) all descend from ONE cutoff Lambda and the SAME moments f_0,f_2,f_4 of one cutoff function (Gilkey/Seeley-DeWitt; Chamseddine-Connes); hence both M_PS and M_s are Lambda x dimensionless heat-kernel factor and kappa = sqrt((f_2/f_0) c_PS/c_grav), a ratio of traces over the 96-dim H_F (v252) -- Lambda cancels, kappa is scale-free. (1) [E] M_s exact. (2) [E] kappa pinned by RG: scanning PDG coupling errors x the two E8-allowed contents gives kappa in a tight O(1) interval ~1.3 (1-loop [1.0,1.7], experiment 2-loop [1.0,1.2]) -- pinned, not free. (3) [C] heat-kernel origin: O(1) cutoff moments give O(1) kappa; the RG interval sits inside the structural band [0.3,3]. (4) [X] negative control: 126-type content drives kappa out of the O(1) window (content-selective; 126 E8-forbidden anyway, v247). (5) [O] the exact scheme-independent decimal needs f_2/f_0 fixed; c3 is the candidate normalisation but the f-fixing is unproven.
Recorded status: Numerical (RG interval + heat-kernel) [E]/[C]/[X]/[O]
Dependencies, status detail and source
Canonical status: [E] M_s exact + kappa pinned to O(1)~1.3 by RG interval scan; [C] heat-kernel origin kappa=sqrt((f2/f0)c_PS/c_grav) scale-free; [X] 126 neg control; [O] exact decimal needs cutoff moments fixed
Dependencies: PS.RGTEST.01; PS.DIRAC.02
Scripts: v253_kappa_scalaron_ps.py
Location: tfpt_research_contracts
Original CSV, line 339Original ledger status
PS.NCG.FLUCT.01
INNER FLUCTUATIONS Omega^1_D -> SM HIGGS DOUBLET (no 126, no junk) + sigma (2026-06-17): closes the open obligation v250 #5 (Higgs content, NOT 126), the NO-JUNK obligation, and fixes the sigma quantum numbers (v250 #4 rep-theory half). Inner fluctuation D -> D + A + eps' J A J^-1 with A = sum pi(a)[D_F,pi(b)] in Omega^1_D = span{pi(a)[D_F,pi(b)]}. Because D_F is purely chirality-odd (Yukawa L<->R + Majorana), every 1-form is odd, so the FINITE fluctuations are SCALARS (gauge bosons come from the continuum [D_M,a]). (1) [E] Omega^1_{D_F} purely chirality-odd, gauge part 0. (2) [E] NO JUNK / NO 126: for the SM algebra C(+)H(+)M_3 every 1-form has support exactly inside the Higgs-doublet pattern (chirality-odd, colour-diagonal, no quark<->lepton mixing) = the single SM Higgs doublet (1,2)_{1/2}; no colour/weak triplet, no (10,1,3)/126-type scalar (0 junk entries, machine-checked). (3) [E] sigma NEEDS BEYOND FIRST ORDER: with the SM algebra the sigma (nu_R nu_R^c Majorana) direction is NOT generated by Omega^1_D (consistent with first order HOLDING, v252) but with the Pati-Salam algebra it IS generated -- sigma is precisely the Chamseddine-Connes-van Suijlekom beyond-first-order scalar (v251/v252). (4) [E] sigma quantum numbers: SM singlet (1,1,0) with B-L=2 (nu_R has B-L=-1, so sigma*(nu_R nu_R) is gauge invariant only for B-L(sigma)=+2), giving the nu_R Majorana mass and breaking B-L (the PS (4,1,2)->SM step). (5) [C] sigma=scalaron: sigma is a real SM-singlet scalar at the B-L/PS scale, consistent in quantum numbers and scale with the R+R^2 scalaron (M_PS=kappa M_s, v249/v253); the gravitational coupling proper remains [C]/[O].
Recorded status: Numerical (NCG inner fluctuations) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] finite fluctuations = SM Higgs doublet (no junk/no 126) + sigma beyond first order + sigma B-L=2 singlet; [C] sigma=scalaron; closes v250 #5 + no-junk obligation + sigma QN
Dependencies: PS.DIRAC.02; PS.DIRAC.01
Scripts: v254_inner_fluctuations.py
Location: tfpt_research_contracts
Original CSV, line 340Original ledger status
PS.SPECACT.01
SPECTRAL-ACTION HEAT-KERNEL EXPANSION (a0,a2,a4) + EXPLICIT kappa(f2/f0) (2026-06-17): closes the open obligation v252 #11 (spectral-action expansion) and makes the kappa-dependence on the cutoff moments explicit (v253 #5). The bosonic spectral action S = Tr f(D/Lambda) has the Seeley-DeWitt expansion S ~ 2 f4 Lambda^4 a0 + 2 f2 Lambda^2 a2 + f0 a4 (f4,f2 moments, f0=f(0)). (1) [E] expansion structure: a0=cosmological (N int sqrt g), a2=Einstein-Hilbert + Higgs mass, a4=Yang-Mills + Weyl^2 + R^2 + Higgs quartic + R|H|^2, with N=dim H_F=96 (Gilkey/Chamseddine-Connes). (2) [E] trace inputs a=Tr(Y+Y)=2.96, b=Tr((Y+Y)^2)=2.92, b/a^2=0.333=1/3 (top-Yukawa dominated, the Chamseddine-Connes value). (3) [E] gauge normalisation: the a4 Yang-Mills term fixes g3=g2=sqrt(5/3)g1, sin^2 thetaW=(sum T3^2)/(sum Q^2)=3/8 (cross-check v245). (4) [C] Higgs quartic: lambda(Lambda) ~ g^2 b/a^2 = g^2/3; with the sigma ingredient present (v254) this is the famous ~125 GeV (vs ~170 GeV without sigma) Higgs-mass prediction after RG running (m_H vs 125.25 GeV); exact coefficient + running cited. (5) [E] scalaron from R^2: the gravitational a4 carries a Weyl^2 + R^2 term with coefficient ~ f0 N; the R^2 spin-0 mode IS the Starobinsky scalaron, M_s its mass (R+R^2 gravity emerges from the spectral action). (6) [C] kappa made explicit: a2 carries f2 (Einstein-Hilbert), a4 carries f0 (R^2/scalaron + gauge), so kappa=M_PS/M_s=sqrt((f2/f0) c_PS/c_grav) is an explicit ratio of the two moments x traces over H_F; the RG value kappa~1.3 (v253) fixes f2/f0 to O(1). (7) [O] the exact kappa decimal and exact lambda(Lambda) need the cutoff function f fixed -- the last genuinely open input.
Recorded status: Numerical (spectral action) [E]/[C]/[O]
Dependencies, status detail and source
Canonical status: [E] Seeley-DeWitt expansion structure + trace inputs (b/a^2=1/3) + sin^2 thetaW=3/8 + scalaron from R^2; [C] Higgs quartic ~125 GeV + explicit kappa(f2/f0); [O] cutoff moments; closes v252 spectral-action obligation
Dependencies: PS.DIRAC.02; PS.KAPPA.01; CONTRACT.QFT4D.01
Scripts: v255_spectral_action_expansion.py
Location: tfpt_research_contracts
Original CSV, line 341Original ledger status
PS.NCG.ORIENT.01
ORIENTABILITY + POINCARE DUALITY -- HONEST STRUCTURAL TREATMENT (2026-06-17): the two NCG obligations of v250 that do NOT close to a clean [E] on the simplified v252 representation. NO faked proof. KO-dimension 6 forces the intersection form cap(p,q)=Tr(gamma pi(p) J pi(q) J^-1) to be ANTISYMMETRIC (from J gamma=-gamma J + order-zero + reality), so a skew form on the SM K-theory K_0(C(+)H(+)M_3)=Z^3 (odd rank) is necessarily degenerate -- a genuine structural fact. (1) [E] skew intersection form forced by KO-6, verified numerically on the explicit triple (v252). (2) [E] rank/corank: rank 2, corank 1, null direction ~ (1,1,-1) in (C,H,M_3). (3) [E] orientability local condition [gamma_F,pi(a)]=0 (v252); orientability proper is a degree-4 Hochschild cycle on the PRODUCT geometry M x F (metric dim 4), the finite factor (metric dim 0) only contributing gamma_F -- not a finite-only check. (4) [C] for the full hypercharge-faithful SM/PS spectral triple orientability and Poincare duality DO hold (van Suijlekom 'NCG and Particle Physics'; Chamseddine-Connes-Marcolli) -- the corank-1 of the naive skew form is resolved on the physical sublattice / proper K-theory pairing; established in the literature, not re-derived. (5) [O] a fully self-contained machine proof needs the COMPLETE hypercharge-faithful canonical bimodule; the v252 simplified rep faithfully verifies the five substantive axioms (reality, grading, KO-6, order-zero, first-order) but not these two skew/degree-4 conditions. Honestly open.
Recorded status: Numerical (NCG; honest status) [E]/[C]/[O]
Dependencies, status detail and source
Canonical status: [E] KO-6 skew intersection form + rank2/corank1/null(1,1,-1) + orientability local condition; [C] canonical triple satisfies both (literature); [O] self-contained machine proof needs full canonical bimodule -- no faked closure
Dependencies: PS.DIRAC.02
Scripts: v256_orientability_poincare.py
Location: tfpt_research_contracts
Original CSV, line 342Original ledger status
PS.NCG.ORIENT.02
ORIENTABILITY + POINCARE DUALITY -- HONEST LITERATURE-CONFIRMED RESOLUTION (2026-06-17): corrects the slightly-too-generous typing of v256. The published result (Cacic-Stephan 'Moduli Spaces of Dirac Operators for Finite Spectral Triples' arXiv:0902.2068; Chamseddine-Connes-Marcolli CCM07; Standard-Model-vacuum arXiv:hep-th/0601192) is that the CCM Standard-Model finite triple (KO-dimension 6, with three right-handed neutrinos -- exactly the TFPT case for the seesaw) GENUINELY FAILS strict orientability and strict Poincare duality, and instead satisfies the physically correct WEAKENED versions: quasi-orientability and modified Poincare duality. This is NOT a TFPT defect -- it is a known property of the whole NCG-Standard-Model model class. Proven on the explicit v252 triple: (1) [E] strict orientability FAILS: gamma is NOT in span{lambda(a) rho(b)} (degree-0 Hochschild image; rel residual 0.354) -- non-orientable, matching CCM/Cacic-Stephan. (2) [E] QUASI-orientability HOLDS: the (left,right) A-irrep boxes are DISJOINT by chirality (left-handed = H weak-doublet, right-handed = C singlet), so L^LR(H^even,H^odd)={0} -- the weakened orientability (Paschke-Sitarz/Krajewski formalism) that the CCM SM satisfies. (3) [E] strict Poincare duality FAILS: KO-6 forces the skew (antisymmetric) intersection form, corank 1 (v256) -- per the SM-vacuum analysis exactly the case of EQUAL left/right neutrino numbers. (4) [C] modified Poincare duality holds (CCM); Stephan's alternative finite triple (St06) restores STRICT Poincare duality with identical physical content. (5) [E] correction of v256: 'canonical triple satisfies both' is replaced by this precise statement (strict FAIL, weakened HOLD). Honest -- no faked closure.
Recorded status: Numerical (NCG; honest status) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] strict orientability + strict Poincare duality FAIL (KO-6, 3 nu_R -- known model-class feature) + quasi-orientability HOLDS (boxes disjoint by chirality); [C] modified Poincare duality / Stephan PD-restoring triple; corrects v256 over-claim -- no faked closure
Dependencies: PS.NCG.ORIENT.01; PS.DIRAC.02
Scripts: v257_quasiorient_modpd.py
Location: tfpt_research_contracts
Original CSV, line 343Original ledger status
PS.DIRAC.03
DIRAC AS MODULAR/COVARIANCE INDUCTION (2026-06-17): the finite Dirac operator D_F (v252) is the modular/covariance induction of the boundary seam quasi-free (KMS, beta=1) state restricted to the carrier, not an independent posit -- closes the 'posited Yukawa vs derived operator' gap of v250/v252 (proposal Q2: D_F as Kasparov product, covariance formula as the finite implementation). Quasi-free/modular dictionary: one-particle modular Hamiltonian H=log((1-C)C^-1) (Araki; Casini-Huerta), inverse to the KMS occupation C=(1+e^H)^-1. The v18 Yukawas are the readout of C_Sigma=P_F C_Sigma P_F, not an extra input.
Recorded status: Numerical (NCG/modular) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] inversion identity log((1-C)C^-1)=H exactly + C_F a positive contraction + chirality-odd (Dirac-type) + unitary-equivalent to v252 (same mass spectrum) + Majorana=off-diagonal covariance (seesaw m_D^2/M_R); [C] C_F=P_F C_Sigma P_F so D_F is the KK shadow of the boundary geometry (rests on the open seam realisation).
Dependencies: PS.DIRAC.02; DYN.KMS.01; CONTRACT.QFT4D.DIRAC.01
Scripts: v258_dirac_covariance_induction.py
Location: tfpt_research_contracts
Original CSV, line 344Original ledger status
PS.SPECACT.02
MODULAR/KMS SPECTRAL-ACTION CUTOFF (2026-06-17): the spectral-action cutoff f is the seam's own modular/KMS weight (beta=1 by Tomita-Takesaki/BW, v239; 2pi=1/(4c3), v58), so the heat-kernel cutoff f(u)=e^{-u} is derived not chosen and the moment ratio f_2/f_0=1 exactly -- reducing the last open scheme freedom of PS.KAPPA.01/PS.SPECACT.01 from a free function to a finite-triple trace ratio kappa=sqrt(c_PS/c_grav) (proposal Q4: the cutoff is modular, not chosen).
Recorded status: Numerical (spectral action) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] KMS cutoff moments f_0=f_2=f_4=1 so f_2/f_0=f_4/f_2=1 (sympy) + seam-fixed beta=1 + non-vacuous neg control (Gaussian cutoff gives sqrt(pi)/2 != 1) + kappa=sqrt(c_PS/c_grav) (no cutoff function left); [C] RG kappa~1.3 pins c_PS/c_grav~1.7 (O(1) band [0.3,3]) and lambda(Lambda) likewise a finite-triple number -- the [O] free scheme becomes a finite computation over the built triple.
Dependencies: PS.SPECACT.01; PS.KAPPA.01; DYN.KMS.01
Scripts: v259_modular_cutoff_kappa.py
Location: tfpt_research_contracts
Original CSV, line 345Original ledger status
ARCH.K3.01
KUMMER/K3 UNIFICATION OF SEAM + CARRIER-16 + E8 (2026-06-17): one smooth Kummer/K3 surface carries the seam (pillowcase = the elliptic-involution quotient of the square torus tau=i, j=1728, v214), the carrier-16 (the 16 Kummer nodes = dim S+, v197) and the E8 lattice (the H^2(K3) intersection form U^3(+)E8(-1)^2), so the v1 glue D5(+)A3 + mu4 => E8 is the lattice shadow of a single geometric object (the K3/Kummer proposal). A unification, not a closure.
Recorded status: Lattice (K3/Kummer) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] E8 Cartan even/det1/pos-def + K3 lattice rank22/det-1/even/signature(3,19) with E8(-1)^2 a summand + seam=T^2/(z->-z) 4 fixed points (pillowcase, orbifold Euler char 0) + carrier-16=Kummer nodes |A[2]|=2^4=16=dim S+=4x4 + honesty (the rank-16 Kummer lattice is NOT literally E8(-1)^2, distinct sublattices of H^2(K3)); [C] two faces of E8 in K3 (local du Val C^2/2I v232/v236; global H^2 summand) + the carrier<->node geometric unification.
Dependencies: E8.GLU.01; QGEO.PILLOW.01; ARCH.RRCAR.02
Scripts: v260_k3_kummer_unification.py
Location: tfpt_research_contracts
Original CSV, line 346