Publication history: 2026-06-17

Details

• v258_dirac_covariance_induction.py (PS.DIRAC.03) closes the “posited Yukawa vs derived operator” gap of v250/v252. The finite Dirac operator D_F is the modular/covariance induction of the boundary seam quasi-free (KMS, \beta{=}1) state: a Gaussian state is fixed by its covariance C (a positive contraction) with one-particle modular Hamiltonian H=\log((1{-}C)C^{-1}) (Araki; Casini–Huerta), inverse to the KMS occupation C=(1{+}e^{H})^{-1}. [E]the inversion identity holds exactly; [E]C_F is a positive contraction; [E]the induced operator is chirality-odd; [E]it is unitarily equivalent to the v252 D_F (same mass spectrum); [E]the Majorana/seesaw m_D^2/M_R is an off-diagonal covariance entry. [C] C_F=P_F C_\Sigma P_F, so D_F is the Kasparov shadow [D_\Sigma]\widehat\otimes_{(E_8)_1}[\mathcal K_{\mathrm{car}}] of the boundary geometry and the v18 Yukawas are the readout of C_\Sigma.
• v259_modular_cutoff_kappa.py (PS.SPECACT.02) fixes the last open spectral-action input. The cutoff f is the seam's own modular/KMS weight: by Tomita–Takesaki/Bisognano–Wichmann the seam is KMS at \beta{=}1 (v239) and 2\pi{=}1/(4c_3) (v58) fixes \beta, so the heat-kernel cutoff f(u)=e^{-u} is derived. [E]its moments give f_2/f_0=f_4/f_2=1 exactly; [E]a generic Gaussian cutoff gives \sqrt\pi/2\neq1 (the choice is non-vacuous); [E]hence \kappa=\sqrt{(f_2/f_0)\,c_{PS}/c_{\rm grav}}=\sqrt{c_{PS}/c_{\rm grav}} loses its scheme factor. [C]the open [O]cutoff freedom of v253/v255 becomes a finite trace ratio over the 96-dim H_F (the RG \kappa{\approx}1.3 pins c_{PS}/c_{\rm grav}{\approx}1.7).
• v260_k3_kummer_unification.py (ARCH.K3.01) unifies seam, carrier and lattice on one surface. The pillowcase is the elliptic-involution quotient T^2_{\tau=i}/(z{\mapsto}{-}z) (v214); squaring it gives the Kummer/K3 surface, whose three facets are three TFPT objects. [E]the E_8 Cartan is even/\det1/positive-definite; [E]H^2(\mathrm{K3},\Z)=U^3\oplus E_8({-}1)^2 (rank 22, \det{-}1, even, signature (3,19)); [E]the seam is T^2/(z{\mapsto}{-}z) with 4 marks (orbifold Euler char 0); [E]the carrier-16 are the |A[2]|{=}2^4{=}16 Kummer nodes =\dim S^+ (v197), 16{=}4{\times}4; [E]honest note — the rank-16 Kummer lattice is not the E_8({-}1)^2 summand. [C]so the v1 “glue D_5{\oplus}A_3{+}\mu_4{\Rightarrow}E_8” is the lattice shadow of a single object; E_8 appears both locally (du Val \mathbb C^2/2I, v232/v236) and globally (H^2 summand). A unification, not a closure.
• v261_modular_spectral_closure.py (QFT.MSC.01) is the assembly capstone. The whole QFT round (v238–v260) is certified as one relative object \mathsf{TFPT}_{\mathrm{QFT}}=(\mathcal A_\Sigma,\omega_\Sigma,\Delta_\Sigma,\rho,A_F,H_F,D_F,J,\gamma, S_{\mathrm{rel}}) reduced to one open premise. [E]the cross-consistency invariants agree: 4=[B{:}A]=|\mu_4|=2\chi=|(\Z/2)^2| (index = marks = fixed points = glue order), one carrier-16 (spinor = lattice = triple), one gap 6\log(3/2) (OS = recovery = Lindblad); [E]the three closures re-derived; [E]the ten-component manifest present. [O]the QFT layer reduces to QGEO.SYM.01 (seam realisation), the ambient QG measure QG.AMB.01 kept separate. [C] the Modular Spectral Universality master statement: the unique RP holomorphic index-4 KMS seam state induces the 96-dim KO-6 triple whose inner fluctuations give the gauged Pati–Salam spectral action — the honest sense in which the boundary QFT is complete.

Details

• The canonical changelog now also drives a public /changelog website page. A new generator verification/make_changelog_web.py parses this file (dated \subsection* entries, itemize items, and the inline markup \textbf/\emph/\texttt/\vref, the status markers [E]/[C]/[O]/[X], inline math and accents) into the typed data file website/ lib/ changelog.ts, rendered server-side (math via KaTeX) by website/ app/ changelog/ page.tsx + website/ components/ Changelog.tsx. The generator is wired into bash build.sh gen; its freshness is enforced by audit_sync.py (section A.generated), so the page can never drift from this file – editing the changelog and re-running gen is the only supported path (the .ts mirror is never hand-edited). /changelog is linked from the navbar, footer and sitemap. The tfpt-workflow, website-sync and sync-maps cursor rules were updated to record the new single-source generation step. [E]build (npm run build green, /changelog prerendered static) and bash build.sh audit (AUDIT OK).

Details

• v257_quasiorient_modpd.py (PS.NCG.ORIENT.02) corrects the slightly too-generous v256 typing. The published result (Cacic–Stephan arXiv:0902.2068; CCM07; SM-vacuum hep-th/0601192) is that the Chamseddine–Connes–Marcolli Standard-Model finite triple (KO-dimension 6, with three right-handed neutrinos — exactly the TFPT seesaw case) genuinely fails strict orientability and strict Poincaré duality, and instead satisfies the physically correct weakened versions. Verified on the explicit v252 triple: [E]strict orientability fails (\gamma\notin\mathrm{span}\{\lambda(a)\rho(b)\}); [E]quasi-orientability holds (the left/right A-irrep boxes are disjoint by chirality: L=H weak-doublet, R=\mathbb C singlet, so L^{\mathrm{LR}}(\mathcal H^{\mathrm{even}},\mathcal H^{\mathrm{odd}})=\{0\}); [E]strict Poincaré duality fails (KO-6 skew intersection form, corank 1, the equal-L/R-neutrino case); [C]modified Poincaré duality holds (CCM), Stephan's alternative triple (St06) restoring strict PD with identical physics. A known property of the NCG-SM model class, not a TFPT defect — no faked closure.

Details

• v254_inner_fluctuations.py (PS.NCG.FLUCT.01) closes v250 #5 + the no-junk obligation. The inner fluctuations \Omega^1_D(A_F) of the finite Dirac operator are computed explicitly: [E]they are purely chirality-odd (scalars), and for the SM algebra sit exactly in the one Higgs doublet (1,2)_{1/2} — no junk, no triplet, no (10,1,3)/\mathbf{126}. [E]the \sigma (\nu_R Majorana) direction is absent for the SM algebra but present for Pati–Salam (the CCvS beyond-first-order scalar), and [E]\sigma is an SM singlet (1,1,0) with B{-}L=2. [C] \sigma={}scalaron.
• v255_spectral_action_expansion.py (PS.SPECACT.01) closes the spectral-action expansion (v252 #11). The Seeley–DeWitt expansion S=2f_4\Lambda^4 a_0+2f_2\Lambda^2 a_2+f_0 a_4: [E]structure (a_0 cosmological, a_2 Einstein–Hilbert+Higgs-mass, a_4 Yang–Mills+Weyl^2+R^2+Higgs-quartic, N=96); [E]b/a^2=1/3 (top-dominated); [E]\sin^2\theta_W=3/8; [E]the R^2 spin-0 mode = the Starobinsky scalaron; [C] Higgs quartic \to{\sim}125 GeV with \sigma; [C]\kappa=\sqrt{(f_2/f_0)c_{PS}/c_{\rm grav}} explicit; [O]exact decimal needs the cutoff moments.
• v256_orientability_poincare.py (PS.NCG.ORIENT.01) — honest status, no faked proof. KO-dimension 6 forces the intersection form \mathrm{Tr}(\gamma\,\pi(p)J\pi(q)J^{-1}) to be antisymmetric: [E]skew form (verified), [E]rank 2, corank 1, null \sim(1,1,-1) on the SM K-theory \mathbb Z^3; [E]the orientability local condition [\gamma_F,\pi(a)]=0. [C]the canonical hypercharge-faithful SM/PS triple satisfies orientability and Poincaré duality (van Suijlekom; CCM); [O] a self-contained machine proof needs the complete canonical bimodule.

Details

• v252_full_finite_triple.py (PS.DIRAC.02) advances the Dirac contract v250 from a 7-obligation contract to a concrete object discharging 5 to [E]. The full 96-dim finite spectral triple (A_F,H_F,D_F,J,\gamma) — H_F the particle\oplusantiparticle doubling of three SO(10) \mathbf{16}-plets — is built explicitly and the NCG axioms are checked by direct computation: [E]reality (J^2{=}{+}1, JD{=}DJ), grading (\gamma^2{=}1, [\gamma,a]{=}0, J\gamma{=}{-}\gamma J), KO-dimension 6, order-zero, the first-order condition (HOLDS for the SM algebra including the Majorana; VIOLATED for the Pati–Salam algebra exactly by the Majorana, localized to \{\nu_R,e_R\} — the CCvS beyond-first-order mechanism on the full triple), and the spectrum of D_F reproducing the fermion masses and the type-I seesaw m_{\rm light}{=}m_D^2/M_R. [C]Poincaré duality; [O]orientability, no-junk, spectral-action expansion.
• v253_kappa_scalaron_ps.py (PS.KAPPA.01) discharges residual 6(b) of v249. \kappa in M_{PS}=\kappa\,M_s (M_s=c_3^{7/2}\bar M\approx3.06{\times}10^{13} GeV, exact) is pinned by an RG scan (PDG errors \times the two E_8-allowed contents) to a tight O(1) interval (\approx1.3). [C]heat-kernel origin: \kappa=\sqrt{(f_2/f_0)\,c_{PS}/c_{\rm grav}} is a scale-free ratio of heat-kernel traces over the 96-dim H_F (\Lambda cancels). [X]the \mathbf{126}-content drives \kappa out of the O(1) window (content-selective). [O]the exact decimal needs f_2/f_0 fixed.

Details

• v251_first_order_sigma.py (PS.DIRAC.01) advances the Dirac contract v250 from a contract to an exhibited mechanism. On a faithful lepton-block model (H=H_F\oplus H_F^c, H_F=\mathbb C^3), the real even structure is checked (J^2{=}{+}1, \gamma^2{=}1, D\gamma{=}{-}\gamma D, KO-dim-6 signs), order-zero holds, the first-order condition HOLDS for the Yukawa block, and is VIOLATED exactly by the Majorana (\nu_R\nu_R^c=\sigma) block — the Chamseddine–Connes–van Suijlekom “beyond first order” mechanism by which the inner fluctuations promote the SM to Pati–Salam and generate \sigma. Honest correction to the review: \sigma does not rescue the first-order condition; its controlled violation is the mechanism. [C]\sigma={}scalaron (v249); [O]the full 96-dim triple + spectral action.

Details

• Four scripts turning the gauge-unification finding into typed claims with negative controls. v247_e8_branching_no126.py (PS.E8BRANCH.01): the E_8 hull branches under SO(10){\times}SU(4) as (45,1){+}(1,15){+}(10,6){+}(16,4){+}(\overline{16},\bar4), so the SO(10) content is exactly \{1,10,16,45\} and the \mathbf{126} is absent — the renormalisable 126_H is algebraically forbidden and 10{+}16{+}45 is E_8-supplied (only one 45\Rightarrow proton-marginal). v248_ps_algebra.py (PS.ALGEBRA.01): A_F=\mathbb H_L\oplus\mathbb H_R\oplus M_4(\mathbb C)\to SU(2)_L{\times}SU(2)_R{\times}SU(4)_c, exact hypercharges Y=T_{3R}+(B{-}L)/2, matching \alpha_1^{-1}=\tfrac35\alpha_{2R}^{-1}+\tfrac25\alpha_4^{-1}, \sin^2\theta_W=3/8 (colour SU(4)_c\subset D_5, distinct from the family A_3). v249_ps_unification.py (PS.RGTEST.01): the two-step PS\to SO(10) unification kill-test with negative controls (SM-only fails; 126 breaks the scalaron match; proton decay selects content), M_{PS}/M_s=1.0–1.7. v250_dirac_triple_contract.py (CONTRACT.QFT4D.DIRAC.01): the spectral-triple contract (H_F=3{\times}16, the seven NCG axioms, \sigma=scalaron) that would turn “carrier is gauged” from a fork into a theorem. The root-level qft.txt carries the full analysis.

Details

• v246_unification_data_crosscheck.py (QFT4D.RGTEST.01) — the first hard data confrontation of the spectral-action prediction. TFPT's gauge-charged content is the Standard Model (v159) and the theory adds no new states; the E_8 cascade is an arithmetic spine (v5), not a threshold tower. So the run is the SM run: at one loop the couplings spread {\sim}10^{13}–10^{17} GeV (residual \Delta\alpha^{-1}\approx9 at 2{\times}10^{16} GeV), and at two loops (RK4) the minimal spread is \Delta\alpha^{-1}\approx3.2 at 1.7{\times}10^{14} GeV — the three never meet. Typed [X]/[O]: with “no new state” there is no admissible threshold source, so the spectral-action 4d-GUT route is in genuine tension (same status as NCG-SM), likely falsified unless extended — never a confirmation, and not rescued by a cascade. Two new figures (figures/qft_skeleton.pdf, figures/qft_unification.pdf) added via make_figures.py (PDF + website PNG) and embedded in tfpt_research_contracts.tex; a root-level qft.txt summarises the emergent-QFT skeleton, what each script proves, the data status and the open questions.

Details

• v245_spectral_matter_content.py (QFT4D.MATTER.01) discharges the computable core of the 4d contract v244. The carrier half-spinor (the SO(10) 16) is verified, in exact rational arithmetic, to be one Standard-Model generation: \mathbf{16}=SU(5)(\mathbf{10}{+}\bar{\mathbf 5}{+}\mathbf1), the SM multiplet dimensions sum to 16=\dim S^+ (15 Weyl fermions +\,\nu_R), the electric charges Q=T_3+Y are exactly the SM values, all four gauge anomalies cancel (\sum Y=\sum Y^3=[SU(2)]^2U(1)=[SU(3)]^2U(1)=0), the spectral-action normalisation gives \sin^2\theta_W=3/8 (g_3=g_2=\sqrt{5/3}\,g_1), and the Higgs is the (1,2)_{1/2} inner fluctuation. So the matter-content + tree-relation half of CONTRACT.QFT4D.01 is now [E]; only the seam\toDirac realisation and the run-down couplings stay [O]. Wired into run_all.py, the ledger, the registry, cited in tfpt_research_contracts.tex, mirrored to the website.

Details

• New v238_modular_lindblad_dynamics.py (DYN.SEMIGROUP.01, cluster registry). A companion to the modular round (v196/v198) and the recovery code (v221/v64) that reads the same seam objects dynamically rather than statically. [E]the modular flow \sigma_t=e^{i t\Lambda_\Sigma} is a unitary one-parameter group (Tomita–Takesaki) whose conservation of the \mu_4 clock, \rho\sigma_t\rho^{-1}=\sigma_t\Leftrightarrow[\rho,\Lambda_\Sigma]=0, IS the v198 state-invariance — so the static commutator is the conservation law of modular time. [E]the recovery channel's generator L=\log T is a doubly-stochastic/CPTP-classical Markov generator (e^{L}=T, e^{L\tau}\to the democratic projector = the arrow of time). [E]keystone: the dissipative gap =-\log((2/3)^6)=6\log(3/2)= the OS mass gap \Delta (v64) — the static recovery bound (v221) and the dynamical mass gap (v64) are one number, one exponentiated. [E]the GKSL split G=i\Lambda_\Sigma+L (imaginary + real\le0 spectrum). [O] the residual is the geometric modular flow (Connes–Rovelli “thermal time = physical time”) on the full seam algebra = QGEO.SYM.01 (v198) — a target, not a closure. Wired into run_all.py, status_ledger.csv, script_registry.csv and cited in tfpt_research_contracts.tex; mirrored to website/public/verification.

Details

• Red Team Target A re-synced to tfpt_5_redteam (rt_A_e8net.py, redteam_table.txt, redteam/README.md, infrastructure). The Python red-team surface still emitted the historical three-residual “A reduced, not closed” form, lagging the reader-facing note. It now encodes the sharper narrowing already in the paper: the residual collapses to the single statement “the seam–Calderón boundary net is holomorphic with c=8” (\Leftrightarrow the index-4 simple-current extension), with the index-4 (D5)_1\times(A3)_1\to(E8)_1 glue realised at Lie level (v143) and its odd sectors twisted/Ramond (E_8=120_{\mathrm{NS}}+128_{\mathrm{R}}, v148), while the q(A_3) normalisation collapses into the one declared anchor (v152). Two exact adversarial checks added (120{+}128=248; glue index =|\mu_4|=\Nfam{+}1=4); status stays [C]/[O](reduced, not closed).
• Shadow export now mirrors figures/ but not website/ (scripts/export-shadow.sh, verification/make_script_index.py, infrastructure). export-shadow.sh ships figures/*.pdf (so make_manifest.py -{}-check passes literally on the exported tree) while still excluding website/ and the compiled paper PDFs; the SHADOW_MIRROR.md table is corrected accordingly. make_script_index.py now detects a missing website/ and skips its ScriptIndex.tsx mirror, so build.sh notes runs on the subset without crashing.

Details

• Both CP phases are \mu_6 powers of one hexagonal CM unit, split by the sheet (v231_cp_mu6_phases, tfpt_5_redteam, [E]/[C]). Sharper than v220/v225 (which only located the CP residual). With \rho=e^{i\pi/3} the primitive 6th root (the j{=}0 Eisenstein CM unit): \delta_{\mathrm{CKM}} leading =\arg(\rho^1)=\pi/3 (quark) and \delta_{\mathrm{PMNS}}=\arg(\rho^4)=4\pi/3 (lepton, the phase lattice), with \rho^4=-\rho, so \delta_{\mathrm{PMNS}}=\delta_{\mathrm{CKM,lead}}+\pi=(\text{CM unit})\times(\Z_2\text{ sheet}, \rho^3{=}{-}1). The two CP phases are one hexagonal unit on the two sheets, not two independent inputs; the dual-frame Jarlskog orientations are sheet-flipped (\pm21\sin(\pi/3)). A structural reduction of Target D (removes one of its two phase inputs); the quark seam correction 3\lambda_C^2 stays the v88 misalignment, the deck stays \Z/4, and the physical CP derivation stays [C](CP.MU6.01).
• The seam as the E_8 Kleinian singularity (v232_e8_kleinian_seam, origin_theory, [E]/[C]). The du Val side of the McKay correspondence (v219) gives the open seam-realisation premise QGEO.REALIZE.01 a canonical algebraic-geometric model: dropping the trivial-rep node of the affine-E_8 McKay graph of 2I leaves the finite E_8 Dynkin = the dual intersection graph of the eight exceptional \PP^1's of the minimal resolution of \mathbb C^2/2I; the eight curves =\operatorname{rank}E_8=g_{\mathrm{car}}{+}\Nfam (a fourth reading of the 8), their negated intersection form is the E_8 Cartan (\det1, even, (-2)-curves), and the link is the Poincaré homology 3-sphere S^3/2I (2I perfect \Rightarrow H_1{=}0, \pi_1{=}2I order 120). The raw seam (\PP^1 with marks) is canonically a du Val exceptional curve — a model for the bedrock, not a P2 proof (it presupposes E_8; TOPO.E8.01, bedrock stays [O]).
• CP is the universal family/triality phase, only sheet-split (v233_cp_triality_phase, tfpt_5_redteam, [E]/[C]). One step past v231 in the \mu_6=\mu_3(family/triality)\times\mu_2(sheet) factorisation: \rho=\omega^2(-1) with \omega=e^{2\pi i/3} the \Z_3 triality centre (the 2/3 cusp weight). Since \rho^4=\rho^1\rho^3 with \rho^3\in\mu_2, the quark (\rho^1) and lepton (\rho^4) CP phases share the same family/triality class and differ only by the sheet — so CP is the universal triality phase, sheet-split, not a free power choice (CP.TRIALITY.01).
• The Seam-Holomorphy selection certificate: the structural residual is one gate (v234_seam_holomorphy_selection, tfpt_research_contracts, [E]/[O]). The whole structural residual is one condition — “the seam carries no nontrivial abelian sector” — with three provably-equivalent faces, all forcing E_8: (i) holomorphy (\mu-index 1; Target A), (ii) the seam link is a homology 3-sphere (\Gamma perfect \Leftrightarrow2I; v232), (iii) exactly one 1-dim irrep (v219). All equal \#(\text{mark-}1)=|\Gamma^{\mathrm{ab}}|=|H_1|, which is 1 only for E_8; and holomorphic c{=}8{=}\gcar{+}\Nfam gives the unique even unimodular rank-8 lattice E_8. So P2, G_{\mathrm{net}}, Target A and the Kleinian seam are one gate. The stated closing theorem ([O]): RP+gap+chirality \Rightarrow quasi-free bulk (v160) \Rightarrow holomorphic boundary \Rightarrow E_8; the single residual analytic step is “free bulk \Rightarrow holomorphic boundary” (GATE.HOLO.01).
• The closing step in Chern–Simons language: holomorphic \Leftrightarrow\det K{=}1 (v235_seam_chern_simons, tfpt_research_contracts, [E]/[O]). A genuine reduction (not a closure) of GATE.HOLO.01 into standard anyon-condensation physics: a free gapped bosonic 2{+}1d bulk is an even integer K-matrix with \#anyons =|\det K|, edge c={\rm signature}(K), and the edge net holomorphic \Leftrightarrow|\det K|{=}1. The TFPT extension tower (v92) is the condensation tower read by determinant: carrier D_5{\oplus}A_3 (\det16, 16 anyons) \to SO(16)_1{=}D_8 (\det4) \to(E_8)_1 (\det1, holomorphic) — all rank 8, c{=}8{=}\gcar{+}\Nfam — so holomorphic =E_8= the Kitaev E_8 quantum-Hall state. Holomorphy \Leftrightarrow condensing the order-|\mu_4| Lagrangian glue (\det16{\to}1, vs. a partial order-2 isotropic \to4{=}D_8). The residual narrows to one integer condition “the free RP seam condenses the Lagrangian glue (\det{=}1)” = the sheet/QGEO selection, now with holomorphy\Leftrightarrow\det1 a theorem (GATE.HOLO.02).
• The (2,3,5) Brieskorn singularity is the one generator of the skeleton (v236_brieskorn_capstone, origin_theory, [E]). Capstone of the icosahedral round: the singularity x^2{+}y^3{+}z^5, whose exponents are the atoms (|\Z_2|,\Nfam,\gcar){=}(2,3,5), has Milnor number \mu{=}(2{-}1)(3{-}1)(5{-}1){=}8{=}\operatorname{rank}E_8 (a fifth origin of the “8”), Milnor monodromy = the order-30 E_8 Coxeter element (eigenvalues the primitive 30th roots = the E_8 exponents, charpoly \Phi_{30}), Milnor lattice E_8, link the Poincaré sphere; the two clocks are facets of this one monodromy — the \mu_3 triality (CP, v233) is h^{10} in \langle h\rangle{=}\Z/30, the \mu_4 clock (v223) is the order-4 Galois automorphism of \mathbb Q(\zeta_{30}) (TOPO.BRIESKORN.01).
• The closing step as a physical condition: no topological degeneracy \Leftrightarrow\det K{=}1 \Leftrightarrow the seam is SRE (v237_seam_sre_closure, tfpt_research_contracts, [E]/[O]). Sharpens GATE.HOLO.02's residual from definitional to falsifiable: the K-matrix ground-state degeneracy on a genus-g surface is |\det K|^g (torus: carrier 16, D_8 4, E_8 1), so “no topological degeneracy on any closed surface” \Leftrightarrow\det K{=}1; among rank-8 even positive-definite K (c{=}8{=}\gcar{+}\Nfam) only E_8 is short-range-entangled (the Kitaev E_8 phase). A unique vacuum on the plane (RP+gap+cluster) is necessary but not sufficient (the plane cannot see \det K); so the one open step is the physical statement “the free RP seam is SRE (no topological order)” (GATE.HOLO.03).