Publication history: 2026-06-21

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• v318_arithmetic_capstone.py (ARITH.CAPSTONE.01, [E]/[O]). The synthesis of the v313–v317 arc. [E]the SM structural sector (the 3 generations, the 2 CP phases, their orbit/hierarchy) is the cyclotomic field \mathbb Q(\zeta_{30}) with Galois group \mu_4\times\Z_2 (degree \varphi(30)=8= rank E_8, 30=|\Z_2|\Nfam\gcar=h(E_8)), forced by the anchor atoms \{2,3,5\}=e(a); [E]the magnitude seed \varphi_0=(|\mu_4|/\Nfam)c_3+\Omega_{\mathrm{adm}}c_3^4=\tfrac43 c_3+48 c_3^4 (c_3=1/(8\pi)) is a pure function of \pi (no free parameters, anchor-derived coefficients), so the magnitudes (\sin^2\theta_{12}=\tfrac13-\varphi_0/2) are \{a,\pi\}-determined, not free; [E]free inventory: zero dimensionless free parameters, \pi the unique transcendental primitive (ARCH.CORE.01), v_{\mathrm{geo}} the unique dimensionful input (No-Unit Theorem) — complete input \{a,\pi,v_{\mathrm{geo}}\}. [O]honest residual: this rigidity holds given the axioms and does not realise the structure on the raw seam (QGEO.SYM.01/SEAM.EQUIV.01 stay [O]). Cited in origin_theory; ledger row ARITH.CAPSTONE.01.

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• v317_galois_family.py (GALOIS.FAMILY.01, [E]/[C]). The v316 follow-up to the generation structure. [E]the three cusp weights \{0,\tfrac13,\tfrac23\} (with \mathrm{Spec}(Q_+)=\{1,2,3\}, the A_3 exponents) are the cube roots of unity \{1,\zeta_3,\zeta_3^2\} under w\mapsto e^{2\pi i w}, and the cyclic family symmetry w\mapsto w{+}\tfrac13 is a transitive \mu_3 orbit (the family IS the cube-root orbit); [E]the transfer eigenvalues (1{-}w)^6=\{1,(2/3)^6,(1/3)^6\} attach to the weights (attractor = eigenvalue 1 at w{=}0); [E]the family Galois \Z_2 (\zeta_3\mapsto\zeta_3^2=\bar\zeta_3, = the v316 CP conjugation) refines the orbit to 1{+}2 — fixing w{=}0 (rational) and swapping w{=}\tfrac13\leftrightarrow\tfrac23; [E]so the Galois-FIXED generation is the recovery ATTRACTOR (the law) and the Galois-conjugate pair the decaying hierarchy. [C]verdict: the family STRUCTURE is the same arithmetic as the CP phase (a \mu_3 orbit, Galois-refined 1{+}2), the mass MAGNITUDES remaining the analytic seed. Cited in origin_theory; ledger row GALOIS.FAMILY.01.

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• v316_galois_readout.py (GALOIS.READOUT.01, [E]/[C]). The v315 follow-up to phenomenology: does the Galois coupling \mu_4\times\Z_2 reach the physics? [E]it does, through the family factor — the hexagonal CP unit \rho=\zeta_6=\zeta_{30}^5 is exactly the order-6=2\Nfam factor c^5 of v315 (the carrier being \zeta_5=\zeta_{30}^6); [C]the leading CP phases are its cyclotomic arguments \delta_{\mathrm{CKM}}^{\mathrm{lead}}=\pi/3=\arg\zeta_6 and \delta_{\mathrm{PMNS}}=4\pi/3=\arg\zeta_6^4 (power 4=|\mu_4|), with \zeta_6^4=-\zeta_6 the \Z_2 sheet flip and \mu_6=\mu_3\times\mu_2 (inherits v231/v233); [E]the family Galois \mathrm{Gal}(\mathbb Q(\zeta_6)/\mathbb Q)=\Z_2 (\zeta_6\mapsto\zeta_6^5=\bar\zeta_6) is CP conjugation \delta\mapsto-\delta; [E]the carrier factor \zeta_5 (golden \sqrt5) carries no phase (\mu_4 static), and the magnitudes (\sin^2\theta_{12}=\tfrac13-\varphi_0/2, transcendental \varphi_0) are not cyclotomic. [C]verdict: the Galois bridge reaches phenomenology in the phase sector predicted by the v314 field split. Cited in origin_theory; ledger row GALOIS.READOUT.01.

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• v315_coxeter_coupling.py (COX.COUPLE.01, [E]/[C]). Answers the v314 question — does 30=\gcar(2\Nfam)=5\cdot6 couple the carrier (5-fold, golden, \mathbb Q(\sqrt5)) and family (6=2\cdot3, rational) sectors, or just factorize? [E]the cyclic group factorizes: \gcd(5,6)=1\Rightarrow \Z/30=\Z/5\times\Z/6 (direct product, group-decoupled, in line with the v314 field obstruction); [E]the E_8 exponents = totatives(30) split as totatives(5)\times totatives(6), \varphi(30)=4\cdot2=8= rank E_8; [E]but the field couples: \mathbb Q(\zeta_{30}) is the compositum of the carrier \mathbb Q(\zeta_5)\ni\sqrt5 and the family \mathbb Q(\zeta_3), degree 8= rank E_8; [E]the Galois group is exactly (\Z/5)^\times\times(\Z/3)^\times=\Z/4\times\Z/2=\mu_4\times\Z_2 (order 8, max element order 4), so the \mu_4 seam clock is the Galois group of the carrier's golden field and \Z_2 that of the family field (refining v223); [E]\sqrt5=\phi+1/\phi is the quadratic Gauss sum in \mathbb Q(\zeta_5). [C]verdict: 30=5\cdot6 does not dynamically entangle the sectors — it couples them as the cyclotomic compositum, the static\leftrightarrowdynamic bridge being Galois, not a dynamical map. Cited in origin_theory; ledger row COX.COUPLE.01.

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• v314_rate_translation.py (RATE.TRANSLATE.01, [E]/[C]). Answers, exactly, whether the discrete kernel (the affine-E_8 network, carrying the golden ratio) and the dynamic part (the seam transfer, carrying the recovery rate (2/3)^6) translate. [E]FIELD SPLIT: the adjacency spectrum is the rational part \{0,\pm1,\pm2\}\subset\mathbb Q plus the golden part \{\pm\phi,\pm1/\phi\}\subset\mathbb Q(\sqrt5), while the transfer spectrum \{1,(2/3)^6,(1/3)^6\} is entirely rational; [E]the family/sheet rates DO translate (the rational angles 2\cos(\pi/2){=}0=|\mathbb Z_2| and 2\cos(\pi/3){=}1=\Nfam build (2/3)^6=(|\mathbb Z_2|/\Nfam)^{2\Nfam}\in\mathbb Q, same field, same atoms), but [E]the carrier 5-fold (golden, \mathbb Q(\sqrt5)) has NO rational dynamic image – a field obstruction (2/3 is neither an adjacency nor a lazy-update eigenvalue; heat-kernel t=\Delta/(2-\phi)\approx6.37 not clean), so the carrier is static-only; [E]the only object holding both is the order-30 Coxeter element 30=\gcar(2\Nfam)=5\cdot6, and the \phi-quasicrystal E_8=H_4{+}\phi H_4 (240=2\cdot120) carries the carrier but not the (rational, 3-fold) cusp-weight family dynamics. Cited in origin_theory; ledger row RATE.TRANSLATE.01.

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• v313_golden_atoms_spectral.py (GOLD.ATOMS.01, [E]/[C]). A follow-up to v312's observation, made exact. [E]the affine-E_8 adjacency characteristic polynomial factors as x(x^2{-}4)(x^2{-}1)(x^2{-}x{-}1)(x^2{+}x{-}1), so the golden minimal polynomial x^2{-}x{-}1 divides it and \phi=2\cos(\pi/5)=(1{+}\sqrt5)/2 is an exact eigenvalue; [E]the non-trivial eigenvalues are 2\cos(\pi/k) for exactly k\in\{2,3,5\}=\{|\mathbb Z_2|,\Nfam,\gcar\} (2\cos(\pi/2){=}0, 2\cos(\pi/3){=}1, 2\cos(\pi/5){=}\phi, 2\cos(2\pi/5){=}1/\phi), so the golden ratio is specifically the \gcar{=}5 (5-fold) signature and \gcar{=}5 gains a new spectral witness distinct from Pascal/Riemann–Roch (raising the v305 multiplicity); [E]the (2,3,5) spherical excess 1/2{+}1/3{+}1/5=31/30>1 (the condition selecting the finite 2I\Rightarrow E_8) ties the v63/v76 gap-margin numerator 31=2^{\gcar}{-}1=248/8=1{+}h(E_8) to the Coxeter 30=h(E_8) — a unification of already-anchored quantities, NOT new evidence (honest anti-numerology, v305); [E]honest null: \phi is a structural lattice/network quantity, not a phenomenological readout. [C]direction: \phi is the E_8-quasicrystal (Elser–Sloane cut-and-project) ratio, a candidate aperiodic substrate for the v312 question. Cited in origin_theory; ledger row GOLD.ATOMS.01.

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• v308_seam_equiv_chain.py (SEAM.EQUIV.CHAIN.01, [E]/[O]). TOE attack point 1: the keystone “the raw seam-Calderón net is the holomorphic c{=}8 (E_8)_1 net” assembled into ONE well-typed logical chain \text{gap}{>}0\to\text{invertible/SRE (Kitaev)}\to\text{holomorphic }c{=}8\ (\text{M\"uger/KLM})\to(E_8)_1\ (\text{Conway--Sloane}), with the exact discriminators |\det\mathrm{Cartan}(E_8)|{=}1 vs |\det\mathrm{Cartan}(D_8{=}SO(16))|{=}4, the carrier tower 4{\cdot}4{=}16\to4\to1 and c{=}\gcar{+}\Nfam{=}8 all machine-checked; the input gap is the derived recovery gap 6\ln(3/2){>}0. The only undischarged premise is the cited OS step — an assembly/reduction certificate (a Lean target), not an end-to-end closure. Cited in tfpt_research_contracts; ledger row SEAM.EQUIV.CHAIN.01.
• v309_modular_bedrock.py (QGEO.MODHAM.01, [E]/[O]). TOE attack point 2: the bedrock \omega{\circ}\rho{=}\omega via an explicit modular Hamiltonian, non-circular — the four marks come from Gauss–Bonnet (v216) and the order-4 clock from cross-ratio 2 (v214), so \mu_4 is derived; the quasi-free modular Hamiltonian K=\log((1{-}C)/C) round-trips to the seam operator and [\rho,K]{=}0 \Leftrightarrow\omega{\circ}\rho{=}\omega (a \mathbb Z_2 profile breaks it). Reduces the bedrock to the Bisognano–Wichmann intrinsicality of the raw collar; not a closure of QGEO.SYM.01. Cited in tfpt_research_contracts; ledger row QGEO.MODHAM.01.
• v310_carrier_sm_anomaly.py (CAR.SM.ANOM.01, [E]). TOE attack point 3: the carrier half-spinor 16=\Lambda^{\mathrm{even}}(\mathbb C^5)=1{+}10{+}5 is exactly one anomaly-free Standard-Model generation (SU(5)\subset SO(10): 10{+}\bar5{+}1), with \sum Y=\sum Y^3=[SU(2)]^2U(1)=[SU(3)]^2U(1)=0 (exact) and the one-loop (b_1,b_2,b_3)=(\tfrac{41}{10},-\tfrac{19}{6},-7); a fourth family shifts b_1, so \Nfam{=}3 pins the RG. Closes the spectrum/anomaly/\beta sub-question; the interacting Epstein–Glaser S-matrix and the Yukawa sector stay [C]/[O]. Cited in tfpt_research_contracts; ledger row CAR.SM.ANOM.01.
• v311_gap_decoupled_measure.py (QGAMB.CLUSTER.01, [E]/[C]/[O]). TOE attack point 4: the physical QG measure as a gap-decoupled, exponentially-clustering object — the transfer spectrum \{1,(2/3)^6,(1/3)^6\} gives geometric clustering at rate (2/3)^6 (correlation length 1/(6\ln(3/2)), finite susceptibility 729/665), and the v76 margin 6\ln(3/2)-31/(4\pi^2){\approx}1.648{>}0 decouples the physical sector from the un-built ambient. QG.AMB.01 stays [O](gap-decoupled, inert). Cited in tfpt_research_contracts; ledger row QGAMB.CLUSTER.01.
• v312_hypergraph_substrate.py (HYP.INJECT.01, [E]/[O]). TOE attack point 5: the hypergraph substrate sharpened — the affine-E_8 Kac marks are the Perron eigenvector (A\,m{=}2m) and the subleading eigenvalue is exactly 2\cos(\pi/5)= the golden ratio, but the recovery rate (2/3)^6 is not any eigenvalue and \varphi_0 is not a rational edge fraction; so a full-structure rewrite must inject exactly two non-graph-spectral data — the cusp weight 2/3{=}|\mathbb Z_2|/\Nfam and the analytic seed \varphi_0. The open question is precisely delimited, not closed. Cited in introduction; ledger row HYP.INJECT.01.

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• v305_witness_independence.py (WITNESS.ECON.01, [E]). The structural anti-numerology firewall complementing the v100 phenomenological null test, against the sharpest self-deception worry (“the same integer counted many times as if independent”). Every headline integer (|\mu_4|{=}4, g_{\mathrm{car}}{=}5, |\mathbb Z_2|{=}2, N_{\mathrm{fam}}{=}3, \dim S^+{=}16, \operatorname{rank}E_8{=}8, 240, 248, |2I|{=}120, h(E_8){=}30, |W(D_5)|{=}1920, \Omega_{\mathrm{adm}}{=}48, the gap exponent 6) is a documented image of the single anchor a{=}(1,1,2) with \pi the only transcendental primitive (13 atoms from 2 free primitives, 6.5\times compression made explicit), so the recurring 2/3{=}|\mathbb Z_2|/N_{\mathrm{fam}} and (2/3)^6 are one ratio; the E_8 spine is overdetermined (\ge3 structurally independent witnesses) while the pure-seed readouts are single-witness (scored jointly by v100, never multiplied); the negative control shows \alpha^{-1}'s “137” is foreign to the anchor family (an independent witness via the F_{U(1)} cubic, v3). Cited in the tfpt_5_redteam body; ledger row WITNESS.ECON.01. Python-only.
• v306_seed_crossval.py (SEED.CROSSVAL.01, [E]). A leave-one-out cross-validation turning the “many observables, one seed \varphi_0” worry into a falsifiable out-of-sample test: back-solving \varphi_0 from each of the five seed-grammar observables (\sin^2\theta_{12}, \sin^2\theta_{13}, \beta, \Omega_b, \lambda_C; NuFIT 6.0 / ACT DR6 / Planck / PDG) and error-weighted-averaging reproduces the axiom seed \varphi_0=1/(6\pi)+48c_3^4=0.0531720 to 0.04\%; leave-one-out predicts every held-out observable within 3\sigma out of sample (most-tensioned =\sin^2\theta_{13}, {\sim}2\sigma); a data\leftrightarrowformula shuffle explodes the \chi^2 by >100\times. Cited in tfpt_5_redteam; ledger row SEED.CROSSVAL.01. Python-only.
• v307_data_watchdog.py (DATA.WATCHDOG.01, [E]). The operational decision pipeline over the frozen registry (v84): it classifies all twenty registry entries pass/watch/tension/kill by live n_\sigma against the repo-documented current bests (CODATA 2022, NuFIT 6.0, ACT DR6, Planck 2018, PDG 2024, BK18). No decidable prediction is in kill; the watch items are \alpha^{-1} (1.9\sigma), s_{23} (1.8\sigma) and the most-tensioned core prediction \sin^2\theta_{13} (2.0\sigma, the honest current pressure point); \sin^2\theta_{12}=0.306747 is compatible (0.02\sigma, the JUNO falsifier), the r\in[0.0033,0.0048] band is below the BK18 limit and the kill, and w{=}{-}1 is flagged as the DESI dark-energy watchdog. Cited in origin_theory; ledger row DATA.WATCHDOG.01. Python-only.
• Target-A holomorphy kill test (freeze_file.csv). A new seam_holomorphy freeze row records the sharpened physical kill criterion for Target A: a raw quasi-free seam bulk showing topological ground-state degeneracy on the torus (|\det K|>1, the same-c rival SO(16)_1) would break holomorphy and kill “seam net =(E_8)_1” — since holomorphic c{=}8\Leftrightarrow E_8 is machine-proven (v83/v143) and \det K{=}1\Leftrightarrow invertible/SRE (v237/v301/v302).