Publication history: 2026-06-22

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• v355 (E8.UNMAPPED.BANDWIDTH.01) — the next step after v354, run honestly. The obvious follow-up (“search those five degrees for hits we overlooked”) is itself the numerology temptation v354 warns about: a rich object like E_8 returns a numerical match for almost anything. So the scan uses a STRICT discriminator – a candidate counts only if it is a FORCED identity (a theorem about E_8), never a coincidence. [E]FORCED, SET-LEVEL (the genuinely overlooked structure, collective not per-number): \sum\deg=128= \dim S^+, the spinor half of 248=120+128 (theorem: \sum\deg=\#pos roots+rank=120+8) – the invariant budget of E_8 equals the fermion content the carrier reads; \sum\exp=120=\#pos roots; \prod\deg=|W(E_8)|=696{,}729{,}600; exponents = the \varphi(30){=}8 totatives of 30 (v223), so the unmapped degrees' exponents are part of the forced set; \max\deg=30=h(E_8). [C]SUGGESTIVE, not a theorem: 12{=}h(E_6), 18{=}h(E_7), 30{=}h(E_8) are all degrees of E_8 (the 2T/2O/2I McKay ladder, v219), but “\deg(E_8) contains its subalgebras' Coxeter numbers” is not a general theorem – flagged, not claimed. [O]DECLINED (the numerology temptation): the per-degree atom matches 14{=}\dim G_2, 20{=}\det L, 24{=}|W(A_3)|, 12{=}|R(A_3)|, 18{=}p_4(a) (v66 already flagged 12,18 as fittable) each admit \ge2 readings, and the lone extra prime in |W(E_8)|=2^{14}3^5 5^2\,7 “=” the scalaron exponent is one of many readings of 7 – all declined. [E]RECONCILES v66 (the 8 degrees coincide with the load-bearing integers) and v354 (only 2,8,30 are primary readouts): the search finds NO new forced physical hit, and the disciplined decline IS the anti-numerology result. (This also sharpens v354: the five are “no PRIMARY readout / hull overhead”, not literally “unmapped” – they coincide with structural atoms, but unforced.) Cited in tfpt_5_redteam (with a full audited degree table); mirrored in Wolfram (301/301).

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• v354 (E8.REVERSE.AUDIT.01) — the reverse complement of v305. TFPT's forward discipline (v305) is one-directional: every physical READOUT must map onto an E_8 structure (anti-fitting). The sharp adversarial question runs the OTHER way — how much E_8 structure maps onto NOTHING, and is the recurring golden ratio numerology? [E]THE REVERSE AUDIT: of E_8's eight Casimir invariant degrees \{2,8,12,14,18,20,24,30\} (exponents +1), exactly THREE feed a readout — 2 (metric), 8 (rank \to\cthree {=}1/(8\pi)), 30 (Coxeter \to\gcar{=}5 and the order-30 cycle) — and FIVE carry no primary readout \{12,14,18,20,24\} (they coincide with structural atoms per v66, but unforced – reconciled in v355); so 3/8 are primary readouts, 5/8 hull overhead. [E]ECONOMY, NOT PROMISCUITY: a small fixed set (\operatorname{rank}8, h{=}30, \det K{=}1, D_5{\oplus}A_3, the 16 half-spinor) generates the ENTIRE readout set, and the higher Casimirs are never reached into to fit — a numerological use would be PROMISCUOUS (mine the rich structure), TFPT is ECONOMICAL (few invariants, many readouts) and leaves most of E_8 untouched, the anti-numerology signature being precisely the large unused remainder. [E]THE GOLDEN RATIO IS UNMAPPED \Rightarrow NOT NUMEROLOGY: \varphi=2\cos (\pi/5) appears only in internal structure (the affine-E_8 attractor eigenvalue v312, the icosian lattice v348) and in NO physical readout; numerology means a number FITTED to an observation, \varphi is fitted to nothing — it is the algebraic signature of the 5-fold (h{=}30{=}2\cdot3\cdot5), the carrier announcing it is icosahedral. [O]the 5/8 unmapped is the HULL OVERHEAD (E_8 is the minimal even-unimodular hull, not the world group); where a future higher-invariant readout could live, or genuine over-capacity — stated plainly, not hidden. Cited in tfpt_5_redteam; mirrored in Wolfram (300/300).

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• v353 (TFPT.SELFLOOP.01) — refining v352. Stepping all the way back: TFPT is not a LINEAR theory (axioms \to theorems) but a CLOSED SELF-CONSISTENT LOOP whose outputs fix its own inputs. [E]it is a loop: \gcar{=}5{=} max prime of the output h(E_8){=}30, and the 8 in \cthree{=}\operatorname{rank} E_8 – the outputs fix the inputs (v6), with a unique fixed point (v56); “which is the axiom?” is the wrong question for a loop. [E]ZERO free adjustable dimensionless dials: every load-bearing number is forced/over-determined. [E]\pi is a FIXED geometric constant (the boundary 2-sphere, \oint K{=}4\pi), not a dial. [C]this refines v352's “framework + \pi”: \pi is not a free input and the framework is a SELECTION PRINCIPLE (the unique self-consistent loop), not a free meta-axiom – the honest characterisation is “a unique self-consistent loop with zero free dials”. [O]the one residual is the CONTINUUM closure of the loop (the chiral-edge existence, v336/v351); structural-realism (loop complete) vs constructivism (continuum open) is a philosophical, not numerical, fork – no free numbers either way. A capstone synthesis.

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• v351 (SEAM.EQUIV.CONTINUUM.02) — the which-net ambiguity falls. The long-standing “c{=}8 ambiguity” (E_8 det 1 vs SO(16){=}D_8 det 4, both c{=}8, flagged open in v277/v344) is RESOLVED, non-circularly, by the seam's ORDER-4 \mu_4 clock: the index-4 simple-current extension of D_5\!\oplus\!A_3 is E_8 (v154), the index-2 would be SO(16), so the order-4 clock (the four Gauss–Bonnet marks, v216) selects E_8 (full \mu_4 condensation 16{\to}4{\to}1) not SO(16) (partial \Z_2 16{\to}4). The bootstrap agrees (h(E_8){=}30 max prime 5{=}\gcar vs h(D_8){=}14 max prime 7). [E]so \det K{=}1 is a consequence, not an open input. [O]the residual of SEAM.EQUIV.01 is then PURELY the chiral-edge / scaling-limit existence (v336), not which-net.
• v352 (TFPT.IRREDUCIBLE.01) — the inputs are not axioms. The deepest answer to “the inputs are back-determined”: BOTH P1 and P2 are bootstrap-forced fixed points. [E]\gcar{=}5 is forced three ways (v6); the 8 in \cthree four ways (\operatorname{rank}E_8=h(D_5)=\phi(30)=\det R, v54); the E_8 hull is forced (unique even unimodular rank-8, v83/v154). [E]so the ONLY irreducibles are the FRAMEWORK (a discrete reflection-positive boundary with a finite carrier) and the transcendental \pi. [O]the framework is the META-AXIOM — not math-reducible, the foundational stance (like “spacetime is a manifold”), minimal, tested by its consequences not proved. TFPT's true input is ONE framework + \pi. Wolfram mirror +2 (297 \to 299).

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• v350 (SEAM.EQUIV.BOOTSTRAP.01) — correcting v349. An honest correction prompted by the right objection that the inputs are not axioms but are back-determined by the theory. [E]the inputs ARE bootstrap-forced (v6): \gcar{=}5 is over-determined three ways – rank-fill (\gcar{+}\Nfam{=}8), Coxeter-match (\gcar{=} max prime of h(E_8){=}30), Pascal (2^4{=}C(5,0){+}C(5,1){+}C(5,2)). [E]so \gcar{=}5 IS the icosahedral 5-fold (the 5 in h(E_8){=}30{=}2{\cdot}3{\cdot}5), not the D_5 rank; the atoms (2,3,5) ARE the primes of the icosahedral Coxeter number 30. [E]the golden ratio is EMERGENT from the glue: D_5 (h{=}8)/A_3 (h{=}4) are non-golden alone, but D_5\!\oplus\!A_3\to E_8 (\mu_4 index-4, v154) gives h(E_8){=}30 (golden) – the seam's own \mu_4 produces the 5-fold, so v349 checked the un-glued pieces, the wrong object, and its pentagon-vs-count dichotomy was false. [O]the residual relocates: the golden is bootstrap-forced, the only open thing is the physical continuum realisation (v336), NOT the golden ratio. Honest caveat: self-consistency within the framework, not from-nothing. Wolfram mirror +1 (296 \to 297).

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• v349 (SEAM.EQUIV.GOLDEN.01). The honest, computed test of the v348 question, and a clarifying NEGATIVE: the raw seam does not carry the golden ratio. [E]the golden ratio 2\cos(\pi/5) is the signature of a 5-fold rotation (5\mid h); the carrier D_5 has h{=}8 (eigenvalue \sqrt2) and A_3 has h{=}4 – no 5-fold; the dynamical data (cusp \{0,1/3,2/3\}, recovery (2/3)^6, seed \varphi_0) are rational/transcendental. [E]5\mid h holds only for A_4{=}SU(5) (h{=}5), H_3 (h{=}10), E_8 (h{=}30) – the output/icosahedral side, so \varphi is the icosahedral INPUT. [E]\varphi alone would not suffice (2I\neq C_5\times\mu_4, 20\neq120). [O]the resolution: the keystone reduces to one question – is \gcar{=}5 a PENTAGON (a 5-fold rotation \Rightarrow golden \Rightarrow icosahedral 2I=E_8) or just a COUNT (the rank of D_5\Rightarrow\sqrt2)? The “absurdly simple solution” is real but is a reading of axiom P2 (the golden ratio is the pentagon content of \gcar{=}5, not derivable from \gcar{=}5-as-a-count); there is no hidden \varphi. L2 is foundational: P2-as-pentagon closes it (mode C), else a rigidity theorem must produce the 5-fold (mode B); not closed. Wolfram mirror +1 (295 \to 296). A NEGATIVE/clarification; prevents a false closure.

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• v348 (SEAM.EQUIV.RIGID.01). Executes the rigidity route (mode B of v347) and lands on a strikingly simple statement. [E]rigidity closes the crystallographic part: the order-4 clock has the unique normal form z\mapsto iz (Nielsen/Kerckhoff, v180), Troyanov gives the unique flat pillowcase metric (v284). [E]the downstream is a LATTICE identity, not just a graph: by Conway–Sloane the ring of icosians (120 unit icosians =2I, golden-ratio quaternion norm) IS the rank-8 E_8 lattice. [E]the single bridge is the golden ratio: the crystallographic restriction allows only orders \{1,2,3,4,6\} (the 5-fold absent), so \varphi=2\cos(\pi/5) is exactly what extends the crystallographic \mu_4 to the non-crystallographic icosahedral 2I (the quasicrystal). [O]so the entire keystone reduces to ONE question: does the raw seam carry the golden ratio? – if yes, \mu_4{+}\varphi\Rightarrow2I\Rightarrow E_8 and rigidity fixes the rest. [O]honest subtlety: TFPT's \varphi is currently E_8-side (v312/v313); the raw-seam data so far are not manifestly golden, so showing the RAW seam carries \varphi non-circularly is the residual (= L2, sharpest form). Wolfram mirror +1 (294 \to 295). An analysis/reduction; closes nothing.

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• v347 (SEAM.EQUIV.CLOSURE.01). A direct, honest answer to “is there any other way to fully solve the one open arrow L2?”. [E]the precise locus: the seam gives the flat pillowcase base S^2(2,2,2,2) (\chi_{\mathrm{orb}}=2-4(1-\tfrac12)=0, v216) while the 2I/E_8 structure is the Seifert base over the spherical S^2(2,3,5) (\chi_{\mathrm{orb}}=2-(\tfrac12+\tfrac23+\tfrac45)=\tfrac1{30}), so L2 bridges two geometric types – why it needs the carrier (2,3,5) input. [E]as a “physics-realises-geometry” statement L2 has exactly four closure modes: A construct (the analytic wall), B rigidity (a unique-realisation theorem – the only theorem-route w/o new input, pursued v284/v300/v331), C axiom P_3 (then TFPT is complete relative to 3 axioms), D empirical. [E]the “no new state” discipline (v246) blocks deriving L2 from extra physics, so no route makes it a free theorem. [E]verdict: full closure is either a rigidity theorem (B) or axiom status (C), validated either way by (D); only B closes it as a theorem. [O]NOT closed – classifies the modes, locates the difficulty. Wolfram mirror +1 (293 \to 294). An analysis/roadmap; closes nothing.

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• v346 (SEAM.EQUIV.GEOM.01). The honest capstone of the SEAM.EQUIV.01 investigation: it lays out the full path raw-seam \Rightarrow\det K{=}1 as six links and shows exactly one is open. [E]/[C]L1 raw seam \to\mu_4 + four marks (Gauss–Bonnet, v216); [O]L2 \mu_4 + the (2,3,5) atoms \to the 2I orbifold \mathbb{C}^2/\Gamma (THE ONE OPEN ARROW = the geometric content of SEAM.EQUIV.01); [E]L3 2I\to affine E_8 McKay (v219); [E]L4 2I\to E_8 du Val (v232); [E]L5 \mathbb{C}^2/2I\to Poincaré link H_1{=}0 (v345); [E]L6 H_1{=}0\to\det K{=}1. [E]five of six links are theorems; [E]the group is FORCED by the seam's own atoms (the exceptional SU(2) axis signatures (2,3,3)/(2,3,4)/(2,3,5)\to 2T/2O/2I; (2,3,5) unique to 2I, and (|\Z_2|,\Nfam,\gcar){=}(2,3,5) ARE those axes); [E]everything downstream of L2 is then forced. [O]the irreducible open content is L2 alone (“the raw seam normal bundle is an ADE orbifold point \mathbb{C}^2/\Gamma”) = SEAM.EQUIV.01; not closed, but the bridge is now demonstrably ONE arrow with the group pinned and all downstream proven.

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• v345 (SEAM.DETK.02). Carries out the combinatorial core of the hypergraph-homotopy route (R3, the most promising fresh angle from v344) as far as it computably goes. By plumbing topology (Brieskorn/Hirzebruch) the link of a tree plumbing with the ADE Cartan intersection form has H_1=\operatorname{coker}(\mathrm{Cartan}). [E]the Smith normal form of the E_8 Cartan matrix is the identity, so \operatorname{coker}(E_8)=0 – the E_8-graph plumbing boundary is an INTEGRAL HOMOLOGY SPHERE (the genus-1 meaning of \det E_8{=}1); among ADE only E_8 does this (A_n\!\to\!\Z/(n{+}1), D_n\!\to\!\Z/4, E_6\!\to\!\Z/3, E_7\!\to\!\Z/2). [E]\pi_1=2I=SL(2,5) is PERFECT (commutator subgroup = the whole order-120 group, computed directly), so H_1=0 and the link is THE Poincaré sphere, |2I|=120=|\mu_4|\,h(E_8). So R3's combinatorial half is [E]-complete: the E_8 attractor link is forced to be the Poincaré sphere (\det K{=}1), uniquely among ADE. [O]what stays open is only the geometric realisation bridge (the raw rewrite seam attractor is this E_8 du Val plumbing) = the SEAM.EQUIV.01 content; not closed, but the keystone is now reduced to that single geometric identification. Wolfram mirror +1 (292 \to 293).

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• v344 (SEAM.DETK.01). A bird's-eye, full-spectrum map of SEAM.EQUIV.01's only open residual, the holomorphy bit \det K{=}1. [E]it is ONE statement with six equivalent faces (holomorphy / homology-sphere link H_1{=}0 / one 1-dim irrep / \det Cartan =1 / full \mu_4 condensation / \tau{=}i CM), unified by |\det\mathrm{Cartan}|=|H_1(\text{link})|=\#(\text{1-dim irreps}) across ADE (A_n{\to}n{+}1, D_n{\to}4, E_6{\to}3, E_7{\to}2, E_8{\to}1) – only E_8 gives 1. [E]the GENUS-1 OBSTRUCTION explains why it is hard: every easy argument (plane vacuum, gap, free-fermion) is genus-0, necessary but not sufficient (v87); \det K is the torus degeneracy. [E]THREE genus-1 access routes: R1 continuum-modular (v336), R2 anyon-condensation (v281), R3 hypergraph-homotopy. [C]the HYPERGRAPH route is the most promising fresh angle: the (2,3,5) rewrite attractor is the affine-E_8 graph (v312) and the E_8 du Val link is the Poincaré sphere (v232), so \det K{=}1 becomes a finite combinatorial statement (no continuum limit). [O]none closes it; the gap is the combinatorial-to-geometric bridge = the SEAM.EQUIV.01 content. A synthesis/roadmap; closes nothing.

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• v343 (FOUR.ROUTES.01). A direct, honest answer to “which black-hole-birth route closes ALL problems and is 100% provable”: none individually — but the four converge on the same two residuals as v335. [E](A) the finite causal diamond reframes QG.AMB to a finite thermal trace (de Sitter S_{\mathrm{dS}}=32\pi^4 e^{2\ainv} finite, \chi=729/665) but inherits the GHP contour (v334, [O]); does not touch the seam. [C](B) Carlip–Cardy gives a second route to existence+c (c{=}8 consistent, but the Carlip ambiguity and no holomorphy \det K{=}1) — inherits SEAM.EQUIV.01. [E](C) the modular/thermal flow (v239) is well-defined but its geometric content is QGEO.SYM.01, a corollary of SEAM.EQUIV.01 (v335) — a reduction. [E]/[O](D) the unique attractor (v56) gives parameter-freeness but the cyclic dynamics is [C]interpretation. [E] Verdict: independent closures =0; irreducible residuals after all four =2 (unchanged from v335); the horizon premise makes both more natural but eliminates neither — focus = SEAM.EQUIV.01. An analysis module; closes nothing, fabricates nothing.

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• v342 (EM.WARD.02) — the heat-kernel sharpening of EM.WARD.01. Derives the ORIGIN and STRUCTURE of the F_{U(1)} determinant-line terms from textbook Seeley–DeWitt / Gilkey coefficients, advancing the EM-Ward origin from [C](conjectured) to [E]for the term structure; no new gate. [E]\cthree= 1/(8\pi) is the boundary Gauss–Bonnet coefficient (\oint_{S^2}K=2\pi\chi=4\pi, \chi{=}2, v216). [E]the \alpha^2 (Calderón) term is the Gilkey gauge-curvature coefficient (30\,\Omega^2/360=\tfrac1{12} \Omega^2, \Omega{=}i\alpha F\Rightarrow-(\alpha^2/12)F^2), so it is forced, not an ansatz. [E]the \cthree-orders \{0,3,6\} are an arithmetic ladder and 2\cthree^3=1/(256\pi^3) carries \pi^3 (three boundary insertions). [C] the exact coefficient needs the seam F-normalisation. [O]the cubic \alpha^3 Maxwell moment (a topological/Chern level) + the full “this is the seam U(1) \zeta-determinant” identification stay the EM.WARD.01 residual, tied to SEAM.EQUIV.01. \ainv stays [E](v3). Wolfram mirror +1 (290 \to 291).

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• v341 (ALPHA.QUILLEN.01). Answers the external review's sharpest valid point — “derive the form of \alpha, not more decimals”. The EM closure F_{U(1)}=\alpha^3-2\cthree^3\alpha^2-\tfrac45\cthree^6 M\log(1/\varphi_{\mathrm{seam}})=0 is rewritten as a stationarity of the boundary U(1) determinant line: [\alpha^3-2\cthree^3\alpha^2]_{\,\delta\log\det\Delta_{U(1)}}=[8b_1\cthree^6]_{\,\text{counterterm}}\cdot [\log(1/\varphi_{\mathrm{seam}})]_{\,\text{seam response}}. [E]every ingredient is a NAMED object, not a knob: M{=}41 is a U(1) index (\tfrac45 M=8b_1, b_1{=}41/10 the SM U(1)_Y beta, EM Ward v48); \cthree{=}1/(8\pi) the Gauss–Bonnet boundary coefficient (8{=}\operatorname{rank}E_8); -5/4=-q(D_5) a discriminant form and \varphi_{\mathrm{base}}{=}1/(6\pi){=}\tfrac43\cthree carries the other, \cthree/\varphi_{\mathrm{base}}{=}3/4{=}q(A_3); \delta_{\mathrm{top}}{=}\Omega_{\mathrm{adm}}\cthree^4. The Quillen split is verified at the \alpha^{-1}{=}137.0359992 root. No new gate: \ainv stays closed [E]; v341 adds no open item — it sharpens the pre-existing [O]/physical-origin obligation EM.WARD.01 (“why this function”, already conjectural). [O]the one still-open part is that residual: the from-first-principles AQFT proof that F_{U(1)} is the EXACT Quillen functional (the variational principle derived, not assembled), unchanged in scope. Wolfram mirror +1 (289 \to 290).

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• v340 (AS.RECONCILE.01). Answers “why don't we have A_s / the old versions derived it” and “can a ratio define mass itself”. [E]A_s is predicted and dimensionless: A_s=N_\star^2(M_{\mathrm{scal}}/\bar M_{\mathrm{Pl}})^2/(24\pi^2)=N_\star^2\cthree^7/(24\pi^2) with M_{\mathrm{scal}}/\bar M_{\mathrm{Pl}}=\cthree^{7/2}=1.26\times10^{-5} a pure ratio, so A_s pins the e-fold count N_\star, NOT \bar M_{\mathrm{Pl}} — it cannot define an absolute mass (No-Unit, v153, stands). [C]the -11.4\sigma at the slow Higgs channel (N_\star{=}51.4) is the slow-vs-fast reheating dichotomy; the A_s-preferred N_\star{=}56.2\sim the instantaneous bound. [C]the archived TFPT v4.5/v4.6 reported a clean A_s=2.124\times10^{-9} using the algebraic N_\star=3/\phiz-1=55.42; at that N_\star the current normalisation gives 2.047\times10^{-9} (-1.9\sigma) — the old clean A_s is near-instantaneous reheating, not the slow channel. [E]nothing lost: the current theory replaced the posited algebraic N_\star with one derived from reheating physics and exposed honestly that A_s requires fast (pre)heating. Record band N_\star\in[50,60] (v84) untouched.

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• v339 (FIRSTPRINCIPLES.BOUNDARY.01). An honest boundary audit answering two head-on questions and fabricating nothing. [E]The Planck mass cannot be computed — v_{\mathrm{geo}}/M_{\mathrm{Planck}} is forbidden by theorem (No-Unit, v153): a mass is mass-dimension 1 while every TFPT datum is dimension 0, so no dimensionless map outputs an absolute scale. It is not a gap but over-determined (gravity = dark energy to 0.11\%, v274); every dimensionless ratio is derived, only the one unit is not. [E]F_{\mathrm{transfer}} cannot be made first-principles either: for m_p/m_e the structure is in place (v262, carrier b_3{=}{-}7 + closed electron) but two inputs stay external. [X] \alpha_s(M_Z) is external because of a tension — the SM (= TFPT content, no new states) does not unify (1-loop spread {\sim}8.8 at 2{\times}10^{16} GeV, v246). The lattice C_p\!\sim\!2.83 is parameter-free but lattice-only (v35); \eta_B/axion ride on cosmological initial conditions; Koide Q{=}2/3 is structural [E]. [E]The residuals classify into four honest kinds: forbidden-by-theorem, external-and-tensioned, external-but-parameter-free, cosmological. A classification, not a solution.

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• v336 (SEAM.EQUIV.CONTINUUM.01) — the one open lemma, literature-anchored. SEAM.EQUIV.01's only open input (the continuum OS reconstruction of the gapped quasi-free collar) is split into a CITABLE part and a narrow open part. [C]continuum leg: Osborne–Stottmeister (CFT from Lattice Fermions, doi:10.1007/s00220-022-04521-8, CMP 398 (2023) 219) give, building on the MMST (Morinelli–Morsella–Stottmeister–Tanimoto) operator-algebraic (wavelet) renormalization framework (CMP 387 (2021) 299; PRL 127 (2021) 230601), the chiral-CFT scaling limit of free lattice fermions (Koo–Saleur \to Virasoro, correct c); WZW range \mathrm{rank}\le c\le D, and (E_8)_1 (c{=}8, rank 8, D{=}16 Majoranas, SO(16)_1\!\to\!(E_8)_1) is inside it. [C]OS leg: Adamo–Moriwaki–Tanimoto (arXiv:2407.18222, 2024) prove the conformal OS axioms for unitary lattice VOAs. [E]target pinned by c{=}\gcar{+}\Nfam{=}8, \det\mathrm{Cartan}(E_8){=}1. [O]narrowed residual = “the raw collar's massless scaling limit is the holomorphic (E_8)_1 net” (\det K{=}1), not a from-scratch construction.
• v337 (DECOUPLING.THEOREM.01) — the Decoupling Theorem. [E]every dimensionless readout factors through the gapped admissible transfer spectrum \{1,(2/3)^6,(1/3)^6\}; finite susceptibility \chi=1/(1-(2/3)^6)=729/665 gap-suppresses the ambient contribution (margin 6\ln\tfrac32-31/(4\pi^2)\approx 1.648>0, with 2\dim(E_8)c_3^2=31/(4\pi^2) exactly), so no readout depends on the ambient measure \Rightarrow QG.AMB.01 decoupled; negative control (gap closed \Rightarrow\chi diverges) makes it non-vacuous. This is the theorem that makes the QG.AMB.01 reclassification (v335) honest; consolidates v76/v311/v330/v332.
• v338 (THETA13.BUDGET.01) — the \theta_{13} budget, fenced. The +2\sigma pressure point (v328) quantified against both NuFIT 6.0 NO variants: +2.0\sigma vs the no-SK central (0.02195) but +1.7\sigma vs the with-SK best fit (0.02215\pm0.00057), a {\sim}0.3\sigma systematic spread; [E]0.023108 lies inside the with-SK 3\sigma band (upper 0.02388), an upper-edge pressure point not a falsification; [X]JUNO's sub-1% precision turns it into a {\sim}6\sigma kill-or-confirm.
• Lean (FORM.CARTAN.DET.01). CartanDeterminants.lean extended with the explicit D_8=SO(16) Cartan matrix and a fully-proven (kernel-only) holomorphy discriminator: D_8 is even and symmetric just like E_8 (same c{=}8) but 4=\det\mathrm{Cartan}(D_8) is genuinely not a unit while \det\mathrm{Cartan} (E_8) is — so unimodularity (one anyon) is the load-bearing selector. lake build green.

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• v335 (SEAM.EQUIV.UNIFY.01) — the keystone unification. A bird's-eye consolidation of the v308/v323/v325/v329/v333 arc that collapses the “two open structural items” narrative. [E]QGEO.SYM.01 is a COROLLARY of SEAM.EQUIV.01 (no extra premise): a chiral conformal net has, by axiom, a Möbius-covariant vacuum (the unique invariant positive-energy vector); rotations U(1) are the compact subgroup of the Möbius group, so the vacuum is rotation-invariant, and the \mu_4 clock = rotation by \pi/2 ((e^{i\pi/2})^4{=}1) fixes it \Rightarrow\omega\circ\rho{=}\omega. So SEAM.EQUIV.01 \Rightarrow QGEO.SYM.01 with NO extra premise (the v323 “rotation-invariant vacuum” is a net axiom) — the two bedrock items collapse to ONE. [E]the one keystone theorem: SEAM.EQUIV.01 = a construction theorem (c{=}\gcar{+}\Nfam{=}8, \det\mathrm{Cartan}(E_8){=}1, gap 6\ln\tfrac32>0 derived) with EXACTLY one open continuum lemma (the rigorous continuum OS reconstruction + invertibility of the raw collar). [E]QG.AMB.01 decoupled, not a gate: margin \Delta-31/(4\pi^2)\approx1.648>0 (v76/v311/v330/v332) \Rightarrow no TFPT prediction depends on the ambient measure; it is the general Euclidean-QG conformal-factor problem (Gibbons–Hawking–Perry 1978), inherited not TFPT-specific — reclassified from “TFPT structural frontier” to “decoupled general QG problem”. [E]collapsed status: open structural items 2\to1; live residual =v_{\mathrm{geo}}+G_{\mathrm{net}}({=}\texttt{SEAM.EQUIV.01})+ F_{\mathrm{transfer}}. [O]the one residual = SEAM.EQUIV.01's continuum OS reconstruction.
• Lean (FORM.QGEO.BW.01, qgeoSymIsCorollary). BWKeystone.lean extended with the cited conformal-net vacuum axiom conformal_net_vacuum_rotation_invariant; with it, StateInvariance (=QGEO.SYM.01) follows from SeamIsE8 alone, so QGEO.SYM.01 is a COROLLARY of SEAM.EQUIV.01 with no independent open premise (#print axioms confirms only the chain + cited BW/net axioms). lake build green (3362 jobs).
• Status deep-sync. The ledger (QG.AMB.01 reclassified to “decoupled general QG; non-TFPT”; QGEO.SYM.01 marked a corollary; new SEAM.EQUIV.UNIFY.01 row), the canonical status card (tfpt_status.tex), introduction (status core + gate table), origin_theory, tfpt_4_frontier, tfpt_5_redteam, tfpt_research_contracts, README, next.txt and the website (OpenGates, FAQ, papers.ts) all moved to “one solvable theorem (SEAM.EQUIV.01) + one decoupled foreign problem (QG.AMB.01)”. No falsifiable prediction changed; this is a status-framing sharpening backed by v335 + the Lean corollary.

Details

• v333_cm_selection.py (QGEO.CMSELECT.01, [E]/[O]). The CM-selection mechanism for the last keystone residual: \tau{=}i (the square) is the unique order-4 modulus — S{=}\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr) has S^4{=}I and fixes i, and an order-4 deck selects j{=}1728 (vs the order-3 hexagon j{=}0) — AND the square is the Fekete attractor of symmetric mark-equilibration (four points minimizing the log-energy converge to equal \pi/2 spacing). So given the \mu_4 deck, \tau{=}i is forced; the residual sharpens from “why the square” to “the deck acts geometrically” (= QGEO.SYM.01). NOT a closure.
• v334_conformal_resolution.py (QGAMB.CONFORMAL.01, [E]/[C]/[O]). The conformal-mode resolution for the QG metric sector: the Gibbons–Hawking–Perry contour rotation \phi\to i\phi flips the wrong-sign kinetic term, making the divergent conformal Gaussian \int e^{+|c|\phi^2} into the convergent \int e^{-|c|\phi^2}{=}\sqrt{\pi/|c|}; the entire IDG form factor e^{p^2/M^2} dresses the propagator pole-free (the polynomial R{+}R^2 ghost as control). Together they give a conditional mode-by-mode construction of the metric-sector measure; the residual is only the contour's nonperturbative validity (gap-decoupled, blocks no test). NOT a closure.
• CartanDeterminants.lean extended (FORM.CARTAN.DET.01, [E]). The Conway–Sloane lattice side: the E_8 Cartan matrix is proved to be an EVEN (diagonal =2), symmetric, unimodular (|\det|{=}1) integer Gram matrix — the defining data of an even unimodular rank-8 lattice — all by kernel decide / the integer-inverse argument. The remaining content (Witt's uniqueness: E_8 is the only even unimodular lattice in dim <16) is the cited classification residual.

Details

• v331_necessity_of_H.py (QGEO.NECESS.01, [E]/[O]). The necessity companion to v329: a concrete Steklov (DtN) computation shows the seam \Lambda{=}|k|{+}M_f commutes with the \mu_4 clock iff the four marks sit at the square (\tau{=}i) — moving a mark off the square breaks \Z_4 and [\rho,\Lambda]\neq0. The square has cross-ratio 2 (j{=}1728), the unique 4-config with an order-4 automorphism (v214/v267), so H\Leftrightarrow square \Leftrightarrow\mu_4 and “necessity of H” sharpens to: the raw dynamics places the four Gauss–Bonnet marks (v216) at the order-4 CM point. Residual = that dynamical selection.
• v332_qgamb_metric_sector.py (QGAMB.METRIC.01, [E]/[C]/[O]). The open metric sector of QG.AMB.01 is characterized precisely as the Gibbons–Hawking–Perry conformal-factor problem: the conformal mode has a negative kinetic coefficient -(D{-}1)(D{-}2)/4 ({=}-\tfrac32 at D{=}4), so the bare Euclidean measure diverges over it. The obstruction is in the conformal/gauge direction, gap-insulated from the admissible sector (v330/v76); the only route is the conditional IDG taming (v304), with the polynomial R{+}R^2 ghost as the control. Open, but pinned to one named obstruction.
• CartanDeterminants.lean extended (FORM.CARTAN.DET.01, [E]). The rank-8 holomorphy discriminator |\det\mathrm{Cartan}(E_8)|{=}1 (E_8 has one anyon, vs D_8's four) is now proved principally and kernel-only: cartanE8 * cartanE8inv = 1 by decide on the 8\times8 PRODUCT gives IsUnit (det E8) via \det E_8\cdot\det E_8^{-1}{=}\det 1{=}1 — no 8! determinant, no native_decide (Mathlib itself leaves E\textsubscript{8}_det = 1 a proof_wanted).
• Mu4Commutation.lean (FORM.MU4COMM.01, Lean 4, [E]). The v201/v323 mark-locality criterion as exact linear algebra over the Gaussian integers \Z[i]: i^4{=}1, i^2{=}-1, and (on concrete 6\times6 matrices) an offset-4 coupling commutes with the \mu_4 clock while an offset-2 coupling does not — all by kernel decide. Built lake build on Lean v4.29.1 + Mathlib.

Details

• v329_os_gap_reduction.py (QGEO.OSGAP.01, [E]/[O]). The one analytic input the v308 keystone chain still cited — “the OS-reconstructed bulk gap = the transfer gap” — is discharged at the many-body level: the OS/Euclidean Hamiltonian is the second quantization d\Gamma(-\log T), whose spectrum is the set of sums of single-mode energies, so the bulk gap = the smallest positive single-mode energy =6\ln\tfrac32, independent of system size. Together with H\Rightarrow\omega\circ\rho{=}\omega and H\Rightarrow SRE \Rightarrow(E_8)_1, this exhibits the rotation-invariant flat-pillowcase premise H as the single sufficient premise of both bedrock IDs; residual = the necessity of H on the raw collar.
• v330_qgamb_admissible_measure.py (QGAMB.MEASURE.01, [E]/[O]). The constructive step v275's roadmap calls for: on the gap-decoupled admissible sector the ambient measure is built as a bona fide OS object — reflection-positive (symmetric PSD transfer), exponentially clustering (C(n)/C(n{-}1)\to(2/3)^6), with finite susceptibility \chi=729/665 so \mathrm{Var}(S_\Lambda)/\Lambda\to\chi converges \Rightarrow uniform tightness (Prokhorov) \Rightarrow the projective limit exists; OS reconstruction gives H_{\mathrm{OS}}\ge0 with gap 6\ln\tfrac32. The full \mathrm{QG}_{\mathrm{amb}} (the metric sector) stays open, gap-decoupled — a genuine partial construction, not a closure.
• CartanDeterminants.lean (FORM.CARTAN.DET.01, Lean 4, [E]). The carrier piece of the holomorphy/anyon discriminator is grounded in real Matrix.det computations (kernel decide, no extra axiom): |\det\mathrm{Cartan}(A_3)|{=}4{=}|\mu_4| (3\times3, det_fin_three) and |\det\mathrm{Cartan}(D_5)|{=}4 (5\times5, decide), giving the carrier |\det\mathrm{Gram}|{=}4\cdot4{=}16 (the v281 anyon count). The rank-8 E_8/D_8 case stays the Nat discriminator — Mathlib's own Data.Matrix.Cartan leaves E\textsubscript{8}_det = 1 as a proof_wanted. Built lake build on Lean v4.29.1 + Mathlib.

Details

• BWKeystone.lean (FORM.QGEO.BW.01, Lean 4, [E]/[O]). The v323 Bisognano–Wichmann linkage formalised: the nodes \textsf{RotationInvariantVacuum}\to\textsf{GeometricModularFlow}\to \textsf{Mu4ModularSymmetry}\to\textsf{StateInvariance} (=QGEO.SYM.01) composed over the v308 chain. The theorem bedrockReducesToRotationInvariance shows the WHOLE bedrock QGEO.SYM.01 reduces to the SINGLE shared open premise RotationInvariantVacuum — the two open bedrock items collapse to one — with #print axioms confirming the residual = the cited chain steps + recovery gap + the named BW steps + that one premise. Plus a decide-proven discriminator (1\mid4)\wedge\neg(4\mid1) (BW geometric flow strictly stronger than v309 clock invariance). Built lake build on Lean v4.29.1 + Mathlib.
• v325_pillowcase_keystone.py (QGEO.KEYSTONE.01, [E]/[O]). The keystone as ONE citable theorem: IF the raw seam state is the flat \tau{=}i pillowcase (rotation-invariant) state THEN the whole bedrock closes (4 square marks, \mu_4 deck, \omega\circ\rho{=}\omega, holomorphic c{=}8\Rightarrow(E_8)_1, with the recovery gap as input); all pieces machine-checked (incl. the pillowcase \chi_{\mathrm{orb}}{=}0), the single open lemma shared by both bedrock IDs and Lean-pinned. An assembly certificate, not a closure.
• v326_ftransfer_suite.py (FR.SUITE.01, [E]/[C]). The four F_{\mathrm{transfer}} readouts productised into one runnable harness, each a gapped relaxation to a unique attractor: F_{\mathrm{pole}} (Koide, seam rate (2/3)^6), F_{\mathrm{Boltzmann}} (\eta_B, H-theorem), F_{\mathrm{relic}} (axion adiabatic invariant), F_{\mathrm{QCD}} (m_p/m_e, RG to the UV fixed point), plus the F_{\mathrm{RareKaon}} bridge. Only F_{\mathrm{pole}} carries the seam rate; the four external rates are honestly fenced (v187/v303).
• v327_hypergraph_rewrite.py (HYP.REWRITE.01, [E]/[O]). The cusp weight 2/3 is DERIVED from a minimal rewrite: a 3-channel/1-absorbing family rule has spectrum \{1,2/3,0\}, so 2/3{=}(\Nfam{-}1)/\Nfam{=}|\Z_2|/\Nfam emerges from the rule arity and (2/3)^6{=}64/729 is the recovery gap; with a PROOF that 2/3 is not a root of the affine-E_8 charpoly (p(2/3){=}-3520/19683\ne0, sharpening v312). v324's injected datum is reduced to the anchor arity \{2,3\}.
• v328_theta13_pressure.py (FLAV.TH13.PRESSURE.01, [C]/[X]) + v307 hardening. The honest weak spot named and quantified: \sin^2\theta_{13}{=}\phiz e^{-5/6}{=}0.023108 is +2.0\sigma above NuFIT 6.0 NO, a SEPARATE carrier-trace channel (v268), not relieved by any freeze-respecting correction (RG <1\%, exponent 5/6 exact), pre-registered as the kill. The v307 watchdog gains a data provenance/date check and the rare-kaon F_{\mathrm{transfer}} bridge row.
• Wolfram mirror 285\to288. v325 (\chi_{\mathrm{orb}}{=}0) and v327 (rewrite spectrum \{0,2/3,1\}, (2/3)^6{=}64/729, and the non-graph-spectral proof) added to tfpt_readouts_extension.wl; GATE.WOLFRAM.02 and the Wolfram README updated to 288/288.

Details

• Keystone naming reconciled to SEAM.EQUIV.01. Following v323/QGEO.BW.01 (the deck premise is downstream of the seam being a chiral net), the single named open keystone is now SEAM.EQUIV.01 everywhere, with QGEO.SYM.01 stated as its downstream conformal-deck face. The ledger row QGEO.SYM.01 was updated (it no longer calls itself “the single structural residual of the whole theory”; it now depends on SEAM.EQUIV.01/QGEO.BW.01), and the superseded “GATE.QGEO open gate” / “two research gates (U_{\mathrm{wall}}) and G_{\mathrm{metric}}” framing was replaced by the current \text{Rest}=v_{\mathrm{geo}}\oplus G_{\mathrm{net}} \oplus F_{\mathrm{transfer}} with the two structural items SEAM.EQUIV.01 + QG.AMB.01 (intro status card, tfpt_4_frontier, origin_theory, tfpt_5_redteam, tfpt_research_contracts, and the website).
• De-accretion: the sprint-by-sprint reduction history was moved to this changelog, where it belongs. Several reader-facing sections had grown by accretion (each round appended a clause). They were rewritten to state the current result once, with the round-by-round narrative left here: the introduction closure ledger (“honest bottom line”) and hard-reduction subsection; the origin_theory “residual inventory” (the five “Update/…/Terminal” passes collapsed to one current inventory); the tfpt_5_redteam Target-A Outcome; and on the website the FAQ “What is still open?”, the OpenGates closing-condition block, the papers.ts research-contracts/red-team entries, the verification-DAG qft node, the glossary G_net entry, the Rosetta lexicon, and the verification/review pages. No scientific content or \veri citation was dropped; only the embedded version-stamp logs were removed.
• Stale facts corrected. Version label 5.0\tov5.2 (intro, single-sourced via \TFPTversion); Lean build 1886\to3357 jobs (FORM.SEAMEQUIV.01); README Python suite v213/ 212\tov324/ 322 modules; Wolfram extension 269/269 / 249/249\to285/285 (README, website replication page, zenodo_description.html); residual “Paper-3” old-paper attributions in tfpt_2 neutralised.

Details

• Bug fix — \delta_{\mathrm{CKM}} display value. The predictions table (tex-artefacts/predictions.tex) showed \delta=\tfrac\pi3+3\lambda^2=66.96^\circ, but that formula evaluates to 68.65^\circ (the canonical frozen DELTA_CKM_RAD=1.198232 rad, used by v88/v202/FLAV.CP.01 everywhere else). Corrected to 68.65^\circ. An external review correctly flagged the inconsistency; it was an isolated display typo (the website was already clean).
• v202_rare_kaon.py — NA62 2016–2024 update ([E]/[C]). The charged-mode comparison is moved from the historical NA62 2016–2022 observation (13.0^{+3.3}_{-3.0})\times10^{-11} (\sim1.2\sigma above) to the full 2016–2024 combination \mathrm{BR}(K^+)=(9.6^{+1.9}_{-1.8})\times10^{-11} (La Thuile 2026), on which the prediction 9.45\times10^{-11} lands at +0.08\sigma. Framed honestly: a strong consistency hit for the closed CKM point, but an F_{\mathrm{transfer}} bridge — not a unique TFPT-vs-SM discriminator (the SM (8.6\pm0.4)\times10^{-11} is also inside the NA62 error). Propagated to the ledger (FR.RAREKAON.01), freeze_file (new rare_kaon kill row), the watchdog (v307, a new F_{\mathrm{transfer}} bridge row), the forward board (v321, rank 9), and tfpt_2_standard_model + the website.
• CP-phase prose hygiene. The Galois CP lock is tightened from “a relation to the measured quark phase” to “a relation to the leading (\pi/3) component of the quark phase”: the lock is to the structural \pi/3, not the full measured \gamma=\delta_{\mathrm{CKM}}^{\mathrm{lead}}+3\lambda_C^2\approx68.7^\circ, so \delta_{\mathrm{PMNS}}=240^\circ (not 248.7^\circ). Fixed in tfpt_2_standard_model, origin_theory, papers.ts and predictions.ts.
• v305_witness_independence.py — the No-Golden-Dynamics rule ([E]). The number-field split is made an explicit anti-numerology firewall rule: \phi=2\cos(\pi/5)\in\mathbb Q(\sqrt5) is the static carrier (\Z/5) field, every dynamic rate is rational (\mathbb Q, the \Z/6 hand), so a dynamic rate explained by \phi is a cross-field false friend (v314/v315).
• v203_eht_achromatic.py — the \mu_4 Fourier fourth null ([X]). The EHT residual pre-registration gains a fourth null: the azimuthal residual must carry power only at m\equiv0\bmod4 (the \mu_4 fingerprint of the four marks, v201); a \mu_4-orbit profile has \sim0 off-mode power while a \Z_2 control leaks at m=2.
• Seed-hyperplane figure (in the introduction). A new figure visualises v306: every scorecard observable inverts independently to the one latent seed \varphi_0, the error-weighted mean matching the axiom to 0.04\%, with \sin^2\theta_{13} the \sim2\sigma pressure point. Added to make_figures.py, tfpt_5_redteam and the manifest.

Details

• Wolfram mirror of the cyclotomic/Galois arithmetic arc (the independent second path). The exact algebraic results of the arc are now mirrored in verification/wolfram/tfpt_readouts_extension.wl (ten new checkExact blocks, 275\to285, all passing): v313 (the affine-E_8 characteristic polynomial with the golden factor x^2{-}x{-}1; the (2,3,5) atoms as 2\cos(\pi/k); the 31/30 selection), v314 (the rational kernel vs \mathbb Q(\sqrt5) dynamic-rate split), v315 (30{=}5{\cdot}6, \varphi(30){=}8, the Galois group \mu_4\times\Z_2, the \sqrt5 Gauss sum), v316 (\rho{=}\zeta_6{=}\zeta_{30}^5, \rho^4{=}-\rho, \mathrm{Gal}{=}\Z_2{=} CP conjugation), v317 (\Nfam{=}3 as the \mu_3 orbit, the 1{+}2 Galois split), v318 (\phiz{=}(4/3)\cthree{+}48\cthree^4 a pure function of \pi) and v320 (the CP lock \delta_{\mathrm{PMNS}}{=}240^\circ). Ledger GATE.WOLFRAM.02 and the Wolfram README updated to 285/285.
• v322_cp_lock_sharpened.py (GALOIS.CP.SHARPEN.01, [E]/[C]/[X]). The v320 CP-lock sharpened into a quantitative, pre-registered kill test. [E]the leading phase sits on the six hexagonal nodes \arg\rho^k{=}k\cdot60^\circ, and which one is selected is the deck order: \delta_{\mathrm{PMNS}}^{\mathrm{lead}}{=}|\mu_4|\,\delta_{\mathrm{CKM}}^{\mathrm{lead}}{=}4(\pi/3) (node 4); [C]the sub-leading correction is bounded by the quark analogue 3\lambda^2\approx8.7^\circ, so the prediction is the band 240^\circ\pm\sim9^\circ; [C]against NuFIT 6.0 (NO, \delta_{CP}{=}212^{+26}_{-41}{}^\circ) the leading value is +1.08\sigma, the band edge +0.74\sigma – compatible today; [X]the nearest wrong node is 60^\circ away, far beyond the budget, so DUNE/Hyper-K (\sim5–15^\circ) discriminate cleanly. Cited in origin_theory; ledger GALOIS.CP.SHARPEN.01.
• v323_bw_geometric_modular.py (QGEO.BW.01, [E]/[C]/[O]). The keystone pushed one honest step via Bisognano–Wichmann. [E]the \mu_4 clock is a geometric rotation \rho{=}\mathrm{diag}(i^n){=}\exp(i\tfrac\pi2 L), \mu_4\subset U(1)_{\mathrm{rot}}; [E]for a rotation-covariant covariance C{=}f(L) the modular Hamiltonian K{=}\log((1{-}C)/C){=}g(L), so [K,L]{=}0 (the modular flow is geometric) and the clock is a modular symmetry for free (the discriminator: a period-4 curvature preserves the clock but its flow is not geometric, [K,L]\ne0); [C]given the seam is the (E_8)_1 chiral net (v308), BW/Hislop–Longo make the vacuum modular flow geometric, so QGEO.SYM.01 (\omega\circ\rho{=}\omega) is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum, not an independent axiom; [O]residual: the raw collar vacuum is rotation-invariant (cleaner than v309, shared with v308). Cited in origin_theory; ledger QGEO.BW.01.
• v324_hypergraph_fiber.py (HYP.FIBER.01, [E]/[C]). The constructive v312 follow-up: the minimal substrate that carries both the E_8 skeleton and the recovery gap is a product. [E]the carrier network T_{\mathrm{net}}{=}(A{+}2I)/4 (attractor = Kac marks = skeleton) fibred by the 3-node family cusp T_{\mathrm{cusp}}{=}\mathrm{diag}((1{-}w)^{2\Nfam}), w\in\{0,\tfrac13,\tfrac23\} (spectrum \{1,(2/3)^6,(1/3)^6\}); the fibred T_{\mathrm{net}}\otimes T_{\mathrm{cusp}} has top eigenvalue 1 (eigenvector \mathrm{marks}\otimes(w{=}0)) and (2/3)^6 as a genuine eigenvalue (\mathrm{marks}\otimes(w{=}\tfrac13)) – the \Nfam{=}3 cusp is the minimal fibre and the substrate = carrier \times family = the 30{=}\gcar(2\Nfam){=}5{\times}6 of v315; [C]honestly (v312) the cusp weight is still injected, not derived from the graph – a consistent realisation, not a derivation of 2/3 from a pure rewrite. Cited in origin_theory; ledger HYP.FIBER.01.

Details

• v321_killtest_board.py (KILL.BOARD.01, [E]). The forward companion to v307 (the current-data board): the decisive upcoming measurements, ranked and each locked to a freeze_file.csv kill row – #1 JUNO \sin^2\theta_{12}{=}0.306747 (fastest), #2 CMB-S4/LiteBIRD r (kill r{>}0.01), #3 DESI/Euclid w (kill w{\ne}{-}1), #4 DUNE/Hyper-K \delta_{\mathrm{PMNS}}{=}240^\circ (the new v320 Galois-CP relation), #5 DUNE/LEGEND/nEXO ordering+m_{\beta\beta}, #6 PSI n2EDM, #7 JUNO \sin^2\theta_{13} (\sim2\sigma now), #8 LiteBIRD \beta. A forward decision ledger (publicly-stated experimental reach), complementing v307; no new physics. Cited in origin_theory; ledger row KILL.BOARD.01.

Details

• SeamEquivChain.lean (FORM.SEAMEQUIV.01, Lean 4, [E]/[O]). A Lean 4 mirror of the v308 closing chain: the four nodes \textsf{GapPositive}\to\textsf{InvertibleSRE}\to\textsf{HolomorphicC8} \to\textsf{SeamIsE8} assembled as a well-typed composition theorem seamEquivChain, with the three cited literature steps (Kitaev free-fermion invertibility; Müger / Kawahigashi–Longo–Müger holomorphy; Conway–Sloane E_8 uniqueness) as named axioms and the carrier recovery gap (the OS step) as the single open input. [E]#print axioms seamEquivChain confirms the residual is exactly those named steps + recovery_gap (no hidden assumption); [E]a decide-proven discriminator \textsf{primariesE8}=1\ne\textsf{primariesD8}=4 (the |\det\mathrm{Cartan}| that separates the holomorphic E_8 from the same-c rival D_8{=}SO(16)). Built end-to-end (lake build, 3357 jobs) on Lean v4.29.1 + Mathlib. [O]The value is the machine-checked composition + axiom audit (pinning the keystone residual), not a from-scratch proof — it does NOT close SEAM.EQUIV.01. Ledger row FORM.SEAMEQUIV.01; mirrors v308.

Details

• v320_galois_cp_relation.py (GALOIS.CP.PREDICT.01, [E]/[C]/[X]). The v316 follow-up turned into a testable cross-prediction. [E]both leading CP phases are powers of the one hexagonal unit \rho=\zeta_6 of the family Galois factor: \delta_{\mathrm{CKM}}^{\mathrm{lead}}=\arg\rho=\pi/3 (60^\circ), \delta_{\mathrm{PMNS}}=\arg\rho^4=4\pi/3 (240^\circ), and \rho^4=-\rho locks them as \delta_{\mathrm{PMNS}}=\delta_{\mathrm{CKM}}^{\mathrm{lead}}+\pi (the \Z_2 sheet flip); [C]this upgrades the previously assigned \delta_{\mathrm{PMNS}}=240^\circ to a Galois-forced relation to the measured quark phase (the quark and lepton leading CP phases are not independent); [X]kill test: a \delta_{\mathrm{PMNS}} robustly incompatible with 240^\circ (>3\sigma at DUNE/Hyper-K/JUNO) falsifies the Galois-CP organisation; 240^\circ is currently within the broad NuFIT 6.0 range. A genuine new falsifiable consequence of the v313–% v319 arithmetic arc (also a freeze_file.csv cp_phase_relation kill row). Cited in origin_theory; ledger row GALOIS.CP.PREDICT.01.

Details

• v319_translation_clock.py (TRANSLATE.CLOCK.01, [E]/[C]). A synthesis of the v313–v318 arc with v124/v55, answering the question “is the static\leftrightarrowdynamic translation a clock running 0,1,2,3,4,5 or 1,2,3,4,5?”. [E]the only object holding both the static carrier (\mathbb Q(\sqrt5)) and the dynamic family (rational) data is the order-30 Coxeter element (v314); 30=h(E_8)=\gcar(2\Nfam)=5\cdot6 and \gcd(5,6)=1\Rightarrow\Z/30=\Z/5\times\Z/6 (two coprime hands, v315). [E]the dynamic hand is \Z/6=2\Nfam (ticks 0,\dots,5): the recovery exponent is exactly 6=2\Nfam, rate (2/3)^6=(|\Z_2|/\Nfam)^{2\Nfam}, order-6 generator c^5 (\zeta_6=\zeta_{30}^5, v316). [E]position 0 is the conserved law — the resummed clock \mathrm{rate}(n)=-p_2\ln(1-n/\Nfam) (v124) has \mathrm{rate}(0)=0 (the attractor, eigenvalue 1, v56), the live decaying modes from n{=}1, and the E_8 live phases (totatives of 30) start at 1 (\{1,7,\dots,29\}), the 0 phase not a totative — so “0,1,2,3,4,5” is law-inclusive, “1,2,3,4,5” live-only. [E]the static hand is \Z/5=\gcar (ticks 1,\dots,5): the golden field \mathbb Q(\sqrt5) (\varphi=2\cos(\pi/5)), field-obstructed (no rational dynamic image, v314) — static-only; \zeta_5=\zeta_{30}^6 carries no phase. [E]one full turn =\operatorname{lcm}(5,6)=30=h(E_8), rank E_8=8=\varphi(30) live phases pairing into |\mu_4|=4 invariant planes (v55). [C]verdict: the discrete\todynamic translation is the order-30 clock = (static 5-ring) \times (dynamic 6-ring), the dynamic hand carrying the rate (exponent 6=2\Nfam), the static hand the golden carrier (no rate), and “0 vs 1” = law-inclusive vs live-only; the arithmetic is [E], the “it is one clock” reading [C]. Python-only (sympy), like the rest of the arc. New figure figures/translation_clock.pdf (concentric 5- and 6-hands). Cited in origin_theory; ledger row TRANSLATE.CLOCK.01.