Publication history: 2026-06-24

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• New module v410 (SHEET.GEN.BINARY.01). The rank-2 sheet axis V=Q\operatorname{diag}(0,1,1) of the centered cross (v95/v218) is a binary internal compiler: its powers print the carrier spine. [E]the closed form V^n=\big(\begin{smallmatrix}0&2^{n-1}&0\\0&2^n&0\\0&2^{n+1}-2&1\end{smallmatrix}\big) (proved by the inductive step F(n)V{=}F(n{+}1)), so V^n\mathbf 1=(2^{n-1},2^n,2^{n+1}-1)=(1,2,3),(2,4,7),(4,8,15),(8,16,31); [E]the four bilinear families \mathbf 1^TV^n\mathbf 1=7\cdot2^{n-1}{-}1, \mathbf 1^TV^na=7\cdot2^{n-1}, a^TV^na=11\cdot2^{n-1}, a^TV^n\mathbf 1=11\cdot2^{n-1}{-}2 give (6,7,9,11) at n{=}1 and \mathbf 1^TV^2\mathbf 1{=}13{=}\Delta_Q, \mathbf 1^TV^3\mathbf 1{=}27{=}\mathbf 1^TRa, \mathbf 1^TV^4\mathbf 1{=}55, \mathbf 1^TV^4a{=}56{=}\dim\mathbf{56}_{E_7}; [C]the binary spine is FORCED by \operatorname{Spec}(V){=}\{0,1,2\}, so the carrier/Lie matches (\mathrm{rank}\,E_8{=}8, \dim S^+{=}16, 248/8{=}31) and the theta cross-link (\sigma_3(2){=}9{=}a^TV\mathbf 1, \sigma_3(3){=}28{=}\mathbf 1^TV^3a{=}\det(I{+}R), \sigma_3(5){=}126) are audit-typed (the v95 G_2-center typing).
• New module v411 (FLAV.UD.VPOWER.01). The quark ratio is a pure V-power readout, c_u/c_d=(\mathbf 1^TV^4\mathbf 1)/((a^TV\mathbf 1)(\mathbf 1^TV^2\mathbf 1))=55/(9\cdot13)=55/117, identical to the established \gcar\|\mathrm{Pl}(K)\|_1/(\Nfam^2\Delta_Q)=5\cdot11/(9\cdot13) (v94). [E]the identity; [C]an exact re-encoding (no new physical input) — the value and the u/d identification stay conditional, coupled to the (U_{\mathrm{wall}}) readout rigidity.
• New module v412 (DIAMOND.SHEET.SOURCE.J.01). Names the operator on the \Z_2 wall (t{=}{-}2, \det{=}2, v135), J{=}M(1,{-}2){=}C{-}2V, completing the sheet axis J,K,C,F{=}M(1,{-}2..1). [E]\chi_J=\lambda^3-6\lambda^2+3\lambda-2\Rightarrow(\mathrm{tr},e_2,\det){=}(6,3,2); a^TJa{=}30{=}h(E_8), \det(I{+}J){=}12{=}\dim\mathfrak g_{\mathrm{SM}} and \det(2I{+}J){=}40{=}|R(D_5)|; [C]the Lie readings audit-typed like the G_2/F_4 shadows.
• New module v413 (DIAMOND.SHEET.CALCULUS.01). On M_t{=}C{+}tV the elementary-symmetric coefficients encode the atoms as difference orders: e_1{=}3t{+}12, e_2{=}2t^2{+}15t{+}25, e_3{=}4t^2{+}14t{+}14 give \Delta e_1{=}3{=}\Nfam, \Delta^2 e_2{=}4{=}|\mu_4|, \Delta^2 e_3{=}8{=}\mathrm{rank}\,E_8 (the e_3 curvature is the v218/v224 sheet quadratic; the full triple is new); [E]the anchor energy a^TM_ta{=}52{+}11t (30{\to}41{\to}52{\to}63, step 11{=}a^TVa); [C]atom/Lie readings audit-typed.
• New module v414 (DIAMOND.CENTER.RESOLVENT.01). The center C{=}M(1,0) is a resolvent portal: [E]\det C{=}14{=}\dim G_2, \det(I{+}C){=}52{=}\dim F_4, \det(2I{+}C){=}120{=}|R^+(E_8)|, the exceptional shell ladder G_2{\to}F_4{\to}E_8^+ as \det(C{+}kI), k{=}0,1,2 (with \mathrm{tr}\,C{=}12{=}\chi_{E_8}(i), \sum C{=}31{=}2^{\gcar}{-}1, v95); [C]the portal reading.
• Surfaces synced. Integrated into the Sheet-Diamond section of tfpt_2 (new keybox, five \veri citations) and the E_8 theta-raster of tfpt_3; run_all.py, script_registry.csv and the ledger updated; Wolfram extension mirror 303\to308.

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• New module v408 (HYP.PHI0.PUNCTURE.01). Names the \varphi_0 puncture term 48c_3^4 as the order-|\mu_4|{=}4 local boundary heat-kernel (Seeley–DeWitt) contact term of the \mu_4 puncture divisor, summed over \Omega_{\mathrm{adm}}{=}\Nfam\dim S^+{=}48 admissible states — the analytic complement of the icosahedral tree term (v396). [E]the exact split \varphi_0{=}1/(6\pi){+}48c_3^4, the puncture count, the closed form 3/(256\pi^4) and the hypergraph negative (reused from v396/ARCH.QUAD.01); [O]the from-first-principles heat-kernel proof; [C]a FACE of the seam boundary analysis (the c_3 Gauss–Bonnet datum), not a new gate. Cited in origin_theory; no new number.
• New module v409 (RES.COXETER.SYMMETRY.01 closed as a structural lemma). prime-2 (the seam sheet involution) is necessary but not autonomous: \sigma{=}\operatorname{diag}(1,-1,-1) has spectrum \{\pm1\} and projector (I{+}\sigma)/2 spectrum \{0,1\}, none in (0,1), so prime-2 carries no contraction rate, whereas the genuine rates 2/3 and (1/3)^6 lie in (0,1) and carry prime-2 only as a factor. [E]no prime-2-only attractor exists; [C]dynamics begins only after coupling to family-3 or carrier-5 (|\mu_4|{=}2^2 the Galois bridge). Resolves the v394 open question's falsifiable corollary; cited in tfpt_research_contracts; the ledger row RES.COXETER.SYMMETRY.01 retyped [O]\to [E]/[C].
• Axion branch consistency (tfpt_4_frontier, website predictions.ts, v185). The dark-matter section's opening + keybox are realigned with the finite-T result already stated later in the same section: the robust dominant-DM branch is the spine angle \theta_i{=}\pi\Nfam/\gcar{=}3\pi/5{=}108^\circ (untuned, in band), while the seam hilltop \theta_i{\approx}170^\circ over-produces (\sim5.4\times) and is disfavoured; the misalignment angle is retyped from a closed [E]input to a [C]branch decision. The website prediction card now leads with the spine angle.
• Lean closure of the prime-2 corollary (FORM.COXETER.PRIME2.01). The v409 result is now also machine-proved in Lean 4 / Mathlib (new module TfptCarrier/CoxeterPrime2.lean): an involution eigenvalue (r^2{=}1) is \pm1, a projector eigenvalue (r^2{=}r) is 0 or 1, neither lies in the open interval (0,1), while 2/3 does — so no prime-2-only attractor exists. Built clean (lake build); the new theorems are axiom-pure (\#print axioms = propext, Classical.choice, Quot.sound only) and pass the AuditCheck/AuditContract signature locks. This lifts RES.COXETER.SYMMETRY.01 to the machine-proof tier.
• Verified, no change (Plücker 11). The quark constant 11{=}\|Pl(K)\|_1{=}\det(\mathbf1|a|\sigma) is already typed honestly as an [E]identity with “why 11” the lone [O]residue (v407, tfpt_1); the repo carried no overstatement, so no edit was required.

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• Ledger hygiene. Resolved a duplicate active claim_id: GATE.UWALL.06--09 was carried by two distinct chains. It is an ID collision (not a supersede): the monodromy/residue chain v114–v117 keeps .06--.09 (as asserted by v123_inventory_update.py and run_all.py); the older exterior-leg chain v42–v45 is renumbered to GATE.UWALL.15--.18 (internal depends_on carried along). No claim content changed.
• Website. papers.ts document count 8\to9 (the Red-Team paper was missing from the tally); Red-Team subtitle/highlight Targets A–E\toA–F with a new Target F section + contribution bullet (the abstract already carried F); StatusMatrix.tsx Doc 7\toDoc 8 for the research contracts and the residual interfaces re-typed v_{\mathrm{geo}} [O], G_{\mathrm{net}} [C], F_{\mathrm{transfer}} [C]; objections Paper 1\toDoc 1.
• Papers (marker/wording alignment, no canonical status change). tfpt_1 Strong CP [E]\to[E]/[C](given RP) and \theta_{12} “genuinely open”\to“conditionally derived”; tfpt_2 Starobinsky A_s [E]\to[E]/[C](M exact, A_s a band over N_\star\in[50,60]) and the Part-II abstract “opens the one remaining gap”\to“closes the last structural gap”; tfpt_research_contracts the QFT4D “nonperturbative measure stays open” line and the “only U_{\mathrm{point}} stays open” headline reconciled with the [C]redundancy / v_{\mathrm{geo}} reduction stated alongside; origin_theory class C re-typed to the ambient-measure [C]redundancy; introduction (U_{\mathrm{wall}})\to v_{\mathrm{geo}}. Forbidden old-paper attributions (“Papers 1–7”, “Papers 2/6”, “Paper-7”, “Paper-1”) replaced by current document / neutral wording (the tfpt_3 cascade bridge stays exempt).
• Suite README. The residual-gates table (Alessandro-5.0 vintage) brought to the current model (v_{\mathrm{geo}} [O]; G_{\mathrm{net}}/SEAM.EQUIV.01, F_{\mathrm{transfer}}, N_\star, QG.AMB.01 [C]; Wolfram 116/116+303/303).
• Figures. Regenerated status_heatmap (ledger-driven) and extended the narrative figures script_timeline/residual_chain/qft_skeleton through the v303–v407 arc (typed solvers, parameter-free local Einstein equation v359, QG.AMB.01 [C]redundancy, seam closed modulo cited theorems); the qft_skeleton “Stelle ghost” open box is retyped (resolved truncation artefact). Added qft_skeleton/qft_unification to the manifest figure list.

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• [E]/[C]/[O]v405_seam_equiv_omega.py (SEAM.EQUIV.02): the dual keystone at \tau{=}\omega – the family/flavor sector as the order-3 Eisenstein/A_2 face of E_8, dual to the seam =(E_8)_1 at \tau{=}i (v404). A named contract (genre of v286/v398), NOT a closure. [E]\chi_{E_8}(\omega)=0 (the family CM point degenerates E_8) vs \chi_{E_8}(i)=12; [E]A_2 is the \tau{=}\omega Eisenstein lattice (Cartan \det=3=\Nfam, h=3=\Nfam); [E]R lives in E_6{\times}A_2 (248=78+8+2{\cdot}27{\cdot}3, \det R=8=\dim A_2=\operatorname{rank}E_8); [E]\delta_{\mathrm{PMNS}}=4\pi/3 is the \tau{=}\omega CM phase; [E]the cusp weights \{0,1/3,2/3\} are the order-3 deck =\Nfam=\det Q. [C]the family sector is the order-3 Eisenstein/A_2 deck; the two keystones sit at the only two elliptic points of \mathrm{PSL}(2,\mathbb Z). [O]residual = the canonical \mathbb Z_3 deck map (v140) + the v_{\mathrm{geo}}-class scale. Cited in origin_theory; Python-only.
• [E]/[C]/[O]v406_pmns_phase_origin.py (FLAV.PMNS.CLOSE.01): the PMNS dynamics closed modulo the \tau{=}\omega keystone – the CP-phase ORIGIN is the family CM point and the absolute scale is one v_{\mathrm{geo}}-class unit, so PMNS has zero free flavor dials beyond v_{\mathrm{geo}}. [E]the three angles (v9/v268) \to unitary U_{\mathrm{PMNS}} (v270); [C]\delta_{\mathrm{PMNS}}=4\pi/3 is the \tau{=}\omega hexagonal CM phase (v405), upgrading v270's [O]“assembled input” to a keystone consequence; [C]J_{\mathrm{PMNS}}=-0.0297 derived; [E]the absolute scale = one seesaw ratio =v_{\mathrm{geo}}-class (v272/v153). [O]residual = the seesaw operator realisation (M_R, v184) + the \Sigma m_\nu floor kill test. Cited in tfpt_2_standard_model; Python-only.
• [E]/[C]/[O]v407_dn_pairings_omega.py (FLAV.SELECTOR.CLOSE.01): the R4' selector residue folds into the \tau{=}\omega keystone – the three (d,n) pairings ARE the \tau{=}\omega family-slice atoms. [E]the frame (\mathbf 1,a,\sigma) has \det=11=\|Pl(K)\|_1 and n=(5,-9,6) is the unique covector with n{\cdot}\mathbf1=2, n{\cdot}a=8, n{\cdot}\sigma=121=11^2 (\Rightarrow(d,n)\Rightarrow R, v136/v139); [E]n{\cdot}a=8=\operatorname{rank}E_8=\dim A_2=\det R (the A_2 block reading R in E_6{\times}A_2, v405); [E]the pairings \{2,8,121\}=\{|Z_2|,\dim A_2{=}\det R,\|Pl(K)\|_1^2\}. [C]deriving (d,n) folds into SEAM.EQUIV.02 (d anchor-derived, v139); [O]the lone residue “why 11{=}\|Pl(K)\|_1”. Cited in tfpt_1_architecture_e8; Python-only.

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• [E]/[C]v403_facet_compositum.py (ARITH.HULL.01): the three single-prime number-field facets of the order-30 clock (v390/v394, used so far only one at a time) compose to one field K=\mathbb Q(i,\sqrt{-3},\sqrt5). [E]the squarefree parts -1,-3,5 are independent in \mathbb Q^\times/(\mathbb Q^\times)^2, so [K:\mathbb Q]=2^3=8=\operatorname{rank}E_8=\gcar{+}\Nfam with Galois group (\mathbb Z/2)^3 (order 8); [E]exactly 7 quadratic subfields \{-1,-3,5,3,-5,-15,15\}, splitting 4 imaginary =|\mu_4| (CM/dynamic) + 3 real =\Nfam (RM/static); [E]ramified only over \{2,3,5\}= the atoms = the primes of h(E_8)=30; [E]2^{\omega(h)}=\operatorname{rank} is E_8-special (the E_7 control h=18 gives 2^2=4\neq7). [C]K is the abelian (\mathbb Z/2)^3 arithmetic hull dual to the E_8 lattice hull; the 7 quadratic subfields are a candidate for the 7 E_8 audit slices. A synthesis composing the established facets, no new number; cited in origin_theory; Python-only by the synthesis-module convention (cf. v390/v394).
• [E]/[C]v404_e8_char_cm_duality.py (E8.CMDUAL.01): the (E_8)_1 vacuum character \chi_{E_8}=E_4/\eta^8=j^{1/3} read at both CM points, extending v282 (\chi_{E_8}(i)=12) to the second elliptic point. [E]\chi_{E_8}^3=j (q-series); [E]SEAM \tau{=}i (\mathbb Q(i), order 4): cross-ratio 2\Rightarrow j=1728\Rightarrow\chi_{E_8}(i)=1728^{1/3}=12 (E_8 present, v214/v282); [E]FAMILY \tau{=}\omega (\mathbb Q(\sqrt{-3}), order 6): the equianharmonic \lambda (\lambda^2{-}\lambda{+}1=0) \Rightarrow j=0\Rightarrow\chi_{E_8}(\omega)=0 (E_4 vanishes, v220/v267); [E]\tau{=}i,\tau{=}\omega are the only elliptic points of \mathrm{PSL}(2,\mathbb Z), so the values are forced. [C]the seam/flavor rigidity duality: the seam is rigid where E_8 is present (\chi=12, the keystone SEAM.EQUIV.01 closes) while the flavor/CP sector is the soft residual where E_8 degenerates (\chi=0: U_{\mathrm{wall}}, v_{\mathrm{geo}}, the CP texture), a candidate dual keystone at \tau{=}\omega. Cited in origin_theory; core values j=1728/j=0 already mirrored (v214/v220/v267/v282), Python-only.

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• [E]/[O]v398_seam_state_rigidity.py (SEAM.RIGIDITY.01, Paper A): the Seam State Rigidity Theorem as ONE named target, consolidating the scattered state-invariance reductions (v177/v199/v308/v309/v335) and hardening the finite-block evidence to a multi-mode sweep. [E]RP-definability hinge (OS canonical transfer T{=}e^{-H} from RP + reflection alone, v54, quasi-free v155, no \mu_4 normal form); [E]band-limited character-block-diagonal core [\rho,|k|]{=}0 swept N{=}4..64 (beyond the 3-dim H^1); [E]holomorphy \Rightarrow(E_8)_1 (c{=}8, |\!\det\mathrm{Cartan}|{=}1 vs 4, index 4{=}|\mu_4|). [O]the one residual: [\rho,\Lambda_\Sigma]{=}0 on the full L^2 (intrinsic Bisognano–Wichmann). NOT a closure of SEAM.EQUIV.01 (continuum = cited MMST v336).
• [E]/[C]/[O]v399_gravity_complete.py (GRAVITY.COMPLETE.01, Paper B): the explicit verdict Gravity-complete = Boundary-complete + Ambient-redundancy, combining the parameter-free local equation (v358/v359) with the ambient redundancy (v369). [E]the five gravitational readout classes (Newton 8\pi, \Lambda, scalaron \cthree^7, horizon 1/|\mu_4|, recovery (2/3)^6) are all boundary/seam/gap, so \mathcal O_{\mathrm{phys}}\subset\mathcal A_\Sigma; [O]residual = SEAM.EQUIV.01 + intrinsic BW (no new gate).
• [E]/[O]v400_grav_nonlocal_action.py (GRAV.NONLOCAL.01, Paper C): the nonlocal spectral-gravity action with the local R{+}R^2 as its IR projection (consolidating v304/v370/v380/v386, adding the IR-matching check). [E]entire a{=}e^u\Rightarrow one pole (ghost-free), every truncation carries a Stelle zero (monotone, v380), and the IR order reproduces R{+}R^2 with the atom-fixed scale M_{\mathrm{scal}}^2/\Mbar^2{=}\cthree^7 (v36/v253). [O]the entire-a assumption + perturbative-only.
• [E]/[O]v401_metrology_closure.py (METROLOGY.CLOSURE.01, Paper D): the final input set \{a,\pi,v_{\mathrm{geo}}\} certified — 0 dimensionless dials + 1 unit (consolidating v153/v274/v364/v384). [E]dimensional bookkeeping mass/1G/\Lambda/U_{\mathrm{point}}/(m/\mu){=} number \times v_{\mathrm{geo}}^{1,2,4,1,0}; [E]the G-vs-\Lambda over-determination 0.11\% (v274); [O]v_{\mathrm{geo}} theorem-forbidden (\mathrm{TOE}_{\mathrm{strict}}{=}\mathrm{TOE}_{\mathrm{dimensionless}}{+}[v_{\mathrm{geo}}]).
• [E]/[C]v402_ftransfer_suite.py (FTRANSFER.SUITE.01, Paper E): the four frontier sectors as ONE functor F_{\mathrm{transfer}}, the composition F_{\mathrm{obs}}\circ F_{\mathrm{thr}}\circ F_{\mathrm{RG}} (consolidating the FR.*.SOLVE solvers), each with an exact atom source, a committed kill test, and the firewall (physical prediction stays [C]/[O], only the algebraic source [E]; never re-pressed into the compiler, v187).
• Ledger rescope (repo hygiene): GATE.METRIC.01 no longer reads “full covariant metric-sector field equation open” — the local covariant equation is closed parameter-free (GRAV.NONLINEAR.01/v359); the gate is rescoped so the open item is only the global nonperturbative ambient measure (QG.AMB.01, a [C]redundancy v369), removing the stale contradiction with the status card.

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• [E]/[C]v396_phi0_icosahedral.py (HYP.PHI0.01): the tree-level seed \varphi_0^{\mathrm{tree}}=1/(6\pi) is exactly F/(4h\pi)=F/(|\Aut|\pi)=(F/(\gcar\Nfam))c_3 on the icosahedral hypergraph (F=20 triangular hyperedges, h=30, |\Aut|=120); F/h=|Z_2|/\Nfam=2/3 (the same survival ratio as v327). [E]Gauss–Bonnet consistent with c_3=1/(|Z_2|4\pi) (v216/v342). [E]honest negative: the puncture 48c_3^4 is still analytic, not a hypergraph fraction. [C]\varphi_0^{\mathrm{tree}} is a combinatorial reinterpretation, not a new independent injection; the puncture stays [O].

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• [E]v395_hypergraph_coupled.py (HYP.COUPLED.01): one coupled local rewrite on a 9\times3 labelled grid (affine-E_8 nodes \times family channels) unifies v299 (carrier growth), v327 (branching rule M) and v324 (fiber product). Micro-step = network lazy diffusion on each cusp column + M on each node's family vector (T_{\mathrm{micro}}=T_{\mathrm{net}}\otimes M); joint attractor marks\otimes e_0; one clock hand (6 family steps) gives recovery (2/3)^6; v324's T_{\mathrm{net}}\otimes T_{\mathrm{cusp}} emerges as the cusp-readout basis. [O]the 3-arm seed (P_2) and analytic \varphi_0 remain irreducible (v312).

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• v397 (FR.CLOCK.PROBE.01) — do the “external” frontier scheme-numbers land on the \{2,3,5\} clock? A falsifiable, anti-numerology probe (strict No-Free-Pattern, v100: “on the clock” means an exact match, not a wide-band fit). [E]the \eta_B decuple A_\Lambda=10=|Z_2|\gcar is ON the clock (v212); [E]the Koide rate (2/3)^6=(|Z_2|/\Nfam)^6 is ON the clock (v303); [E]the axion spine angle \theta_i=\pi\Nfam/\gcar=3\pi/5 is ON the clock (v211/v373); [E]the proton C_p\sim2.83\pm0.15 is not a discriminating clock value — within its \pm5.3\% band several simple values fit (2^{3/2},17/6,14/5), so by No-Free-Pattern it is declared EXTERNAL (the firewall v187/v374 is correct there). [C]verdict: 3 of the 4 frontier scheme-numbers land exactly on the clock, so the irreducible “external physics” residue shrinks to essentially the proton's non-perturbative QCD number C_p. Cited in tfpt_4_frontier; a falsifiable probe whose answer could have been “none on the clock”; Python-only.

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• v393 (CORRECTIONS.NUMERIC.01) — the typed correction budget (v388) as actual numbers. The short-term, testable harvest of v388: instead of merely typing each prediction's correction, this gives the magnitude, so a parameter-free central value can be quoted as “value \pm computed first correction”. [E]FIXED-POINT band (the \varphi_0-derived \theta_{12},\theta_{13},\lambda_C,\beta_{\mathrm{rad}}) = the explicit \varphi_0 puncture term 36c_3^4/c_3 = 9/(128\pi^3)\approx0.227\% (exact; =\theta_{12}'s \varepsilon first-correction, \varepsilon=(3/4)\varphi_0=c_3+36c_3^4); [E]SEAM-GAPPED band (Koide F_{\mathrm{pole}}, recovery) =(2/3)^6\approx8.78\%, the QG-decoupling bound capped by \chi-1=64/665\approx9.62\% (v337); [E] EXACT-IDENTITY band =0 structurally (\det R=8, N_\Phi=1, \theta_{\mathrm{eff}}=0); [E]EXTERNAL-RATE band = external physics, quoted (m_p/m_e\,{\pm}7.5\% v374; n_s/r the N_\star{\in}[50,60] band; \eta_B\,1.07\times v212), fenced by v187. [C]the assembled budget gives each prediction a computed theoretical-uncertainty band. Cited in origin_theory, mirrored on the website prediction surface; a harvest of v388, no new number; Python-only.

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• v392 (SEAM.S3.SCALINGLIMIT.01) — an honest attack on SEAM.EQUIV.01's one open lemma (continuum existence, v336). It does not close v336 (the abstract existence is the cited MMST theorem); it strengthens the case with the existence-relevant observable v376 did not test: that the finite-size data converge. [E]the Calabrese–Cardy entanglement slope c_1(L) at L=66,130,258,386,514 is a monotone Cauchy-like sequence approaching 1 (|c_1(L)-1| strictly decreasing, 4.6\times10^{-4}\to7\times10^{-6}); [E]a 1/L Richardson fit extrapolates to c_1(\infty)\approx1.000, so the 16-Majorana collar c=8c_1\approx8.00 =\gcar{+}\Nfam is reached in the limit; [E]the successive differences are themselves strictly decreasing (a numerical Cauchy sequence). [O]convergent data are evidence, not a proof of existence (the cited MMST theorem, v336), and do not pin (E_8)_1 over the same-c SO(16)_1 (the holomorphy discriminator, v377/v378). Strengthens the residual, does not close it. Cited in tfpt_research_contracts; Python-only.

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• FORM.SPECTRALGAP.01 — the multi-dimensional case, reduced to the scalar core. The Lean module TfptCarrier/SpectralGapAttractor.lean now also proves the finite-dimensional gapped-operator statement (lake build OK, no sorry): [E]component_tendsto (a non-dominant eigencomponent |r|<1 decays, r^n c\to0); [E]deviation_tendsto (with the dominant eigenvalue normalised to 1, the deviation vector — all non-dominant components — tends to 0, so the iterate converges to the dominant eigendirection, the unique attractor, for any start); [E]deviation_unique (two starts give deviation vectors both \to0, so the selected eigendirection is independent of the start — parameter-freeness in finite dimensions); [E]flavor_deviation_tendsto (a concrete instance on the cusp-transfer spectrum \{1,(2/3)^6,(1/3)^6\}, v56/v82/v383). Honest scope: this is the finite-dimensional diagonalisable case (the genuine multi-dimensional content, reduced to the scalar core); the general non-diagonalisable / infinite-dimensional Perron–Frobenius theorem stays the cited statement. Referenced in origin_theory.

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• Website — the UniversalGapLab animation on /verification. A new client component (Framer Motion, no new dependency) with three linked, module-grounded views: (A) every sector as the gap contraction \mathrm{iter}_n=x_\star+r^n(x_0-x_\star) racing to its attractor at r=(2/3)^6 (seam-gapped, v303/v387/v388) or the golden (\varphi+2)/4 (v312), with a gapless r=1 track that never forgets its start (the Lean theorem FORM.SPECTRALGAP.01) — the rate is the first-correction size (v388); (B) the entire-form-factor graviton ratio e^{-u} falling faster than any power, ghost-free and UV-soft (v386/v389); (C) the residual matrix as certification, not construction — every open item external, zero open internal mechanisms (v384). Mirror-only: every number is grounded in a module; no theory change.

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• v391 (ALPHA.QUILLEN.PROGRESS.01) — an honest attempt at the external target ALPHA.QUILLEN.EXACT.01 (v382, door 5). It REDUCES the open step to a sharper named sub-target via a solvable model but does not close it: ALPHA.QUILLEN.EXACT.01 stays [O](external math, type A, v384). [E]a solvable model \det_\zeta(-\partial_\tau^2+m^2) on a circle of circumference \beta equals (2\sinh(\beta m/2))^2, whose m-variation \beta\coth(\beta m/2)=2/m+(\beta^2/6)m-(\beta^4/360)m^3+\dots is a closed heat-kernel/Laurent series (the m^3 coefficient -1/360 verified). [E]so a \zeta-determinant's coupling-variation is structured heat-kernel data, and v382's counterterm 8b_1\cthree^6 (a Gauss–Bonnet heat-kernel power) is the right kind of object; the v382 split is re-verified at \ainv=137.0359992. [E]the reduction: the open step now reduces to the named sub-target “the seam U(1) operator's Seeley–DeWitt coefficient =8b_1\cthree^6”. [O]not closed: the actual seam-operator \zeta-determinant proof stays open, and SEAM.EQUIV.01 continuum existence (MMST, v336) stays the other external [O]; the value \ainv stays [E]. An attempt that reduces, does not close. Cited in tfpt_1_architecture_e8; Python-only.

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• FORM.SPECTRALGAP.01 — a Lean 4 proof of the gap \Rightarrow unique-attractor \Rightarrow parameter-free core (door 4). New module TfptCarrier/SpectralGapAttractor.lean (lake build OK, no sorry, only the three standard kernel axioms), formalising the meta-theorem of v303/v383/v387 at the one-dimensional reduction — the affine gap contraction x\mapsto x_\star+r(x-x_\star) with multiplier r=\lambda_2/\lambda_1. [E]iter_sub: the closed form \mathrm{iter}_n-x_\star=r^n(x_0-x_\star) (general n); [E]iter_tendsto: 0\le r<1 (the gap) \Rightarrow\mathrm{iter}_n\to x_\star (the unique attractor); [E]starts_agree: two starts differ by r^n(a-b)\to0, so the attractor is independent of the initial state (parameter-freeness); [E]no_gap_no_forget: r=1 (no gap) leaves \mathrm{iter}_n-x_\star=x_0-x_\star constant — the initial condition is then a free parameter, so the gap is exactly what removes it; [E]the seam rate (2/3)^6 and the golden (\varphi+2)/4=(5+\sqrt5)/8 both lie in [0,1). Honest scope: the scalar reduction (the contraction core v303/v383 iterate numerically); the full multi-dimensional Perron–Frobenius statement stays the cited theorem. Referenced in origin_theory; ledger row FORM.SPECTRALGAP.01.

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• v390 (COXETER.PRIME2.01) — the prime-2 facet of the order-30 Coxeter clock. v383/v387 named two number-field facets of the clock 30=2{\cdot}3{\cdot}5=h(E_8) — the golden \varphi\in\mathbb Q(\sqrt5) (prime-5, carrier) and (2/3)^6\in\mathbb Q (prime-3, family). This names the prime-2 facet and identifies it with structure already in the theory: the Gaussian field \mathbb Q(i), i.e. the CM point \tau=i and the order-4 \mu_4=\mathbb Z/4 square seam deck (v282/v288/v333). [E]h(E_8)=30 factors into exactly three primes; one quadratic facet field per prime, each ramified at its prime: \operatorname{disc}\mathbb Q(i)=-4 (2 ramifies), \operatorname{disc}\mathbb Q(\sqrt{-3})=-3 (3 ramifies), \operatorname{disc}\mathbb Q(\sqrt5)=5 (5 ramifies); the ramified primes are \{2,3,5\} with product 30=h(E_8). [E]the prime-2 (Gaussian) facet is the \mathbb Z/4 seam deck: min. poly x^2+1, j(i)=1728=12^3, \tau=i fixed by the modular S of order 4=2^2, |\mu_4|=4=2^2 — the static, even, seam-orientation facet. [C]so the three primes of the clock now each carry a facet (square/hexagon/pentagon = seam/CP/carrier); the prime set \{2,3,5\} is forced by h(E_8)=30, not chosen. Cited in origin_theory; a synthesis, no new number; Python-only.

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• v389 (GRAV.LOOP.FINITE.01) — the entire-form-factor graviton is UV-finite at the loop level. The honest next step past v386 (which established the tree amplitude), extending v304/v370/v380/v386 from the propagator to the superficial degree of divergence of loop graphs. This is the standard infinite-derivative / nonlocal-gravity finiteness statement (Tomboulis; Biswas–Mazumdar–Siegel; Modesto), not a full proof of perturbative quantum gravity. [E]super-polynomial falloff: p^k e^{-p^2/M^2}\to0 for every degree k. [E]one dressed line turns a quadratic divergence into the finite (M^2/2)e^{-p_0^2/M^2}, while the GR \int p^3/p^2\,dp diverges quadratically. [E]the superficial degree \to-\infty at every loop order L (the form factor multiplies the integrand by e^{-(\#\mathrm{internal})p^2/M^2}; dressed cutoff-ratio \to1 vs GR \to4). [E]the regulator is the seam scale M=M_{\mathrm{scal}}=\cthree^{7/2}\Mbar (v253/v259), not a dial. [C][O]loop unitarity of the nonlocal form factor (Lorentzian continuation / macro-causality, Pius–Sen / Briscese–Modesto) stays the honest open point — tree unitarity is v386; not claimed solved. Cited in tfpt_4_frontier; Python-only.

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• v388 (CORRECTIONS.BUDGET.01) — the gap-driven correction (v387) is a typed budget. A machine-checked answer to “does the (\lambda_2/\lambda_1)^n correction apply to all predictions?” — the answer is no, and for one whole class a band would be wrong. Because the size is the distance from a gapped operator's leading eigenvector, it is defined only where a subleading \lambda_2 exists, so the prediction surface splits into four classes. [E]seam-gapped (Koide F_{\mathrm{pole}} v303, recovery v221, the QG bound capped by \chi=729/665 v337): rate (2/3)^6; the discrete compiler: golden (\varphi+2)/4 (v312). [E]exact identity (\det R=8, N_\Phi=1, \theta_{\mathrm{eff}}=0, \sin^2\theta_{13}=\varphi_0 e^{-5/6}): no \lambda_2, so the correction is 0 structurally and a band would contradict the [E]status. [E] external-rate (\eta_B washout, axion freeze, m_p/m_e RG): the gapped shape with an external rate — only F_{\mathrm{pole}} has the seam rate (v303), firewalled (v187). [E]fixed-point: the texture is already explicit (e.g. \sin^2\theta_{12}, \varepsilon=(3/4)\varphi_0=c_3+36c_3^4). [E]the Koide first correction (door 2): F_{\mathrm{pole}} is the one seam-gapped prediction, rate (2/3)^6\approx0.0878 (v371), the source quotient 0.33\% below 2/3 — the rate is the suppression scale, not the residual. [C]so a uniform band on all predictions is wrong; the size is keyed to the mechanism class. A typing of v387 (as v384 is of v383); cited in origin_theory; Python-only.

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• v385 (UVBRANCH.KILLTEST.01) — the carrier-Pati–Salam UV branch on the all-order footing. A consolidation that ties the all-order EG/BRST closure (v381) to the optional UV branch and computes the one genuinely new thing: the proton-decay safety hierarchy. [E]the carrier content is the SM with no new states, so the SM-non-unification (v246) is now all-order-robust; [X]the plain-SM 4D-GUT stays killed (\sin^2\theta_W=3/8 vs 0.231); [C]the optional carrier-PS branch sits at M_{PS}=\kappa M_{\mathrm{scal}}\approx 3\times10^{13} GeV (\kappa\in[1.0,1.2], v253); [E]minimal PS gauge bosons are SU(4) leptoquarks that mediate rare LFV (K_L\to\mu e), not p\to e^+\pi^0, so the branch clears the binding bound by M_{PS}/M_{\mathrm{bound}}\sim10^7 – proton-safe, with no fake \tau_p window (v265). Cited in tfpt_5_redteam; Python-only.
• v386 (GRAV.AMPLITUDE.01) — the graviton-exchange amplitude is finite and UV-soft. Extends v304/v370/v380 from the propagator to the actual tree amplitude. [E]the dressed spin-2 propagator e^{-p^2/M^2}/p^2 has its only pole at p^2=0 (residue 1>0, ghost-free), is exponentially softened in the UV (the ratio e^{-u} falls faster than any power) and reduces to GR + a calculable -p^2/M^2 correction at low energy (M=M_{\mathrm{scal}}, v253); [C]a single positive-residue pole gives tree unitarity, so perturbative graviton scattering A\sim(T{\cdot}P_2{\cdot}T')e^{-p^2/M^2}/p^2 is an explicit finite computation (perturbative-only; QG.AMB.01 stays a [C]redundancy, v369). Cited in tfpt_4_frontier; Python-only.
• v387 (CORRECTIONS.GAP.01) — the gap-driven correction calculus. The practical harvest of v383: the same spectral gap that makes a sector parameter-free also sets the size of its first correction (\mathrm{correction}_n\sim(\lambda_2/\lambda_1)^n). [E]flavor, horizon recovery and the QG bound all share the (2/3)^6 rate (the family-3 facet; QG susceptibility \chi=729/665), while the discrete compiler transient decays at the golden (\varphi+2)/4 (the carrier-5 facet) — the two number-field facets of v314/v383. [C]the gap does double duty (why no free parameter and how big the first correction), so sub-leading magnitudes are organised, not free. Cited in origin_theory; Python-only.