Original claim records.

Original ledger status

FLAV.RIGID.01

FLAV.RIGID.01

Readout Rigidity (Theorem U2): c_u/c_d is CONSTANT on the discrete selector stratum S_{8,Q}={det R=8, SNF(R)=(1,1,8), Spec(Q+)={1,2,3}}, independent of the continuous D4 position => c_u/c_d=55/117 needs NO point-uniqueness. (U_wall) splits into U_unitary (v40, Phi=0 finite) + U_H2 (v39, R/Q/K from bundle) + U_Lambda2 (v42/v45, exterior Plucker 55/117) + U_point (open, only for full U_f* matrix). D4 fixes the chamber, selectors cut the point. The flavor bridge is a finite exterior readout-rigidity theorem, NOT a PDE gate

Recorded status: Identity [I] (rigidity) + [A] (U_point)

Original ledger status

FLAV.QGEO.02

FLAV.QGEO.02

Q geometry (Theorem Q): Sigma-split Q_+=(Q+SigQSig)/2=[[3,0,0],[0,2,0],[0,2,1]] (chi=(t-1)(t-2)(t-3), Spec={1,2,3}=A3 exponents), Q_-=[[0,1,0],[3,0,0],[3,0,0]] (chi=t(t^2-3), Q_-^2|supp=3=N_fam). UNIQUE nonneg-integer Q with rows(4,5,6), cols(9,5,1), chi_Q+, chi_Q- (enumeration) => K=R+Q*Sigma is the second Sigma-shadow of the same parabolic flavor operator. Algebraic normal form [I]/[L]; D4-equivariant parabolic realisation of Q_+/Q_- on P1\\mu4 remains [P] (GATE.QGEO)

Recorded status: Identity/Lattice [I]/[L] + [P] geometry

Original ledger status

SEL.THM.01

SEL.THM.01

Boundary Carrier Selection Theorem (Theorem A): admissible boundary-carrier data SELECT (D5+A3)+mu4=E8 uniquely. Boundary mu4->H1=Z^3->A3,N_fam=3; minimal 16-spinor->D5 (only n=5); disc(D5)=disc(A3)=Z4 cyclic anti-isometric, q(D5)+q(A3)=5/4+3/4=2=E8 root norm, glue index 4=|mu4|, E8 unique even unimodular rank 8. Four simultaneous conditions (16-spinor, N_fam=3, cyclic Z4, q-sum 2) met ONLY by D5+A3; D8 (non-cyclic, no family), E7+A1 (1 fam), E6+A2 (2 fam, Z3), A4+A4 (Z5), A8 (single) excluded. Lattice facts [L]; physical selection (P1 seam + P2 carrier => these inputs) is [P]/[A] selection axiom, not zero-input derivation

Recorded status: Lattice [L] + selection [P]

Original ledger status

EM.WARD.01

EM.WARD.01

EM Boundary Ward decomposition (Theorem C): F_U(1)=0 read as reduced boundary U(1) Ward identity in three terms - Maxwell alpha^3 + Calderon -2c3^3 alpha^2 (2=|Z2|) + transport -8b1 c3^6 log(1/phi_seam). Coefficient identities EXACT [I]: 8b1=(4/5)M=(4/5)(sumL+N_Phi)=164/5; seam exponent -5/4=-q(D5); transport power c3^6=C6 hexagon. Root reproduces alpha^-1=137.0359992 (v3). Physical Ward ORIGIN (EM1/EM2/EM3: d_tau log Z = each term) is [P] interpretation, NOT machine-proven. SHARPENED 2026-06-22 (v341/ALPHA.QUILLEN.01): the three terms are reassembled as the stationarity of a U(1) determinant line (delta log det Delta_U(1) + counterterm + seam = 0), every coefficient identified as index (8 b1) / heat-kernel (c3^{3,6}) / discriminant (q(D5), q(A3)); the [P] residual is unchanged in scope -- the from-first-principles AQFT proof that this is the EXACT Quillen functional -- only stated more sharply (still [P]/[O], no new gate) UPDATED 2026-08-27 (v974): the cited EM1 inflow step now carries a computed lattice witness -- on the v367 collar model pumped charge == edge spectral flow == Streda count == Chern == 1 per flux quantum (hybrid-Wannier winding residual 0.0; Streda DeltaN = C n_phi as exact integers; the complex-conjugation mutant flips all four readings simultaneously, the wrong-band projector breaks quantization). Single complex copy computed; the 16-copy bookkeeping (c_- = 16 x 1/2 = 8 = g_car + N_fam, kY = 5/3, (1/kY) 41/6 = 41/10 = b1) re-derived as exact Fractions. The physical Ward ORIGIN keeps its [P]/[O] scope -- nothing upgrades here.

Recorded status: Identity [I] + physical origin [P] (sharpened to a Quillen form by v341; to a heat-kernel origin by v342)

Original ledger status

SEAM.EQUIV.CONTINUUM.02

SEAM.EQUIV.CONTINUUM.02

THE CONTINUUM REALISATION SHARPENED BY THE BOOTSTRAP (2026-06-22): the 'c=8 ambiguity' flagged open in v277/v344 (E8 vs SO(16) both have c=8) is RESOLVED, non-circularly, by the seam's ORDER-4 mu4 clock; so the continuum residual of SEAM.EQUIV.01 is no longer 'which c=8 net?' but PURELY the existence of the chiral edge / massless scaling limit. [E] two c=8 lattices: E8 (det Cartan 1, holomorphic, 1 primary) and D8=SO(16) (det Cartan 4, 4 primaries). [E] the order-4 clock resolves it: the seam clock is mu4 (order 4, forced by the four Gauss-Bonnet marks |mu4|=4=h(A3), v216); the index-4 simple-current extension of D5(+)A3 is E8 (v154), the index-2 would be D8=SO(16); so the order-4 clock selects E8 (full mu4 condensation 16->4->1, det 1) NOT SO(16) (partial Z2 16->4, det 4, v281) -- the order (4 not 2) is the discriminator, non-circular (marks are Gauss-Bonnet). [E] the bootstrap agrees: h(E8)=30 (max prime 5=g_car, Coxeter-match v6/v350) vs h(D8)=14 (max prime 7); SO(16) fails the carrier bootstrap. [E] so the which-net ambiguity is RESOLVED (c=8 + the order-4 clock pin E8; det K=1/holomorphy is then a CONSEQUENCE, not an extra open input). [O] the residual is PURELY existence: does the gapped quasi-free collar HAVE a chiral edge / massless scaling limit (the bulk-edge correspondence: a gapped E8 bulk has a chiral c=8 edge)? -- the literature-anchored continuum leg (Morinelli-Stottmeister, Adamo OS; v336). The net and the holomorphy are pinned; only existence remains.

Recorded status: Sharpening [E]/[O] (the order-4 clock + Coxeter-match pin E8 over SO(16); the residual is purely the chiral-edge existence)

Original ledger status

VGEO.SHARPEN.01

VGEO.SHARPEN.01

v_geo IS THE SINGLE DIMENSIONFUL INPUT OF THE WHOLE THEORY (2026-06-23, Direction 6 of next-plan.md), now confirmed across matter AND gravity, and irreducibly primitive. The No-Unit Theorem (v153) already established that the dimensionless compiler cannot output an absolute scale, so v_geo is a forced metrology primitive, with the matter residuals collapsing onto it (U_point~v_geo, 1/G~v_geo^2, m/mu=e^{3/4} a pure ratio). The genuine SHARPENING here, after the parameter-free gravity (v358/v359/v361): the GRAVITY sector adds NO new dimensionful input either. [E] THE COMPILER IS DIMENSIONLESS: c3=1/(8pi) and g_car=5 are mass-dimension 0, so by the No-Unit Theorem an absolute scale CANNOT be an output -- v_geo is a forced INPUT (a chosen unit), not derivable. [E] GRAVITY ADDS NO NEW DIMENSIONFUL INPUT: the Einstein coefficient 1/c3=8pi and the Lambda prefactor (8pi)^2*48c3^4=3/(4pi^2) are dimensionless, and rho_Lambda/Mbar^4=(3/(4pi^2))e^{-2 alpha^-1} is a dimensionless ratio; so 1/G~Mbar^2~v_geo^2 and Lambda~(dimensionless)*v_geo^4 both reduce to v_geo times atoms -- the gravity sector introduces NO scale beyond v_geo. [E] THE FINAL TALLY: every dimensionful quantity in the theory (masses, 1/G, Lambda, U_point, the EW/flavor/Planck scales) = (a dimensionless compiler ratio)*(a power of the ONE unit v_geo); TFPT's complete free content is {v_geo (one mass unit), pi (a fixed math constant)} with ZERO dimensionless dials -- versus the SM's ~26 free parameters, a ~26->1 reduction. [O] v_geo IS IRREDUCIBLY PRIMITIVE: NOT an open gap -- a dimensionless theory provably cannot derive a scale (No-Unit), so v_geo is the one necessary external unit (the choice of ruler, like the metre); 1/G is UV-sensitive (Sakharov induced gravity, v68), a cutoff-dependent metrology readout, not a clean prediction. RESULT: v_geo is the single dimensionful input of the complete theory (matter via v153 + gravity via v358/v359/v361), irreducibly primitive; the theory is 0 dimensionless dials + 1 unit + pi. Direction 6 is clarity (the last dimensionful anchor named and shown to be the only one), not a new derivation.

Recorded status: v_geo capstone [E]/[O] (Direction 6: v_geo the single dimensionful input across matter+gravity, irreducibly primitive by No-Unit; final tally 0 dimensionless dials + 1 unit + pi vs SM ~26)

Original ledger status

GRAV.BACKREACT.01

GRAV.BACKREACT.01

THE MATTER-GRAVITY BACKREACTION WITH THE EXPLICIT J3 FLUX (2026-06-23, Direction 5 of next-plan.md). With the parameter-free Einstein equation (v359, G_ab+Lambda g_ab=(1/c3)T_ab) and the matter boost flux (v358, delta<K_B>=(8 pi^2 R^4/15) delta<T_00>), feed TFPT's carrier stress tensor in and ask what is FORCED about the backreaction of the SM content on the metric. Run with the v354/v355 discriminator: two genuine forced facts (positive) + an explicit decline. [E] THE CARRIER'S GRAVITATIONAL ANOMALY IS FORCED c_-=8: the carrier is 16 Majoranas (=dim S^+=2^(g_car-1)); the chiral central charge / gravitational t Hooft anomaly is 16/2=8=g_car+N_fam (the same c_- as v358) -- the matter content's gravitational anomaly is an atom, not a free number. [E] THE BACKREACTION IS FINITE -> Lambda FROM ALPHA (the CC backreaction): the matter vacuum energy, fed through G_ab=(1/c3)T_ab, does NOT give the M_Pl^4 catastrophe -- the seam gap suppresses it (Decoupling Theorem v337/v76) to the forced rho_Lambda/Mbar^4=(3/(4 pi^2))e^{-2 alpha^-1} (v60), with the 123 orders = 2 alpha^-1/ln10 (119.03, the EM fixed point) + log10(256 pi^4/3) (3.92); the loop (v359 equation + v60 Lambda + the carrier vacuum) closes. [E] THE MATTER COUPLING IS EXPLICIT (J3, v358): delta<K_B>=(8 pi^2 R^4/15) delta<T_00> sources the metric; the carrier stress tensor IS the SM's (v159: reproduces the SM betas, no new gauge-charged state), so the backreaction is GR with the SM content. [C] THE SM TRACE/CONFORMAL ANOMALY IS REPRODUCED, NOT NEW: its coefficients are functions of the field content, and TFPT's content = the SM. [O] DECLINE (the discriminator): beyond c_-=8 (anomaly) and Lambda (vacuum), there is NO new atom-forced gravitational read-out of individual SM quantum numbers -- the backreaction is GR-with-the-SM and the forced pieces are already established (v358/v60); per v354/v355 we DECLINE to manufacture one. HONEST CONCLUSION: the matter-gravity backreaction is finite and consistent (gravitational anomaly c_-=8 + gap-suppressed vacuum -> the forced Lambda), closing the loop; it adds no new dial.

Recorded status: Backreaction result [E]/[C]/[O] (Direction 5: carrier gravitational anomaly c_-=8 forced + finite backreaction -> Lambda from alpha, loop closes; no new SM-quantum-number read-out -> declined per v354/v355)

Later scope correction: the DtN form estimate alone does not prove the transfer gap.

Original ledger status

GRAV.GAPCORR.01

GRAV.GAPCORR.01

THE GAP-INDUCED HIGHER-CURVATURE CORRECTION TO GR -- THE DISCIPLINED RESULT (2026-06-23, Direction 2 of next-plan.md). Question: does the seam transfer gap {1,(2/3)^6,(1/3)^6} yield a NEW, calculable, FORCED deviation from GR (a near-term kill test) beyond the known R^2 scalaron? Run with the v354/v355 anti-numerology discriminator (a coefficient counts only if atom-forced, never a fit), the honest answer is NO -- and this is recorded rather than manufacturing a coefficient. [E] THE LEADING HIGHER-CURVATURE TERM IS THE R^2 SCALARON (already in TFPT): the spectral-action a4=R^2/72 (v36) gives R^2; TFPT fixes its scale M_scal^2/Mbar^2=c3^7 (exponent 7=g_car+N_fam-1), M_scal=c3^(7/2)Mbar~3.06e13 GeV -- the calculable beyond-Einstein correction. [C] IT IS THE ENTANGLEMENT-EQUILIBRIUM NEXT ORDER: the higher-curvature (Wald) entropy correction to the first law gives the f(R)=R+R^2 equation (v28), so the scalaron is the next-order term of the SAME route that gave the Einstein equation (v359) -- a structural confirmation, not a new term. [E] ALREADY FALSIFIABLE NOW: the R^2/Starobinsky attractor predicts n_s=1-2/N_star in [0.960,0.967] and r=12/N_star^2 in [0.0033,0.0048] (N_star in [50,60]) -- a live CMB-S4 kill test; Direction 2's calculable GR deviation already EXISTS and is testable. [E] THE GAP IS A DIMENSIONLESS RATE -> SEAM/PLANCK-SCALE CORRECTIONS: Delta=6 ln(3/2) is dimensionless; a distinct gap-induced curvature^2 term enters at xi*v_geo with xi=1/Delta~0.41 (seam units) and v_geo~M_Pl -- the seam/Planck scale, NOT near-term observable. [O] DECLINE (the discriminator): a NEW dimensionless deviation coefficient distinct from the scalaron needs the subleading heat-kernel/Wald data, which is NOT a forced atom combination (the scalaron exponent 7=g+N-1 and the gap exponent 6=2 N_fam are different structures; c3^7 and (2/3)^6 are unrelated). Per v354/v355 we DECLINE to manufacture an atom-unforced coefficient. HONEST CONCLUSION: Direction 2 yields no new near-term forced prediction; the answer is the existing scalaron + Planck-scale corrections.

Recorded status: Gap-correction result [E]/[C]/[O] (Direction 2 declined: the calculable GR deviation is the R^2 scalaron already falsifiable via n_s/r; a new gap coefficient is not atom-forced -> declined per v354/v355)

Original ledger status

GRAV.NONLINEAR.01

GRAV.NONLINEAR.01

THE FULL COVARIANT EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23, Direction 1 of next-plan.md, extends GRAV.ENTROPY.EQUILIBRIUM.01/v358). v358 ran the entanglement first law at fixed radius -> the Ricci-level (linearised) equation. Jacobson 2015 (PRL 116, 201101) shows that demanding stationarity of the small-ball entanglement entropy at fixed VOLUME brings in the EINSTEIN TENSOR G_ab (not just R_ab) -- the unique symmetric divergence-free second-order metric tensor (Lovelock) -- so the equilibrium yields the FULL covariant G_ab + Lambda g_ab = (1/c3) T_ab with local conservation built in, and TFPT supplies BOTH coefficients that Jacobson's local argument leaves free. [C/E] FIXED VOLUME -> EINSTEIN TENSOR: at fixed volume the small-ball area deficit is governed by G_ab, not R_ab (Jacobson 2015); the structural identity G_ab=R_ab-(1/2)R g_ab has trace g^{ab}G_ab=(1-d/2)R=-R in d=4 -- the trace structure that carries Lambda, upgrading v358's linearised Ricci-level result. [E] LOVELOCK UNIQUENESS -> CONSERVATION: in d=4 the only symmetric divergence-free second-order metric tensor is a*G_ab+b*g_ab, so fixed-volume equilibrium NECESSARILY produces G_ab+Lambda g_ab and nabla^a(G_ab+Lambda g_ab)=0 forces nabla^a T_ab=0 -- matter conservation is an OUTPUT, not an input. [E] COEFFICIENT 1 PARAMETER-FREE (from v358): the equilibrium coupling 2 pi/eta with eta=1/|mu4| gives 8 pi G -> 8 pi = 1/c3 FIXED (no free dimensionless Newton dial). [E] COEFFICIENT 2 FROM ALPHA (Direction-4 fold-in; v60): Jacobson's local argument leaves Lambda FREE; TFPT fixes it -- rho_Lambda/Mbar^4 = (3/(4 pi^2)) e^{-2 alpha^-1}, prefactor (8 pi)^2*48 c3^4 = 3/(4 pi^2) [E exact], the '123 orders' split 2 alpha^-1/ln10 + log10(256 pi^4/3) [N]. So BOTH coefficients of G_ab + Lambda g_ab = (1/c3) T_ab are TFPT-fixed, not free integration constants. [C] ALL TIMELIKE DIRECTIONS -> FULL TENSOR: holding the scalar equilibrium for every unit timelike direction at every point promotes it to the covariant tensor equation; the matter side is the assembled CHM ball modular flux (v358/v323). RESULT: the original B6 'full covariant Einstein-side field equation' is now PARAMETER-FREE at the LOCAL level. [O] RESIDUAL: this is the Jacobson EQUATION-OF-STATE derivation (a [C] physical framework), NOT a from-action quantisation; the global non-perturbative ambient measure (C7/QG.AMB.01) is separate and stays the inherited, gap-decoupled QG problem; v_geo remains the one unit.

Recorded status: Full covariant Einstein equation [E]/[C]/[O] (parameter-free: 8pi=1/c3 + Lambda from alpha; Einstein tensor via fixed-volume equilibrium + Lovelock conservation; B6 closed at local level; residual = equation-of-state status + C7 global measure)

Original ledger status

GRAV.ENTROPY.EQUILIBRIUM.01

GRAV.ENTROPY.EQUILIBRIUM.01

THE ENTANGLEMENT-EQUILIBRIUM DERIVATION OF THE LINEARISED EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23). Carries out the Jacobson-2015 / Faulkner-et-al. 'first law of entanglement => Einstein equation' (delta S = delta<K>) with TFPT's atoms, and closes the two pieces that the bridge analysis (v356) left open (the matter boost flux J3 and the entropy-density coefficient). [E] PARAMETER-FREE EINSTEIN COEFFICIENT: delta S = delta<K> => G_ab = (1/c3) T_ab with 1/c3 = 8pi FIXED; all four c3-roles mutually consistent with NO free dimensionless Newton dial (Unruh factor 1/(2pi)=4c3, EH action 1/(16pi)=c3/2, Einstein coeff 8pi=1/c3, Bekenstein entropy density 1/4=1/|mu4|). [E] THERMO = GEO COINCIDENCE (the new exact result): the first law REQUIRES the Einstein coefficient 2pi/eta (eta=entropy density); the geometry gives 1/c3 = |Z2|*2pi*chi(S^2) (one-sided Gauss-Bonnet, v58); these give the SAME equation iff |mu4| = |Z2|*chi(S^2) = 4 (a TFPT atom identity, v216) -- so the thermodynamic (first-law) and geometric (Gauss-Bonnet) origins of c3 are ONE object, coupled by 4=|mu4|=|Z2|*chi. [C] J3 SOLVED (matter boost flux assembled): the ball modular Hamiltonian is the boost (Casini-Huerta-Myers; geometric via Bisognano-Wichmann, v323) K_B = 2pi int_B w T_00 with w=(R^2-r^2)/(2R) the CHM weight; the 3-ball weight integral int_B w d^3x = 4pi R^4/15, so delta<K_B> = (8 pi^2 R^4/15) delta<T_00> -- the concrete matter-side flux of TFPT's stress tensor (closing the 'ingredients present, flux not assembled' gap). [E] ENTROPY DENSITY ATOM-FIXED (the [A] coefficient pinned): the Bekenstein numerator 1/4=1/|mu4| (exact, v57) and the entanglement normalisation = the central charge c=8=g_car+N_fam (a CFT's universal area-law/entanglement coefficient IS its central charge); the area-law FORM is established for the RP-Gaussian seam (v59). So the dimensionless entropy-density coefficient is atom-determined; the only residual is the absolute area UNIT (v_geo), not a free dimensionless gap. RESULT: the LINEARISED Einstein equation G_ab=(1/c3)T_ab closes PARAMETER-FREE. [O] RESIDUAL: (i) linearised -> full non-linear covariant Einstein (the original B6/GATE.METRIC, all-balls/all-frames + non-linear) and (ii) the absolute scale v_geo (Planck-area unit); NO free dimensionless parameter remains in the gravity coupling.

Recorded status: Entanglement-equilibrium gravity [E]/[C]/[O] (parameter-free linearised Einstein 8pi=1/c3; thermo=geo coincidence via |mu4|=|Z2|chi; J3 assembled; entropy density atom-fixed; residual = nonlinear + v_geo)

Original ledger status

SEAM.EQUIV.CONTINUUM.03

SEAM.EQUIV.CONTINUUM.03

DIRECTION A (2026-06-23): the continuum chain of SEAM.EQUIV.01 assembled theorem-by-theorem, with the residual reduced to ONE lattice-realization input tied to the seam's one-sidedness (the c3 origin). After the reverse-audit/bandwidth work showed no hidden DISCRETE lever remains, the only fundamental progress is to assemble the CONTINUUM chain -- which this does. THE CHAIN: S1 the gapped (Delta=6 ln(3/2)>0, v302) quasi-free (CAR, v155/v160) 16-Majorana collar -> S2 invertible (a gapped free-fermion 2+1D phase has no intrinsic topological order, Kitaev, v301) -> S3 the NONTRIVIAL invertible phase (c_-=8!=0), not the trivial c_-=0 one -> S4 a NON-gappable anomalous chiral edge (anomaly inflow / bulk-edge: a nontrivial invertible phase cannot have a gapped edge), so edge EXISTENCE is a CONSEQUENCE of S2+S3, NOT a separate assumption -- this directly reduces the v336/v351 'does the gapped collar HAVE a chiral edge?' residual -> S5 the edge is 16 chiral Majoranas whose chiral-CFT scaling limit is given by MMST (Morinelli-Morsella-Stottmeister-Tanimoto, v336; range rank<=c<=D = 8<=8<=16, IN range) -> S6 the order-4 mu4 clock selects the index-4 net (E8)_1 (v351) and the AQFT stack invertible->trivial center->holomorphic->(E8)_1 is cited (v297, LIT-A/B/C). [C] the MMST range; [C] edge-existence-from-invertibility (anomaly inflow); [C] chirality = one-sidedness = the c3 ORIGIN: c_-!=0 is the seam being one-sided, the SAME one-sidedness that makes c3=1/(8pi) a one-sided Gauss-Bonnet (|Z2| oint K, oint K=4pi on S^2; v58/v73); [E] the chain closes modulo S3 (the only non-theorem link); [O] the ONE remaining input -- the abstract collar realized as a genuine LATTICE chiral free-fermion invertible phase (the v297 Flat-Away). NET: SEAM.EQUIV.01 reduced to one realization input tied to the constant that DEFINES the theory; ADVANCES, does not close. SEPARATELY, Direction B (the U_wall Hitchin selector) was found ALREADY closed by v40: at the polystable point Mehta-Seshadri makes the bundle unitary, so the harmonic metric is the constant invariant Hermitian form (finite linear algebra) with Higgs Phi=0 -- NOT a transcendental PDE -- and det R=8 is read off; only the absolute amplitude U_point (an anchor) remains. No new B module was shipped (a redundant v357 was built, found to duplicate/contradict v40, and deleted).

Recorded status: Continuum chain assembly [C]/[E]/[O] (edge-existence reduced via anomaly inflow; residual = lattice realization tied to seam one-sidedness; B already closed by v40)

Original ledger status

QGAMB.SADDLE.01

QGAMB.SADDLE.01

Direction 3 (2026-06-23): the QG measure (gate C7/QG.AMB.01) as one-loop GAUSSIAN fluctuations around the PARAMETER-FREE saddle (v358/v359). Before v358 fluctuations around the gravitational saddle were not even dimensionlessly well-posed (the Newton coupling was a free dial); with the parameter-free saddle the path integral organises as a Gaussian one-loop determinant whose convergence and dimensionless content are atom-fixed. [E] PARAMETER-FREE QUADRATIC FORM: the fluctuation inverse-propagator stiffness is the saddle coefficient 1/c3=8pi (no free dial). [E] GAPPED FLUCTUATION OPERATOR: 2||V||=2*248*c3^2=31/(4pi^2) (31=2^g_car-1) and Delta=6 ln(3/2), so M>=Delta_eff=Delta-2||V||~1.648>0 (v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the saddle directions. [E] FINITE ONE-LOOP LOG-DET (model): on the OS spectrum {0,6 ln(3/2),6 ln 3} the regularised Tr' log M=6 ln(9/2) and W=3 ln(9/2) are finite (gapped, no IR divergence). [C] CONFORMAL DIRECTION CONVERGENT: the wrong-sign conformal mode (c_conf(4)=-3/2) is made convergent by the GHP rotation int exp(-|c|phi^2)=sqrt(2pi/3) + IDG (no Stelle ghost), v334. [E] ADMISSIBLE PROJECTIVE LIMIT EXISTS: chi=1/(1-(2/3)^6)=729/665 finite (v330) => the one-loop measure is tight on the admissible sector. [E] NO NEW DIAL: the dimensionless content is a function of {c3,Delta,chi,c=8} only (v364). [O] RESIDUAL: the full non-perturbative projective limit (G6/QG.AMB.01) on the ambient sector remains OPEN. Sharpens C7 from diffuse full QG to a one-loop Gaussian problem around a parameter-free saddle; does NOT close it.

Recorded status: QG measure reduced to a one-loop Gaussian problem around the parameter-free saddle [E]/[C]/[O] (residual = the non-perturbative projective limit)

Later scope correction: the DtN form estimate alone does not prove the transfer gap.

Original ledger status

SEAM.MMST.INCLASS.01

SEAM.MMST.INCLASS.01

Direction 7 (2026-06-23): the seam collar is IN MMST's free-lattice-fermion scaling-limit class, so the chiral scaling limit is a chiral CFT of central charge 8 pinned to (E8)_1 -- reducing SEAM.EQUIV.01 to the single S3 lattice-realisation input. the cited theorem (Osborne-Stottmeister, arXiv:2107.13834, CMP 398 (2023) 219; the internal 'MMST' code-token/claim-id is LEGACY and denotes THIS Osborne-Stottmeister theorem, built on the MMST/Morinelli-Morsella-Stottmeister-Tanimoto OAR framework, arXiv:2010.11121): free lattice fermions (CAR quasi-free) give a chiral-CFT massless scaling limit with the correct central charge, in the range rank<=c<=D (D = number of Majorana copies). [E] CLASS ARITHMETIC / MMST RANGE: D=dim S^+=2^(g_car-1)=16 Majoranas, c=g_car+N_fam=8, rank E8=8 => rank<=c<=D reads 8<=8<=16, IN range (saturating rank=c=8, inside D=16). [E] GAPPED (derived, not assumed): Delta=6 ln(3/2)>0 (v302). [C] QUASI-FREE CAR CLASS: the collar is in the CAR quasi-free class (v155/v160), MMST's hypothesis class. [E] CHIRALITY: c_-=D/2=8=g_car+N_fam!=0 (a non-gappable anomalous edge by inflow, v356 S4). [C] MMST SCALING LIMIT APPLIES: in-class & gapped & chiral & in-range => the MMST theorem gives a chiral CFT of central charge 8 for the 16-Majorana edge (v336). [E] TARGET PINNED TO (E8)_1: c=8 + the order-4 mu4 clock select E8 (det K=1, one primary) over the same-c rival SO(16)_1 (det K=4) -- v351; holomorphy det K=1 is the consequence. [O] RESIDUAL S3: the one open input is the abstract collar realised as a genuine LATTICE chiral free-fermion invertible phase (the one-sidedness / Flat-Away, v297/v356 S3) = the seam one-sidedness that DEFINES c3; given S3 the MMST scaling-limit theorem applies as a cited result and the limit is (E8)_1. Verifies MMST in-class; ADVANCES SEAM.EQUIV.01, does NOT close it.

Recorded status: SEAM.EQUIV.01 reduced to the single S3 lattice-realisation input; MMST in-class verified for the collar [E]/[C]/[O]

Original ledger status

GRAV.SPIN2.UNITARITY.01

GRAV.SPIN2.UNITARITY.01

Track 2 (2026-06-23): the Barnes-Rivers spin decomposition of the graviton propagator -- perturbative graviton unitarity SECTOR-BY-SECTOR. Extends v304 (scalar pole algebra) to the tensor spin structure and combines it with v334 (spin-0 conformal mode); does not re-derive v304 and does not close QG.AMB.01. [E] PROJECTOR COMPLETENESS: tr P2=(d-2)(d+1)/2, tr P1=d-1, tr P0s=tr P0w=1 sum to d(d+1)/2 (the symmetric-tensor dimension); in d=4 the split is 5+3+1+1=10. [E] EH STRUCTURE: the Einstein-Hilbert graviton propagator ~ P2/p^2 - 1/(d-2) P0s/p^2, spin-2 coefficient +1, spin-0 scalar -1/(d-2)=-1/2 (d=4). [E] THE STELLE GHOST IS A SPIN-2 STATE: a local R+R^2/Weyl^2 dressing 1+p^2/M^2 gives the spin-2 propagator a negative-residue pole at p^2=-M^2. [E] ENTIRE FORM FACTOR => SPIN-2 GHOST-FREE: the un-truncated KMS form factor a(p^2)=e^{p^2/M^2} (v304/v259, entire, nowhere zero) keeps the spin-2 propagator's only pole at p^2=0. [C] SPIN-0 VIA GHP: the wrong-sign conformal/trace mode (c_conf(4)=-3/2, v332) is cured by the GHP contour rotation + IDG (v334) -- a different mechanism. [C] => the full PERTURBATIVE graviton is ghost-free sector-by-sector (spin-2 entire form factor + spin-0 GHP), conditional on v304's entire-analyticity assumption. [O] perturbative-only: inherits the unproven entire-a(Box) assumption, and a ghost-free perturbative graviton is NOT the non-perturbative ambient measure (QG.AMB.01). [O] DECLINE: tr P2=5 in d=4 is the spin-2 polarisation count (dimension counting), NOT g_car=5 (v354/v355).

Recorded status: perturbative graviton unitarity sector-by-sector (spin-2 ghost-free via the entire KMS form factor, spin-0 via GHP) [E]/[C]/[O]

Original ledger status

SEAM.S3.LATTICE.01

SEAM.S3.LATTICE.01

Track 1 (2026-06-23): the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1. Turns the abstract S3 residual of v356/v366 ('the collar realised as a genuine lattice chiral free-fermion invertible phase') into a RUNNABLE model with a numerically computed topological invariant, not a re-assertion of det K / c_-. [E] BULK GAPPED: h(k)=sin kx sx + sin ky sy + (M-cos kx-cos ky) sz at M=1 has band gap 2 min|d(k)|=2>0. [E] NON-TRIVIAL CHERN (numerical Fukui-Hatsugai-Suzuki plaquette): |C|=1 at M=1 (topological) and C=0 at M=3 (trivial neg control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2; N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam (the edge gravitational anomaly). [E] EDGE K-MATRIX: the 16-Majorana edge is SO(16)_1 (det Cartan D8=4) condensed to (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the index-4 mu4 simple-current condensation SO(16)_1 -> (E8)_1 (v154/v351) is the seam glue on top. [O] residual: the continuum scaling-limit existence (the lattice model's massless limit IS the chiral CFT, MMST v336) stays open; S3 is now a concrete lattice model, reduced not closed.

Recorded status: S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]

Original ledger status

SEAM.S3.INFLOW.01

SEAM.S3.INFLOW.01

Track 1 (2026-06-23): the bulk-edge / anomaly-inflow leg (S4 of the v356 chain) made CONCRETE on the v367 lattice model -- a finite STRIP of the gapped p+ip model has a REALISED, non-gappable chiral edge, so edge EXISTENCE is a CONSEQUENCE of the bulk being a nontrivial invertible phase (anomaly inflow), NOT a separate assumption. This directly discharges the v336/v351 'does the gapped collar HAVE a chiral edge?' residual on the lattice. [E] TOPO IN-GAP EDGE STATES: for M=1 (|C|=1) the strip spectrum reaches min_kx|E|~0 deep in the bulk gap (a chiral edge branch traverses the gap). [E] TRIVIAL CLEAN GAP (neg control): for M=3 (C=0) min_kx|E|>0.3 stays well above zero. [E] EDGE-LOCALISED: the near-zero in-gap state's weight is concentrated within 4 sites of a boundary (edge>>bulk). [E] BULK-EDGE CORRESPONDENCE: the chiral edge branch count = bulk Chern |C|=1 (one chiral Majorana per edge) => the 16-copy carrier has a 16-Majorana chiral edge, c_-=8. [C] ANOMALY INFLOW: a gapped invertible phase with c_-!=0 cannot have a gappable edge, so the chiral edge of the collar (c_-=8, v301/v356) EXISTS as a consequence of the bulk. [O] residual: the continuum scaling-limit of that lattice edge (MMST v336) stays open; v368 establishes edge EXISTENCE on the lattice, not the scaling limit.

Recorded status: edge existence forced by the invertible bulk (anomaly inflow) -- realised non-gappable chiral edge on the lattice strip [E]/[C]/[O]

Original ledger status

FR.POLE.SOLVE.01

FR.POLE.SOLVE.01

Track 3 (2026-06-23, F_transfer): the Koide source->pole transfer as a TYPED solver -- promotes experiments/ftransfer/koide_source_to_pole into the ledger with a kill test. Does NOT re-prove the 53/54 operator factor (v183, [E]) or Q*=2/3 (v82/v101); answers the one open question: is the physical source->pole transfer standard QED/EW running? A clean NEGATIVE. [E] Q_pole=Koide(PDG poles)=2/3 to 3e-6; Q_src(phi0-ladder, v93)=0.6644638 (0.33% below 2/3). [E] dQ/deps>0 at the pole: more QED running widens the hierarchy and pushes Q ABOVE 2/3 (Q(eps_QED)>2/3). [E] the eps mapping Q_pole->Q_src is NEGATIVE (brentq) while QED gives eps=+(3/2)(alpha/pi)>0 for any alpha in [1/137,1/128] -- the WRONG SIGN. [C] verdict: F_pole is NOT perturbative running; the transfer is the non-QED operator/Moebius generator whose signature is 53/54 (v183, [E]) and the (2/3)^6 Moebius flow (v82). Stays [C], sharper -- plain running is the kill (excluded).

Recorded status: Koide source->pole: standard QED running EXCLUDED (wrong sign) -- the transfer is the non-QED 53/54 operator/Moebius generator [E]/[C]

Original ledger status

FR.BOLTZMANN.SOLVE.01

FR.BOLTZMANN.SOLVE.01

Track 3 (2026-06-23, F_transfer): eta_B from the INTEGRATED Buchmueller-Di Bari-Pluemacher Boltzmann network at the TFPT-frozen heavy scale -- promotes experiments/ftransfer/leptogenesis_boltzmann, replacing the v326 toy washout (kappa=0.3 tuned). Does NOT re-derive the viability band (v169) or the decuple anchoring (v212); integrates the ODE at the frozen inputs and checks eta_B. Frozen (NO free M_R dial): M1=M_scal phi0^2/A_Lambda~8.65e9 GeV (v212), m~1=m3/A_Lambda=5 meV, eps1 Davidson-Ibarra with the predicted delta_CP=4pi/3 (|sin|=0.866, v9). [E] the full-ODE efficiency kappa_f~0.09 (strong washout K~4.6) agrees with the analytic BDP fit to ~25% -- integrated transport, not a tuned constant. [C] eta_B(ODE)~6.5e-10 at the frozen M1 lands within a factor ~1.1 of the observed 6.1e-10; the M1 reproducing eta_B_obs is within a factor ~1.1 of the frozen M1. [X] kill: a proper solve putting eta_B_obs outside [obs/3, 3obs] at the frozen (M1, m~1) for all seesaw-consistent M_R drops the leptogenesis branch (compiler core + Omega_b readout FR.ETAB.01 untouched).

Recorded status: eta_B from the integrated BDP Boltzmann ODE at the frozen M1 -- near-hit (factor ~1.1) with no M_R dial [C]/[X]

Original ledger status

FR.RELIC.SOLVE.01

FR.RELIC.SOLVE.01

Track 3 (2026-06-23, F_transfer): the axion relic abundance from the FINITE-T misalignment ODE -- promotes experiments/ftransfer/axion_relic and DECIDES the two competing TFPT angle branches, replacing the v326 adiabatic toy. Does NOT re-fix the candidate scales (f_a=M_scal/128, m_a; v185/v211); integrates the nonlinear ODE theta'' + (3+dlnH/dN)theta' + (m_a(T)/H)^2 sin theta = 0 with the lattice topological susceptibility m_a(T)=m_a min(1,(T_QCD/T)^{n/2}) (n=8.16) and realistic g_*(T). [I] candidate scales f_a=M_scal/(2 dim S^+ |mu4|)=M_scal/128~2.39e11 GeV, m_a~23.8 ueV, N_DW=1 (determinant-line atoms). [C] the frozen SPINE angle theta_i=3pi/5=108deg=pi N_fam/g_car gives Omega_a h^2~0.12 in [0.08,0.16] (brackets Omega_DM=0.12) with NO tuning -- the robust dominant-DM branch. [X] the HILLTOP theta_i~170deg over-produces (Omega_a h^2~0.65, ~5.4x) -- disfavoured/fine-tuned. [X] kill: a finite-T solve robustly outside [0.08,0.16] for the spine across lattice chi(T)/g_* variations drops the dominant axion-DM branch.

Recorded status: axion relic finite-T solve DECIDES the branch -- spine 3pi/5 lands on Omega_DM (in band), hilltop 170deg over-produces [C]/[X]

Original ledger status

FR.QCD.BUDGET.01

FR.QCD.BUDGET.01

Track 3 (2026-06-23, F_transfer): the m_p/m_e transfer as a TYPED uncertainty-budget + kill test -- the falsifiability companion to v262 (the precise 2-loop point), replacing the v326 toy (which hardcodes 1836.15). Does NOT re-derive Lambda^(3) or the b3=-7 beta (v164/v262, [E]); propagates the two declared externals through the transfer m_p/m_e=C_p Lambda^(3)/m_e. [E] b3=-(11-2 n_f/3)=-7 (n_f=6 carrier). [N] two declared externals (v339): alpha_s(M_Z)=0.1179+-0.0009 (PDG) and C_p=2.83+-0.15 (lattice), entering linearly-in-log so d ln Lambda=(2pi/(b0^(5) alpha_s^2)) d alpha_s~5.3% and d C_p/C_p~5.3%. [C] m_p/m_e central ~1824 (v262 2-loop) with a propagated 1-sigma band +-sqrt(5.3^2+5.3^2)%~+-7.5% -> [~1690,~1960], CONTAINING the observed 1836.15 -- no free dial. [X] kill: a future lattice C_p+-0.03 + tighter alpha_s shrink the band to ~+-2%; a band then EXCLUDING 1836.15 drops the QCD-matching transfer route (b3=-7 carrier + the SU(9)-integer firewall v262 untouched).

Recorded status: m_p/m_e uncertainty budget: the observed 1836.15 is inside the propagated +-7.5% (alpha_s, C_p) band; lattice-tightening is the kill [C]/[N]/[X]