Original claim records.

Original ledger status

SEAM.EQUIV.ONEPARTICLE.01

SEAM.EQUIV.ONEPARTICLE.01

The one-particle RIGIDITY + scaling limit attacking R1 (2026-06-30, A of the A+B round) -- the REALISATION input of SEAM.EQUIV.01. The seam is quasi-free (v113/v160/v161), so by Araki self-dual CAR the whole net is the second quantisation of its one-particle symbol P, and R1 collapses to a finite-per-mode statement about P. [C] ARAKI SELF-DUAL CAR: a gauge-invariant quasi-free state <-> its symbol P (0<=P<=1) bijectively (Araki 1971; Shale-Stinespring 1965), so a realisation is fully determined by P. [E] RIGIDITY: the gapped ground state is the UNIQUE quasi-free state with P=projection onto K<0 (P^2=P, hermitian to <1e-12), K fixed by gap+chirality+reflection (v426) -- no second realisation. [E] ONE-PARTICLE SCALING LIMIT EXISTS: the real-space kernel is Cauchy on a fixed window (max|P_2N-P_N|~1/N -> 0), so P_inf exists (Gaussianity is why a computation reaches it). [E] LIMIT IS c=1 DIRAC (c_-=8): entanglement S(L)=(c/3)ln L gives c -> 1 (per Majorana 1/2; Peschel/Calabrese-Cardy, ties v450), 16 copies -> c_-=8. [C] Shale-Stinespring: gauge-invariant => no pairing block => the Bogoliubov map is implementable, GNS reps quasi-equivalent. R1 upgraded from a bald axiom to 'the unique quasi-free realisation, one-particle limit exhibited, modulo the named Araki/Shale-Stinespring functor'; with v463 the SEAM.EQUIV.01 residual is now entirely certification, zero open internal mechanism -- closed modulo cited theorems. SEAM.EQUIV.01 stays [O].

Recorded status: one-particle rigidity + scaling limit [E]/[C]: via Araki self-dual CAR (state<->symbol P, cited) the gapped ground state is the UNIQUE quasi-free realisation (P^2=P), its kernel is Cauchy (~1/N, limit exists), and the limit is the c=1 Dirac CFT (EE c-fit ~1 => c_-=16*1/2=8); Shale-Stinespring makes the Bogoliubov map implementable, upgrading R1 from a bald axiom to an audited [C]; with v463 the residual is entirely certification, zero open internal mechanism; SEAM.EQUIV.01 stays [O]

Original ledger status

FLAV.THIRDGEN.PATTERN.01

FLAV.THIRDGEN.PATTERN.01

THIRD-GENERATION LADDER-BASE PATTERN (2026-07-02, v467): the suite's THREE ~2 sigma mixing tensions (sin^2 theta13 +2.0 sigma, sin^2 theta23 +1.8 sigma, V_cb +1.8 sigma; v307) are ONE common ~ -phi0 suppression -- quantified and pre-registered as a CANDIDATE, the frozen record (v84/REG.FREEZE.01) is NOT changed. In existing vocabulary the pattern is parameter-free: the seed enters third-generation mixings through the LADDER BASE lambda_C^2 = phi0(1-phi0) (the Yukawa mass-ladder base, tfpt_2) instead of bare phi0, and the 2-3 texture carries the same -phi0/2 shift as theta12: sin^2 theta13 = lambda_C^2 e^{-5/6} = 0.021880 (+2.00 -> -0.12 sigma), sin^2 theta23 = (1-phi0)/2 <=> cos 2theta23 = phi0 = 0.473414 (+1.76 -> +0.20), V_cb = lambda_C^2/(1+lambda_C) = 0.041119 (+1.79 -> +0.01). Scope checks that make it a rule not a fit: V_ub = lambda_C^3/3 has no bare phi0, stays untouched and consistent (-0.23); the 1-2 channels must stay undressed and do (theta12 -0.02, lambda_C -0.25; dressing lambda_C would give -24 sigma). Weighted common deviation of the three affected channels -5.36% +- 1.67% (3.2 sigma non-zero, -0.03 sigma from -phi0); joint chi^2 over the four third-gen observables 10.4 -> 0.11 with ZERO new parameters; Daya Bay alone already prefers the pattern (Delta chi^2 = 4.3). HONEST: post-hoc (5/16 equally-simple dressings fix the two PMNS tensions; (1-phi0)/(1-c3)/resolvent degenerate <1.4%); suppression size phi0 = 5.32% sits INSIDE the seam-gapped correction band (2/3)^6 = 8.78% (v393) but NO seam-transport derivation exists (Bernoulli-variance reading lambda_Y^2 = Var[Bernoulli(phi0)] is an exact identity but only an interpretation). Pre-registered kills: theta13 stable at 0.0231 kills the pattern; at ~0.0219 (sigma_rel<1%) kills the record's bare-phi0 reading; exclusive |Vcb| ~0.0395 kills both; JUNO-era reactors + Belle II decide.

Recorded status: Numerical [N]/[C]/[O] (candidate pattern; record of v84 untouched)

Original ledger status

FLAV.DM2RATIO.01

FLAV.DM2RATIO.01

SPLITTING RATIO = LEPTONIC JARLSKOG (2026-07-02, v468): the neutrino mass-squared splitting ratio equals the Jarlskog invariant of the FROZEN PMNS matrix -- dm21^2/dm31^2 = |J_PMNS| = 0.029653 vs NuFIT 6.0 NO measured 0.029805 +- 0.000792 -> -0.19 sigma; equivalently m2/m3 = sqrt|J| = 0.17220 vs 0.17264(229). A parameter-free CANDIDATE relation for a ratio the compiler does not otherwise predict: |J| is the canonical derived CP invariant of the already-frozen matrix (angles v9/v268 + delta = 4pi/3, assembled v270; zero dials). It SUPERSEDES the informal tfpt_2 table heuristic m2/m3 ~ pi phi0 = 0.16704 (a '~' entry, never frozen), which is now at -2.44 sigma vs NuFIT 6.0. Spectrum consistency: with r = |J| the NO floor Sigma m_nu = m3(1+sqrt|J|) = 0.0588 eV reproduces the documented 0.0586 eV (v272); the absolute scale stays the ONE seesaw ratio (no absolute number added). HONEST: post-hoc; as a free formula the window density is ~1.2e-2 (unremarkable alone) -- the content is structural (canonical invariant, zero dials); NO mechanism (the mu-tau texture gives theta13 = 0 at leading order and cannot carry this relation) -- [O] until an M_nu-level derivation exists. Pre-registered kill: JUNO shrinks the r-window ~4x; a stable offset from 0.029653 at >3 sigma kills the relation; the v467 pattern would shift |J| to 0.028849 (-1.21 sigma today), so the two candidates cross-discriminate at JUNO precision.

Recorded status: Numerical [N]/[C]/[O] (candidate relation; informal pi phi0 table heuristic superseded as comparator)

Original ledger status

SEAM.EQUIV.CROSSEDPRODUCT.01

SEAM.EQUIV.CROSSEDPRODUCT.01

The NET-LEVEL CROSSED-PRODUCT certification of the 128-spinor extension + the R1 -> R1' invariant-level reduction (2026-07-02, closure-route round) -- re-founds BOTH halves of the SEAM.EQUIV.01 residual on harder-to-reject ground; does NOT close it. [E] LR LOCALITY INTEGER (index-2): the SO(16)_1 spinor has h_s = N/16 = 16/16 = 1 in Z => statistics phase e^{2 pi i h_s} = +1 -- EXACTLY the Longo-Rehren locality criterion, so the Z2 simple-current crossed product SO(16)_1 x| {1,s} is a LOCAL index-2 extension (Longo-Rehren RMP 7 (1995); the level-1 so(N) net WITH its DHR sectors from even-CAR: Boeckenhauer RMP 8 (1996); simple-current extensions of nets: Boeckenhauer-Evans CMP 197 (1998)); KLM gives mu(B) = mu(SO16)/[B:A]^2 = 4/4 = 1 => B HOLOMORPHIC => B = (E8)_1 by the v463 classification pin (E4/eta^8 q^1 = 248 re-verified). THE 2024/2025-PREPRINT LEG (AGT arXiv:2506.01008 + AMT arXiv:2407.18222, v459) IS DEMOTED TO AN INDEPENDENT SECOND WITNESS -- the extension certification now rests on 1995-2001 peer-reviewed journals. [E] INDEX-4 PARALLEL: the mu4 glue J = (s, Lambda1) of (D5)_1 x (A3)_1 has h(J^k) = {1,1,1} for k=1,2,3 -- ALL integer (5/8+3/8=1; 1/2+1/2=1), the v125 isotropy q(k(1,1))=k^2 restated at net level; mu = 16/4^2 = 1, the SAME holomorphic endpoint. [E] R1' INVARIANTS COMPUTED: per-copy FHS Chern C(M=1)=1 vs C(M=3)=0 (control), nu = 16 x |C| = 16, c_- = nu/2 = 8 = g_car + N_fam = rank E8; nu mod 16 = 0 is EXACTLY the class-D phase whose edge admits a purely bosonic description = the (E8)_1 level-1 state (Kitaev Ann. Phys. 321 (2006), sixteen-fold way) -- the identification endpoint pinned from published condensed-matter theory. [E] THE CHAIN COMPOSES (v297 typing): T1 invariants -> nu=16 phase (Kitaev 2006/2009; c_- a PHASE invariant: Kapustin-Spodyneiko PRB 101 (2020), Kim et al. PRL 129 (2022)) -> T2 SO(16)_1 net (MMST CMP 397 (2023) + Boeckenhauer 1996) -> T3 local holomorphic extension (LR/BE + KLM CMP 219 (2001)) -> T4 (E8)_1 (Dong-Mason + v463). [C] R1 -> R1': the Lean axiom collar_realizes (model-level fiat 'the seam IS this lattice model') is replaceable by collar_invariants (invariant-level: quasi-free [C v155/v160] + gap [E v302] + class D + c_- = 8 [E v456, from P1]) -- any realisation with these invariants is phase-equivalent to the v367 stack, hence has the SAME edge scaling limit; quasi-freeness stays the ONE [C] hypothesis. [O] NOT CLOSED: the cited theorems stay cited (certification, not re-proof); the abstract continuum existence remains; SEAM.EQUIV.01 stays [O].

Recorded status: Net-level crossed-product certification + R1 invariant reduction [E]/[C]/[O]: h_s = 16/16 = 1 in Z => the LR locality criterion holds, the Z2 crossed product SO(16)_1 -> (E8)_1 is a LOCAL index-2 extension by 1995-2001 PEER-REVIEWED subfactor theory (LR/Boeckenhauer/BE/KLM), AGT/AMT demoted to second witness; mu4 glue h(J^k)={1,1,1} parallel; R1 reduced to computed invariants (FHS |C|=1, nu=16, c_-=8, the Kitaev-E8 16-fold-way class); SEAM.EQUIV.01 stays [O]

Original ledger status

ALPHA.QUILLEN.INFLOW.01

ALPHA.QUILLEN.INFLOW.01

THE ALPHA^3 LEVEL = THE COMPUTED BULK CHERN INVARIANT + THE SEAM F-NORMALISATION = THE AFFINE EMBEDDING INDEX (2026-07-02, closure-route round) -- upgrades BOTH residuals left after v433/v434/v435; does NOT close ALPHA.QUILLEN.EXACT.01. [E] TARGET RE-VERIFIED: alpha^-1 = 137.0359992168 unique root; Quillen split a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at the root < 1e-35 (v341/v382). [E] THE INVARIANT COMPUTED: FHS Chern C(M=1) = 1, C(M=3) = 0 -- the same integers as v367/v461, here read as the quantised U(1) response of the collar phase. [C] k0 = |C| = 1 UNDER THE EM1 READING: the a^3 coefficient EQUALS the computed bulk invariant of the SAME p+ip model that realises S3 -- 'why exactly 1' is answered by COMPUTATION plus the cited inflow identification (TKNN PRL 49 (1982) / Avron-Seiler-Simon PRL 51 (1983): integer quantisation of the U(1) response; Callan-Harvey NPB 250 (1985): boundary level = bulk response; APS 1975 / Alvarez-Gaume-Della Pietra-Moore Ann. Phys. 163 (1985) / Witten RMP 88 (2016): the metric-independent part of delta log det is the eta/CS level; Quillen 1985 / Bismut-Freed CMP 106 (1986): determinant-line curvature integrality) -- REPLACING v435's minimality assumption ('1 is the unit level') by a measurement on the model. [E] k_Y = 5/3 EXACT: tr_5bar(Y^2)/tr(T3^2) = (5/6)/(1/2) = 5/3 (Ginsparg PLB 197 (1987), the string-GUT hypercharge level); (3/5) x (41/6) = 41/10 = b1 -- the 'GUT 3/5' in the closure IS the inverse embedding index, not an imported convention; carrier decomposition 41/6 = 20/3 (3 generations x 16-content) + 1/6 (Higgs doublet). [C] F-NORMALISATION RETYPED: once the seam net is (E8)_1 at level 1 (SEAM.EQUIV.01), the U(1)_Y current sits inside a level-1 current algebra and <J(z)J(w)> = k/(z-w)^2 is FIXED by the embedding index -- level-1 current-algebra rigidity, no continuous dial; the v434 [C] ('how does the gauge curvature couple to the boundary measure') has ZERO independent content: it is a face of SEAM.EQUIV.01 (consistent with the v382 typing), its value the embedding index. [E] THE STRUCTURAL UNIFICATION: the two named targets share ONE object -- the collar phase carries TWO quantised responses, gravitational c_- = 16 x 1/2 x |C| = 8 = g_car + N_fam (feeds SEAM.EQUIV.01/S3, v456) and U(1) C = 1 (feeds the a^3 level, EM1); 'the one theorem and the one number' are two faces of the SAME invertible phase, and closing R1' (v469) feeds both residuals simultaneously. [O] NOT CLOSED: the bridge lemma 'delta log det_zeta(seam) = the inflow response' stays the named cited step; the EM1 reading stays [C]; ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 = 137.0359992 stays [E] regardless. UPDATED 2026-08-27 (v974): the inflow step 'boundary level = bulk response' is upgraded from cited to COMPUTED on the same model (probe laughlin_pump_em1_probe 20/20, PUMP_QUANTIZED_MATCHES_CHERN): Laughlin pump +1 per flux quantum (teleport bookkeeping with exact period closure), edge spectral flow -1/+1 antisymmetric as inflow demands, Streda dN/dPhi = C/(2 pi) exact integers over n_phi = 0..3 -- all equal to the FHS Chern, with the M = 3 controls all zero. The identification of the complex-copy U(1) with the physical Majorana collar stays part of the cited EM1 reading [C]; ALPHA.QUILLEN.EXACT.01 stays [O].

Recorded status: Inflow level + embedding rigidity [E]/[C]/[O]: the a^3 coefficient = the COMPUTED bulk Chern |C| = 1 of the v367 collar (TKNN/ASS quantisation + Callan-Harvey inflow + APS/Witten eta=CS), replacing v435's minimality; the seam F-normalisation = the affine embedding index k_Y = 5/3 (Ginsparg 1987, (3/5)(41/6) = 41/10 = b1) -- zero independent content, a face of SEAM.EQUIV.01; one phase, two responses (c_- = 8, C = 1) feeding both named targets; ALPHA.QUILLEN.EXACT.01 stays [O]

Original ledger status

SEAM.EHMODEL.04

SEAM.EHMODEL.04

SEAM-HORIZON REPLICA CHAIN AT COLLAR LEVEL (the kernel-identification premise of SEAM.THEOREM.01, exercised numerically; SEAM.EHMODEL.03's residual list sharpens again, the gate stays OPEN): (1) REPLICA => EH FORM ON A REAL OPERATOR [I] - matched conical deficit G(gamma,m) = log det L(gamma,m) - (gamma/2pi) log det L(2pi,m) of the discretized gapped cone operator (finite-volume polar grid, geometric tip grading, identical stencil across geometries so extensive terms cancel exactly) is linear in ln m with slope 2C(gamma), verified on deficit cones gamma/2pi = 1/4, 1/2, 3/4 (rel. err 0.01-0.05%) AND on REAL replica sheets gamma/2pi = 2 (0.8%), 3 (1.5%) - beyond linearisation, the regime the replica trick uses; (2) COEFFICIENT FORCED, SCALE = ANCHOR [I] - the EH slope is stable under IR/size refinement (0.5% drift) while the intercept (zeta-scale mu_lat 146.9 -> 153.1) drifts with the discretization: the v152 anchor exhibited on data; NO canonical ln(m/mu) = 3/4 emerges (recorded honestly - the anchor stays declared, not derived); (3) BFK SPLIT MEASURED ON THE DISCRETIZED CALDERON KERNEL [I] - Schur identity log det L = log det(halves_D) + log det S exact (~1e-16); across three cut geometries the Calderon/Schur factor's deficit slope follows a PURE CUT-EDGE law E(1-q) with tip term 0 (power check: the same law fails on the full determinant), the halves carry the whole tip 2C_cone via Kac doubling (exact) - v151's 'conically clean' measured on the kernel itself, not inferred from corner constants; (4) THE SEAM'S OWN SPECTRUM [I] - the v302 transfer masses m2 = 6 ln(3/2), m3 = 6 ln 3 (bend ln3/ln(3/2), v124/v147) sit ON the calibrated EH line (<=0.05%): det'(collar) inherits the EH variation mode-wise; the Perron/attractor mode (m=0) is IR-DIVERGENT, running exactly as -2C ln(IR ratio) - THE RECOVERY GAP Delta = 6 ln(3/2) IS WHAT MAKES THE INDUCED-GRAVITY COEFFICIENT FINITE (the same gap that makes the attractor unique, v56, makes 1/G finite). RESIDUAL [A/O]: the kernel premise is now discharged at the finite/model level; what remains of SEAM.THEOREM.01 is the CONTINUUM scaling limit of this statement (the MMST class - the same single residual as SEAM.EQUIV.01, v336) plus the one dimensionful anchor (v152). NOT claimed: a continuum proof, or a derivation of ln(m/mu) = 3/4

Recorded status: Numerical [I] (lattice determinants vs exact conical constants) + [A/O] (continuum leg + anchor stay open)

Original ledger status

ALPHA.QUILLEN.DETLINE.01

ALPHA.QUILLEN.DETLINE.01

THE OCCUPIED BERRY LINE OVER THE U(1)-TWIST MODULI HAS CHERN NUMBER = THE BLOCH RESPONSE LEVEL (2026-07-03) -- the finite Berry-Chern response, COMPUTED at the collar model; the Quillen/Dai-Freed comparison remains a separate target; does NOT close ALPHA.QUILLEN.EXACT.01. [E] BLOCH LEVEL RE-VERIFIED: BZ Chern C(M=1) = 1, C(M=3) = 0, C(M=-1) = -1 (the v470/v367 integers). [E] THE OCCUPIED BERRY LINE COMPUTED: the same collar h(k) = sin kx SX + sin ky SY + (M - cos kx - cos ky) SZ in real space on an L x L torus with twisted boundary conditions -- the twist torus (theta_x, theta_y) IS the moduli space of flat U(1) connections -- and the Fukui-Hatsugai-Suzuki curvature of the DETERMINANT LINE of the occupied one-particle frame (the free-fermion many-body ground-state LINE; its normalized vectors are chosen locally) over that torus integrates to 2 pi x 1 exactly (integer to 1e-9) at M = 1, for L = 4 AND L = 6 -- size-independent, as an integer must be. [E] CONTROLS: trivial collar M = 3 gives 0; orientation flip M = -1 gives -1 -- the curvature tracks the phase, not the mesh. [E] TWO TORI, ONE LEVEL: the twist-moduli integer equals the Bloch-BZ integer for all three M -- the Niu-Thouless-Wu identification (PRB 31 (1985)), exhibited on the collar model that realises S3 (v460/v461). [E] GAP: the sampled Fermi gap is open at M=1; analytically |d(k)|^2 = 1 + 2(1-cos kx)(1-cos ky) >= 1 gives the global gap >= 2. This defines a smooth occupied projector, not a global nonvanishing section of its topologically nontrivial line. No Dai-Freed filling or global-section test is performed. [C] INTEGRATED RESPONSE: k0 = |C_FHS| = 1 in the collar model. The FHS continuum convention C_Berry=integral F_occ/(2pi i), with nabla=d+A_occ and c1=iF_occ/(2pi), gives c1[B]=-C_Berry; finite-mesh identification and theta-line sign/duality must be checked separately. This integrated integer does not identify individual holonomies or the exact zeta-determinant variation. The open comparison requires a separately defined chiral Fredholm/Quillen line with its zero locus, a common gauge base, relative connection-level comparison and uniform ultraviolet controls; a Dai-Freed vector requires filling data and takes values in the pulled-back inverse boundary line (Quillen 1985; Bismut-Freed CMP 106 (1986); Dai-Freed JMP 35 (1994); FHS JPSJ 74 (2005)). [O] NOT CLOSED: the abstract-seam zeta-determinant identification (the continuum leg, = the SEAM.EQUIV.01 face); ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 = 137.0359992168 stays [E] regardless. Structurally the same manoeuvre as v471 for SEAM.THEOREM.01's replica side: the named open lemma exercised on a real finite operator.

Recorded status: Numerical [N] (finite Quillen shadow; gate unchanged)

Original ledger status

GRAV.ENTROPIC.ACTION.01

GRAV.ENTROPIC.ACTION.01

THE ENTROPIC-ACTION BRIDGE (2026-07-13) -- Bianconi's 'Gravity from entropy' (Phys. Rev. D 111, 066001 (2025); arXiv:2408.14391; DOI 10.1103/PhysRevD.111.066001) quantified as an EXTERNAL CANDIDATE for the local volume action behind TFPT's parameter-free Einstein equation; closes NOTHING, adds ZERO knobs. Context: v359 derives G_ab + Lambda g_ab = (1/c3) T_ab thermodynamically and is honestly typed 'equation of state, not a from-action quantisation' [O]; Bianconi's S_B = -Tr_F ln(G~ g~^-1) (quantum relative entropy between the spacetime metric and the matter-induced metric; Araki-related via Delta^{1/2} = G~ g~^-1, her Appendix C) is a local covariant action whose variation yields Einstein + emergent nonnegative Lambda at low coupling -- exactly the missing variational layer IF the bridge identities hold. [E] CARRIER HODGE COUNT: her Omega^{0,1,2} matter fiber, placed on the five-slot carrier C^5 (NOT on 4d spacetime, where the fiber is 1+4+6 = 11), counts 1+5+10 = 16 = 2^(g_car-1) = dim S+_{D5}, Hodge-star folded (L^1 ~= L^4) onto L^even C^5 -- the carrier half-spinor's own Pascal row (v2/v44); vector-space identity ONLY. [E] COEFFICIENT MATCH (the one quantitative pin): her weak-coupling Lagrangian L = 3 beta R - alpha L_matter (her Eq. (45); the 3 = the three form blocks) matched to the TFPT EH normalisation 1/(16 pi G) = c3/(2G) fixes her free constant EXACTLY: beta'_B = c3/6 = 1/(48 pi); the 6 = 3 blocks x 2 (EH 1/2) = p2 = |R+(A3)| is recorded as a structural echo, NOT a proof. [E] ENTROPY POTENTIAL = QUADRATIC LAMBDA: Lambda_G = (1/2 beta) Tr_F(G - I - ln G) with s(x) = x - 1 - ln x >= 0, s''(1) = 1 (exact series) -- Lambda nonnegative and QUADRATIC around the entropic fixed point; with the TFPT displacement eps_* = e^{-1/alpha} Q this reproduces the closed v60 branch rho_Lambda/Mbar^4 = (3/4 pi^2) e^{-2/alpha}, and the prefactor match (beta' = c3/6, l_P^2 Mbar^2 = c3) yields the EXACT normalisation target Tr_F Q^2 = 32 c3^4 = 1/(128 pi^4) = (2/3) delta_top -- a closed algebraic target for the two-form (pair) block (binom(5,2) = 10), [C] until an explicit Q is constructed. [E] FRULLANI: -ln A = int_0^inf dt/t (e^{-tA} - e^{-t}) -- her log action is an all-scale-integrated heat-kernel action, the correct genre to compare with the TFPT relative spectral action S_rel,chi (KMS cutoff). [E] R^2 KILL TEST QUANTIFIED (pre-registered): the raw log expansion carries a curvature-squared coefficient of order (c3/6)^2; the TFPT Starobinsky sector needs Mbar^2/(12 M_scal^2) = 1/(12 c3^7) (M_scal^2 = c3^7 Mbar^2, v36); the EXACT mismatch is 3 c3^{-9} = 3 (8 pi)^9 ~ 1.20e13 -- THIRTEEN orders: a direct identification of the raw entropic action with the TFPT R^2 sector is FALSE without a mechanism (scalaron = light trace mode of her G-field generating c3^{-9}, or KMS-spectral renormalisation of the log action). [C] COMPRESSION CONJECTURE (the bridge proper, recorded not proven): P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} -- the physical Calderon compression of her relative metric operator equals the TFPT seam modular operator; if proven, S_B is a local volume representation of the TFPT relative spectral action; open together with the operator-level Hodge items (Clifford action, Dirac intertwining, leaf involution, Calderon polarisation). Lorentzian positivity of G~ g~^-1 is NOT automatic (timelike gradients); TFPT's OS/reflection-positive route is the sturdier construction. [O] NOT CLOSED: the v359 equation-of-state status stays [O]; adopting S_B would close it ONLY after the compression conjecture AND an R^2 mechanism; no status marker moves anywhere

Recorded status: Formal/Numerical [E] (five exact identities + quantified gap) + [C] (compression conjecture, Q-target) + [O] (variational completion open)

Original ledger status

GRAV.ENTROPIC.HODGE.01

GRAV.ENTROPIC.HODGE.01

THE OPERATOR LEVEL OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; work packages 1 + 4-algebra of GRAV.ENTROPIC.ACTION.01/v473) -- upgrades the v473 vector-space identity to the operator level on the finite carrier Fock space and DECIDES the Q-block conjecture by exact enumeration; closes NO gate. [E] CAR => CLIFFORD: on Lambda^* C^5 (dim 2^g_car = 32) the ten Gamma_{2k} = a_k^dag + a_k, Gamma_{2k+1} = i(a_k^dag - a_k) satisfy {Gamma_a,Gamma_b} = 2 delta_ab exactly (all 100 pairs, integer matrices) -- the so(10) = D5 Clifford algebra acts on the carrier exterior algebra. [E] S+ = LAMBDA^EVEN AS D5-MODULE: the 45 bilinears [Gamma_a,Gamma_b]/4 commute with gamma = (-1)^N, span exactly 45 = dim so(10), and leak nothing from the 16-dim even subspace -- the carrier half-spinor IS the even exterior algebra WITH its spinor structure (the operator level of v2/v44/v197). [E] HODGE-DIRAC SYMBOL = CLIFFORD MULTIPLICATION: c(xi) = a^dag(xi) - a(xi), the principal symbol of Bianconi's D = d + delta, satisfies c(xi)^2 = -|xi|^2 Id exactly for symbolic real xi. [E] THE FOLD IS THE 5 -> 5bar CONJUGATION: the traceless u(5) weights of Lambda^4 are exactly the negatives of those of Lambda^1, so the Hodge fold *: Lambda^1 ~ Lambda^4 conjugates the vector block -- Bianconi's fiber 1+5+10 becomes the GUT half-spinor 16 = 1 + 5bar + 10, the SM carrier decomposition (not a mere dimension match). [E] HONEST OBSTRUCTION: Omega^{<=2} is NOT Clifford-invariant (c(xi) maps Lambda^2 into Lambda^3), while Lambda^even is invariant under the even Clifford algebra -- the bridge identification must go through the Hodge fold, never through naive degree truncation (a recorded constraint on the AP2 compression). [E] Q-TARGET ENUMERATION (the v473 [C] conjecture DECIDED at the algebraic level): the uniform-block ansatz Q = k c3^2 P_d solves Tr Q^2 = d k^2 c3^4 = 32 c3^4 with integer k for EXACTLY three supports (d,k) = (2,4), (8,2), (32,1) -- the sheet pair |Z2|, the rank g_car + N_fam = rank E8 = 8, and the full Fock space 2^g_car = 32 = dim Lambda^* C^5 (minimal uniform solution: eps_* = e^{-1/alpha} c3^2 per Fock mode, q = c3^2 = 1/(64 pi^2)); the two-form block d = 10 (the problem-note 'pair sector' conjecture) and the fiber d = 16 require irrational multipliers sqrt(16/5) and sqrt(2) -- the naive pair-block reading is KILLED; bonus identity Tr Q^2 / delta_top = 32/48 = 2/3 = dim Lambda^*/Omega_adm (the 2/3 in the v473 target is a mode-count ratio). [C] the physical choice among the three admissible supports (uniform-32 is minimal, not derived). [O] UNCHANGED: the AP2 compression P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} and the R^2 mechanism (v473 pre-registered kill test) stay open; no status marker moves anywhere

Recorded status: Formal/Identity [E] (operator-level Clifford/weights/enumeration, exact) + [C] (physical Q choice) + [O] (AP2 + R^2 unchanged)

Original ledger status

GRAV.ENTROPIC.SCALARON.01

GRAV.ENTROPIC.SCALARON.01

THE R^2 KILL TEST OF THE ENTROPIC-ACTION BRIDGE, EXECUTED (2026-07-13; work package 5 of GRAV.ENTROPIC.ACTION.01/v473) + THE EXPLICIT LORENTZIAN-POSITIVITY WITNESS -- decides v473's interpretation 2 negatively in its naive form; closes NO gate. [E] MAX-SYMMETRIC EIGENVALUES (the Appendix-B flattening of Bianconi PRD 111, 066001 (2025) verified): with R_munurhosigma = (R/12)(gg - gg) on a general diagonal 4d metric the relative curvature operator R~ g~^-1 has eigenvalues EXACTLY {R (scalar); R/4 x4 (Ricci); R/6 x6 (Riemann on flattened 2-forms)}, computed symbolically including the 6x6 two-form flattening. [E] EXACT VACUUM f(R): L_B(R) = -Tr_F ln(I - beta R~ g~^-1) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] with exact series 3 beta R + (17/24) beta^2 R^2 + O(R^3) -- the linear 3 re-derives her Eq. (45) (= the v473 coefficient pin) and 17/24 = (1/2)(1 + 1/4 + 1/6) is the EXACT tensorial R^2 factor (v473 quoted the raw (c3/6)^2 'before tensorial factors'). [E] THE RAW ENTROPIC SCALARON IS TRANS-PLANCKIAN: with the pinned beta' = c3/6 the EH normalisation matches identically and BOTH routes (Starobinsky matching and f'/(3 f'')) give m_raw^2 = 4608 pi^2 Mbar^2/17, i.e. m_raw = pi sqrt(4608/17) Mbar ~ 51.7 Mbar ~ 1.3e20 GeV -- HEAVY, not light; the TFPT scalaron is c3^{7/2} Mbar = 3.06e13 GeV, a factor ~4.1e6 below; the mechanism interpretation 2 requires must supply EXACTLY m_raw^2/m_TFPT^2 = (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 -- the kill-test number now WITH tensorial factors. [E] DOMAIN: positivity of I - beta R~ g~^-1 holds iff R < 1/beta = 384 pi^2 Mbar^2 (first log pole; then 4/beta, 6/beta) -- a trans-Planckian curvature bound, so the raw action is well-defined at all sub-Planckian curvatures (the obstruction is the mass, not the domain). [E] LORENTZIAN-POSITIVITY WITNESS (the v473 [C] caveat made explicit): in her warm-up sector G = g + alpha dphi x dphi on Minkowski, a TIMELIKE gradient gives G g^-1 eigenvalues {1 - alpha v^2, 1,1,1} -- nonpositive for v^2 >= 1/alpha, so ln(G g^-1) is UNDEFINED on an open field region; spacelike and Euclidean stay positive for all v -- Lorentzian positivity is a genuine restriction, TFPT's OS/reflection-positive route is the sturdier construction. [C] VERDICT: of v473's three interpretations, (1) SURVIVES (the entropic action covers the Einstein + Lambda level: beta' pinned v473, Lambda channel closed v473/v474), (2) is DEAD in naive form (this module), (3) KMS-spectral renormalisation of the log action remains the ONLY route to the R^2 sector. [O] UNCHANGED: the AP2 compression and the R^2 gate itself stay open; no status marker moves anywhere

Recorded status: Formal/Identity [E] (exact tensorial factors, scalaron mass, positivity witness) + [C] (verdict/surviving route) + [O] (AP2 + R^2 gate unchanged)

Original ledger status

GRAV.ENTROPIC.COMPRESS.01

GRAV.ENTROPIC.COMPRESS.01

THE AP2 COMPRESSION CONJECTURE MADE WELL-POSED (2026-07-13; the last open work package of the v473 entropic-action-bridge round, exercised on an exactly solvable model) -- decides the READING of the conjecture, does NOT prove it; AP2 stays [O]. Background: v473 recorded AP2 as P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2}. On a quasi-free system everything is computable: the boundary restriction is the covariance compression C_A = PCP (v155), the boundary modular data is Peschel's K = ln((1-C_A)/C_A), and for a thermal bulk the Bianconi-shaped bulk identity is exact: f(C_T) = C_T(1-C_T)^{-1} = e^{-h/T}. [N] PURE GAPPED BULK: staggered-mass chain (N=16), ground-state covariance a projection to 1e-15, gap verified -- pure bulk, thermal boundary (Bisognano-Wichmann shape). [N] COMPRESSION IS THERMAL: spec(PCP) strictly inside (0,1), Peschel K finite, entanglement S > 0. [N] THE ORDER OBSTRUCTION (the reading DECIDED): on the pure bulk spec(C) = {0,1}, so f(x) = x/(1-x) is singular on the spectrum -- the literal operator-side reading P f(C) P is ILL-POSED, while the state-side f(PCP) is finite: the compression identity can only be read STATE-SIDE (build Delta_Sigma FROM the compressed relative metric), which matches Bianconi's own construction (her G~ is a LOCAL field, i.e. already the compressed datum) -- AP2 is retyped, not weakened. [E] BLOCK ALGEBRA (symbolic, exact): [C^2]_AA - (C_AA)^2 = B Bt and [C^3]_AA - (C_AA)^3 = A B Bt + B Bt A + B D Bt -- every term carries the cross block TWICE, first order vanishes identically: for ANY analytic f the two readings agree up to SECOND order in the cross-cut correlations; the modular (entanglement) content lives exactly in that quadratic correction. [N] GAP CONTROL at the bounded (covariance) level: the mismatch d = ||[Fermi(h/T)]_AA - Fermi(h_AA/T)|| obeys the quadratic law d <= x^2 (d/x^2 in [0.4, 1.0]) and falls with the gap in the gapped regime while the cross-cut norm falls monotonically; the decoupled cut ([h,P] = 0) agrees EXACTLY -- the two readings converge in the gap-dominated regime, which is where TFPT operates (Delta_eff > 0, v76/v337). [C] AP2 well-posed: state-side reading at the bounded level, mismatch quadratic in cross-cut correlations, gap-suppressed. [O] NOT proven: the continuum seam statement (the Calderon compression of the actual relative metric operator = the actual seam modular operator); no status marker moves anywhere

Recorded status: Numerical [N] (lattice exhibits) + Formal [E] (block identity) + [C] (retyped conjecture) + [O] (AP2 continuum statement open)

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

Original ledger status

GRAV.ENTROPIC.SCALEFLOW.01

GRAV.ENTROPIC.SCALEFLOW.01

THE SCALE-FLOW REPRESENTATION OF THE ENTROPIC ACTION (2026-07-13; the follow-up to GRAV.ENTROPIC.SCALARON.01/v475) -- the surviving interpretation 3 (KMS-spectral renormalisation) typed PRECISELY: the (72/17)(8 pi)^9 kill-test gap collapses to ONE moment condition on the scale weight, and that condition is satisfied by the SAME KMS moment TFPT already fixes (v36's f0); zero new dials; closes NOTHING. [E] SCALE-FLOW IDENTITY: -ln a = int_0^inf dt/t (e^{-ta} - e^{-t}) (symbolic exact) = 2 int_0^inf dchi/chi (e^{-a/chi^2} - e^{-1/chi^2}) (t = 1/chi^2; verified to 1e-20) -- Bianconi's log action IS the FLAT scale-integral of relative heat-kernel actions, while TFPT's S_rel,chi evaluates ONE KMS scale with a weight f: the bridge between the two action principles is a choice of SCALE MEASURE. [E] MOMENT DICTIONARY: with a general weight w(t) dt/t the maximally-symmetric expansion (v475 eigenvalues {1; 1/4 x4; 1/6 x6}) gives S_w = 3 beta R mu1 + (17/24) beta^2 R^2 mu2 + O(R^3), mu1 = int w e^{-t} dt, mu2 = int w t e^{-t} dt; the flat Frullani weight has mu1 = mu2 = 1 and reproduces the v475 raw coefficients identically -- the raw action is the unit-moment point of a moment family. [E] THE ONE MOMENT CONDITION: the weight-dressed EH pin beta' mu1 = c3/6 gives m^2 = (4608 pi^2/17)(mu1^2/mu2) Mbar^2; demanding m^2 = c3^7 Mbar^2 forces EXACTLY mu2/mu1^2 = (72/17)(8 pi)^9 -- the v475 kill-test number IS a moment ratio -- and the closure identity (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 holds IDENTICALLY: the '13 orders of magnitude' are not a mismatch of the bridge but the statement that the R^2 coefficient is a SCALE-MEASURE datum which the flat measure fixes wrongly and the TFPT KMS measure fixes correctly. [E] v36 CONSISTENCY: the same physics as f0 = 6 (4 pi)^2 / c3^7 <=> M^2/Mbar^2 = c3^7 -- the moment-condition route and the spectral-action route land on the SAME scalaron mass: one KMS moment, two equivalent parametrisations, zero new dials. [C] HONEST TYPING: a consistency statement, NOT a derivation -- the KMS weight (hence mu2/mu1^2) is TFPT input; the entropic action supplies the Einstein + Lambda level and the scale-flow FORM, TFPT supplies the measure; interpretation 3 stays [C]. NEGATIVE SIDE-RECORD: a first attempt to tie the AP2 state-side reading (v476) to an EXACT lattice Bisognano-Wichmann commutator (Peschel-Eisler simple tridiagonal forms) did NOT hold on the half-filled open chain -- kept out of the suite, logged in the research notes. [O] UNCHANGED: the R^2 gate and v359's equation-of-state typing; no status marker moves anywhere

Recorded status: Formal/Identity [E] (scale-flow + moment dictionary + closure identity, exact) + [C] (measure is TFPT input) + [O] (R^2 gate unchanged)

Original ledger status

GRAV.ENTROPIC.LEGS.01

GRAV.ENTROPIC.LEGS.01

FIRST COMPUTABLE STEPS ON THE TWO REMAINING LEGS OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; after v473-v477 the open items were the AP2 continuum statement and the derivation of the KMS scale measure) -- both legs stay [O]; closes NOTHING. LEG A (AP2 CONTINUUM, exhibited on the exactly solvable critical chain, infinite-volume sinc kernel, 50-digit precision): [N] Calabrese-Cardy -- the compressed interval state has S(L) = (c/3) ln L + k with c_est = 1.00089, 1.00038, 1.00021 (pairs (8,16), (12,24), (16,32); monotone convergence to the Dirac c = 1 = 2 Majoranas x 1/2, consistent with the v450 reading), while a gapped control gives c_est ~ 6e-8 (area law) -- the compressed state carries the conformal/BW structure quantitatively. [N] CHM PARABOLA -- the nearest-neighbour couplings of the Peschel modular Hamiltonian K = ln((1-C_A)/C_A) organise on the CHM weight beta(x) = x(L-x)/L with correlation 0.866 -> 0.967 -> 0.986 (L = 8, 16, 24) and shrinking mid-plateau drift; constants deliberately NOT asserted (the lattice EH is only asymptotically local). [N] QUASI-LOCALITY -- at half filling the even-range bands of K vanish identically (particle-hole structure, max |K_{i,i+2}| ~ 1e-36) and the odd tail is small (mean ratio r=3/r=1 = 1.8%) with parabolic envelope. READING [C]: the state-side Delta_Sigma (the reading FORCED by v476) flows to the CHM/Bisognano-Wichmann geometric form in the continuum limit -- exactly the object TFPT's Einstein derivation consumes (the CHM ball modular Hamiltonian, v323/v358): the bridge's modular side and TFPT's gravity side MEET. NOT a continuum proof. LEG B (THE MEASURE): [E] single-scale corollary -- for dmu = delta(t - t0) dt the moments give mu2/mu1^2 = e^{t0} EXACTLY, so the v477 one-moment condition forces t0 = ln(72/17) + 9 ln(8 pi) = 30.460996... -- ONE dimensionless KMS time with an exact closed form; DECLINED explicitly: t0 is NOT h(E8) = 30 (difference 0.461; the near-miss is recorded and rejected per the v354/v355 no-free-pattern rule); no TFPT atom equals t0 -- deriving the measure from the entropic structure stays OPEN. [O] both legs remain the named open research items; no status marker moves anywhere

Recorded status: Numerical [N] (high-precision lattice exhibits) + Formal [E] (single-scale corollary) + [C] (meeting-point reading) + [O] (both legs open)

Original ledger status

SEAM.KRONHEIMER.01

SEAM.KRONHEIMER.01

THE R3 COMBINATORIAL-TO-GEOMETRIC BRIDGE IS KRONHEIMER'S THEOREM (2026-07-14, bird's-eye round): SEAM.DETK.01/v344 named R3's residual 'the rewrite attractor (a graph) IS the du Val singularity' -- that bridge is Kronheimer's ALE hyper-Kaehler quotient construction + Torelli classification (J. Diff. Geom. 29 (1989) 665-683 and 685-697), and ALL of its finite input data are already TFPT compiler outputs. [E] MARKS = NULL = PERRON: C_aff delta = 0 and A delta = 2 delta on the affine-E8 McKay graph; multiset {1,2,2,3,3,4,4,5,6} = 2I irrep dims = the v312 rewrite-attractor Perron vector. [E] KRONHEIMER DIMENSION COUNT: dim_R M - 4 dim_R G = 4(sum_edges d_i d_j - sum d_i^2 + 1) = 4 -- the hyper-Kaehler quotient of the marks quiver is 4-real-dimensional (the C^2/Gamma geometry); sum_edges d_i d_j = sum d_i^2 = 120 forced by the null-vector identity. [E] GAUGE DATUM = |2I|: sum delta_i^2 = 120 = |mu4| h(E8) = 4*30, h(E8) = 30 = |Z2| N_fam g_car. [E] RESOLUTION DATA: 8 = rank(E8) exceptional spheres, intersection form -Cartan(E8) with det = 1 and SNF = identity (the Poincare-sphere link = the det K = 1 face, v232/v344); deformation space h (x) R^3, dim 24. [C] the theorems are CITED, not re-proved (same import class as NPW26/MMST/AGT, v424/v336/v459). [O] the residual premise TRANSFORMS, it does not vanish: 'the raw seam supplies ALE hyper-Kaehler data asymptotic to the (2,3,5) cone' = the SAME single arrow as SEAM.EQUIV.01 (v346 L2); the keystone's open content is now reachable from BOTH sides (net-side scaling limit v336/v458; geometry-side Kronheimer datum). No gate closes, no marker moves. Wolfram mirror added to the extension file (counted with the next verified engine run). LEAN: the finite input data are kernel-checked in TfptCarrier/KronheimerMarks.lean (marks = null/Perron vector over Z, sum delta^2 = 120 = 4*30 with 30 = 2*3*5, hyper-Kaehler dimension count = 4; lake build clean, #print axioms = the 3 standard kernel axioms only).

Recorded status: Reduction/citation [E]/[C]/[O] (graph->geometry bridge discharged to Kronheimer 1989; finite input data compiler-exact; premise transforms, stays open)

Original ledger status

QGEO.MULTILOCAL.01

QGEO.MULTILOCAL.01

THE FOUR mu4 MARKS ARE A FOUR-INTERVAL FREE-FERMION GEOMETRY (2026-07-14, bird's-eye round): the seam circle minus the four Gauss-Bonnet marks (v216) IS the n = 4 disjoint-interval region of a chiral free fermion -- the class whose multi-interval modular theory is EXACTLY KNOWN (Casini-Huerta 2009 mixing term; Longo-Xu multi-interval subfactors CMP 251 (2004); Rehren-Tedesco multilocal fermions LMP 104 (2014); KLM CMP 219 (2001): holomorphic <=> trivial multi-interval inclusion, mu = 1 -- the SAME one bit as det K = 1). Until now the suite used only the single-ball CHM boost (v358). On the exactly solvable antiperiodic critical ring (N = 256, Q = 64, ell = 48): [E-num] BINARY CLOCK: the quarter-turn clock is a signed permutation with rho^4 = -1 EXACTLY -- the fermionic mu4 clock is the order-8 double-cover lift (the same Z2-lift pattern as 2I over I). [E-num] EXACT SECTOR DECOUPLING: the primitive-8th-root character projectors resolve the identity and commute with the reduced covariance at 1e-15 (the Rehren-Tedesco diagonal 4-fold-cover isomorphism at covariance level). [E-num] COVER/TWIST: each sector kernel is the single-interval quarter-ring kernel with boundary twist lambda_k; sector spectra union = full spectrum at 1e-14. [E-num] CLOCK INVARIANCE MANIFEST: rho C rho^-1 = C at 1e-14 => [rho, K] = 0 by functional calculus -- omega o rho = omega HOLDS at the 4-interval state level; NEGATIVE CONTROL: displacing ONE interval by one site re-couples the sectors at 1e-2. [N] CONJUGATE-POINT MIXING: every cross-interval K row peaks on the conjugate diagonal |s'-s| <= 2 at both sizes -- the lattice shadow of the Casini-Huerta mixing term (band energy spreads with size: long-range conformal tails, honestly noted). [C] the continuum multi-interval theory is CITED, not re-derived. [O] the raw-seam premise (QGEO.SYM.01 / SEAM.EQUIV.01) stays OPEN -- this exhibits the mechanism in the exactly solvable class the seam is argued to reach, it does not prove the seam reaches it. Also the natural continuum frame for the AP2 leg (v476/v478). No gate closes, no marker moves.

Recorded status: Mechanism exhibit [E-num]/[N]/[C]/[O] (bedrock omega o rho = omega manifest at the 4-interval state level; multi-interval modular theory cited; raw-seam premise open)

Original ledger status

FLAV.NUSCALE.02

FLAV.NUSCALE.02

ABSOLUTE NEUTRINO SCALE UNDER THE CARRIER YUKAWA NORMALISATION (2026-07-14, bird's-eye round; CANDIDATE class, genre v467/v468): FLAV.NUSCALE.01 typed the absolute scale as one open UV input with the two-parameter trade-off (y_nu, M_R) and recorded the y_nu = 1 inversion at the structureless log_c3(Mbar/M_R) ~ 2.6. But y_nu = 1 is not the carrier normalisation: one SO(10) 16 per family + the MINIMAL Yukawa sector (10_H / PS (1,2,2), the v245/v248/v249 structure) forces y_nu = y_t at the matching scale [P named premise]. With explicit 1-loop RG (gauge/y_t/lambda up; ADKLR Weinberg-operator kappa down, hep-ph/0108005): [N] the trade-off COLLAPSES to M_R alone and the observed m_3 = 0.0503 eV demands M_R = 9.3e13 GeV -- INSIDE the compiler's own PS window [4.2e13, 2.4e15] (v249), at log_c3(Mbar/M_R) = 3.15. [X] INTEGER-RUNG HONESTY GATE: the rung M_R = c3^3 Mbar = 1.53e14 GeV predicts m_3 = 0.030 eV -- 40% LOW at 1-loop (kappa run-down factor 0.794): NEAR the rung (3.15 vs 3), NOT ON -- the ladder pin is DECLINED per v354/v355; the decision computation is NAMED (2-loop SM + explicit PS threshold matching). [N] FALSIFIABLE BAND: over the PS window m_3 spans [0.002, 0.115] eV bracketing the observation; the M_PS end (Sigma ~ 0.124 eV) is already in tension with DESI ~0.072 -- cosmology cuts the window from below; the NO floor Sigma = m_2 + m_3 = 0.059 eV stays the falsifiable edge (v272). [C]/[O] candidate class: the premise is the minimal Yukawa sector [P]; the absolute scale stays formally open until the decision computation lands; the frozen record (v84/REG.FREEZE.01) is untouched. NET: the absolute neutrino scale is now a ONE-parameter statement bracketed by the compiler's own UV window, not a two-parameter trade-off. UPDATE 2026-07-14 (v482/FLAV.NUSCALE.03): the named decision computation is EXECUTED -- the integer rung is EXCLUDED as an unstructured pin (bracketed envelopes reach x1.165 vs required x1.670); the 5/3-proximate structure escape is recorded and declined; the one-parameter window candidate stands.

Recorded status: Candidate [P]/[N]/[X]/[C]/[O] (carrier normalisation collapses the trade-off; window bracketing; integer rung declined at 1-loop; decision computation named)

Original ledger status

FLAV.NUSCALE.03

FLAV.NUSCALE.03

THE v481 DECISION COMPUTATION, EXECUTED (2026-07-14, deep-dive round; bracketed / assumption-robust form): the integer rung M_R = c3^3 Mbar is EXCLUDED as an UNSTRUCTURED exact pin, conditional on the carrier premise y_nu = y_t. [E-num] THE GAP: central m_3(rung) = 0.0301 eV vs observed 0.0503 -- rescue factor 1.670 (+67%, 0.154 c3-rungs). [E-num] BRACKETED EXCLUSION: with m_t(MSbar) +-3 GeV (>3 sigma), alpha_3^-1 -+0.10 (>3 sigma), +-10% on the ADKLR kappa run-down, and the PS-leg Yukawa beta scaled x[0.5,1.5] between M_PS and M_R, the largest single shift is x1.10 and the ALL-favourable combination only x1.165 -- far below x1.670: the v481 1-loop miss is NOT an RG artifact (no exact higher-loop coefficients trusted; pure envelope argument). [C] NAMED ESCAPE, RECORDED AND DECLINED: only a third-generation Majorana STRUCTURE factor r = M_R/M_R3 ~ 1.67 revives the rung; post-hoc r_needed = 1.6696 sits 0.18% from g_car/N_fam = 5/3 -- NO mechanism forces a 3/5 Clebsch on M_R3 today, so this is a named coincidence with a named missing mechanism (a PS Majorana-sector Clebsch), DECLINED per v354/v355 -- the same discipline as the t0 ~ 30 near-miss (v478). [O] the v481 one-parameter window candidate (M_R free in [M_PS, M_GUT], required M_R = 9.3e13 GeV at rung 3.15, NO floor Sigma m_nu = 0.059 eV) never used the integer rung and STANDS. Frozen record (v84) untouched; no gate closes, no marker moves.

Recorded status: Decision executed [E-num]/[C]/[X|P]/[O] (integer rung excluded as unstructured pin; 5/3 structure escape recorded and declined; window candidate stands)

Original ledger status

HYP.PHI0.GEOM.01

HYP.PHI0.GEOM.01

THE GEOMETRIC SIDE OF THE phi0 PUNCTURE TARGET IS EXACT (2026-07-14, deep-dive round; narrows HYP.PHI0.PUNCTURE.01/v408, does NOT close it): every twisted/orbifold heat trace of the flat tau = i pillowcase is a t-INDEPENDENT RATIONAL, so the pi^-4 in dtop = 48 c3^4 cannot come from flat orbifold geometry at ANY order -- the entire analytic content sits in the coupling normalisation of the contact insertions. [E-num] TWISTED TRACES EXACTLY 1: Tr(gamma e^{t Delta}) = 1 for gamma = sigma (z->-z), rho (z->iz), rho^3 at every tested t in [0.05, 3] (position-space Gaussian image sums / erf closed forms + 2d quadrature, residual < 1e-9; mode-space proof: gamma permutes Fourier modes, only n = 0 is diagonal) -- t-independent, no asymptotic tail. [E] ATIYAH-BOTT MATCH: 1 = 4 x 1/|det(1-dsigma)| with det = 4 = |mu4| (four Z2 marks) and 1 = 2 x 1/|det(1-drho)| with det = 2 = |Z2| (two order-4 points). [E] PILLOWCASE DATA EXACT: orbifold heat trace = (1/2) Tr_T2 + 1/2 -- the Z2 contact term is EXACTLY 1/2 = 4 x (1/8 per mark) for all t; the clock-equivariant pillowcase trace is exactly 1 for all t. [E]/[O] THE REDUCTION: dtop = 48 c3^4 = 3/(256 pi^4) decomposes as 4 = |mu4| contact insertions of per-mark weight c3 (giving c3^4) times the multiplicity Omega_adm = 48; the bare 1/(2 pi)-propagator 4-cycle differs by the exact RATIONAL 3/16; the ONE remaining [O] is the rule 'per-mark insertion weight = c3' (seam zeta/Green normalisation), the same residual class as the EM-Ward c3-ladder (EM.WARD.02). The target narrows from 'derive the term' to 'derive the weight'. Complements the experiments-side Donnelly probe (honest negative). No gate closes, no marker moves.

Recorded status: Reduction [E-num]/[E]/[O] (geometric side exact: t-independent rational twisted traces; value reduces to the one per-mark coupling rule)

Original ledger status

SEAM.CONTACT.UNIT.01

SEAM.CONTACT.UNIT.01

THE SEAM CONTACT UNIT -- THE MERGED RESIDUAL OF THE TWO ANALYTIC TARGETS (2026-07-14, deep-dive round II): the shared 'c3 per boundary insertion' rule of ALPHA.QUILLEN.EXACT.01 (c3-ladder {0,3,6}, v3/v341/EM.WARD.02) and HYP.PHI0.PUNCTURE.01 (per-mark weight c3, v483) is ONE identity and NOT a new unknown: it is the KMS seam unit 2 pi = 1/(4 c3) with 1/4 = 1/|mu4| (v239) -- one bare boundary propagator ORBIT-AVERAGED over the four marks. [E] UNIT CHAIN: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3. [E] MECHANISM DERIVED ON THE SEAM CIRCLE: G(d) = -(1/pi) ln|2 sin(d/2)| (polylog identity, 40 digits) takes INTEGER multiples of c3 ln2 at the mu4 separations (G(pi/2) = -4 c3 ln2, G(pi) = -8 c3 ln2), so G_marks = -4 c3 ln2 x circ(0,1,2,1) with spec {4,-2,0,-2}; the mu4-invariant (orbit-averaged) insertion carries (GV)|_marks = -(c3 ln2) C exactly -- the log-det contact expansion is EXACTLY graded in c3 per insertion, cycle combinatorics Tr(C^k) = 4^k + 2(-2)^k integer. [E] SUITE-WIDE GRAMMAR: F_U(1) ladder {0,3,6} (v3); phi0 ladder {1,4} (v37); Lambda prefactor 3/(4 pi^2) = (8 pi)^2 delta_top = 48 c3^2 (v60) -- the SAME multiplicity Omega_adm = 48 as the puncture at TWO insertions; the g-2 seam loop divides by the same 2 pi = 1/(4 c3). [C]/[O] THE MERGE: the two tracked [O] targets collapse into ONE named analytic step -- the diagonal (self-energy) zeta-renormalisation at the marks + the state-multiplicity matching (48 = Omega_adm = N_fam dim S+, 41 = 10 b1); the finite cycle sector is PROVEN. Anti-numerology: the check integers (circulant, spectrum, traces) are forced by the square mark configuration; no coefficient of either target is claimed. No gate closes, no marker moves. UPDATE 2026-07-14 (v485/SEAM.CONTACT.UNIT.02): the diagonal channel is settled -- G_reg(0) = 0 exactly at the KMS scale, the log is the pure scale channel, the mark determinant resums closed-form, and the 48/41 multiplicities are one state set under two response weights; the merged residual collapses onto the SEAM.EQUIV.01 face. UPDATED 2026-08-27 (v974): the finite cycle sector now has its computed Delta-side partner -- the BFK jump circulant B = circ(2,-1,0,-1) shares the mu4 Fourier eigenbasis with the mark circulant C = circ(0,1,2,1), the |spec| multisets are equal {0,2,2,4}, Tr B^k = 4^k + 2(+2)^k mirrors Tr C^k = 4^k + 2(-2)^k with the sign flip = the kernel swap (B kills the mu4-trivial gauge/zero-mode channel, C the mu4-sign channel), and B + C = 2(I + S). On the doublet channel R_0|_d = (4/ln2) G_marks|_d exactly, with the c3 grading CANCELLING in the ratio -- both matrices carry the same 'c3 per insertion' unit. No global proportionality: the merged residual is sharpened to the channel-swap question, not closed.

Recorded status: Unification/reduction [E]/[C]/[O] (the two analytic targets merge; contact unit = KMS seam unit, derived for the finite cycle sector; diagonal renormalisation + multiplicity matching stay open)

Original ledger status

SEAM.CONTACT.UNIT.02

SEAM.CONTACT.UNIT.02

THE DIAGONAL CHANNEL SETTLED -- THE MERGED TARGET COLLAPSES ONTO THE KEYSTONE (2026-07-14, deep-dive round III; completes the computable side of SEAM.CONTACT.UNIT.01/v484): [E] DIAGONAL ZERO AT THE KMS SCALE: the renormalised coincident-point Green function of |D| on a circle of circumference l (zero mode removed, flat short-distance subtraction) is G_reg(0; l) = (1/pi) ln(l/2pi) EXACTLY (symbolic limit + 30-digit numeric checks at l = pi, 2pi, 4pi) -- it vanishes iff l = 2 pi = 1/(4 c3), the v239 KMS seam circumference: the seam scale is the unique zero-self-energy scheme point. [E] THE LOG IS THE SCALE CHANNEL: G_reg(0; 2 pi kappa) = (1/pi) ln kappa -- matching the functional structure: F_U(1) carries exactly ONE log term (the transport 8 b1 c3^6 ln(1/phi_seam), v3) while the phi0 split (4/3)c3 + 48 c3^4 is log-free; [C] reading: the log term is the diagonal/scale channel, the log-free terms are contact data at the KMS point. [E] CLOSED-FORM RESUMMATION (the BFK/gluing route already in the suite, v151): for a finite-rank mark potential the zeta-det ratio reduces to the 4x4 mark determinant, and with the diagonal zero the v484 expansion RESUMS exactly: det(I - u C) = (1 - 4u)(1 + 2u)^2, u = eps c3 ln2, C = circ(0,1,2,1) -- closed form to ALL orders, the linear term absent precisely because the diagonal vanishes (Tr C = 0), series = v484's Tr(C^k) = 4^k + 2(-2)^k term by term. [E] WEIGHT DICHOTOMY (the multiplicity matching): the SAME 48 = 3 x 16 admissible states (v245) carry both multiplicities -- counted FLAT they give the phi0 puncture count 48 = Omega_adm = N_fam dim S+; counted with the U(1) response weights (Y^2 with the standard loop/GUT factors) they give the alpha budget 41 = 10 b1 = 40 fermionic (Tr_16 Y^2 = 10/3 exact, nu^c = 0) + 1 Higgs (Ginsparg (3/5)(41/6) = 41/10, v470) -- one state set, two response weights, no new number. [C]/[O] CONSEQUENCE: with v484 (unit + cycle sector) EVERY finite/computable piece of the merged ALPHA.QUILLEN.EXACT.01 + HYP.PHI0.PUNCTURE.01 target is proven; the single remaining [O] is the identification of the abstract seam zeta-det with this glued mark determinant -- a face of SEAM.EQUIV.01, exactly the v382 typing, now substantiated computationally. The merged residual collapses onto the keystone; no separate analytic unknown remains. No gate closes, no marker moves; no coefficient fitted (all integers forced by the square configuration and the SM charge table). UPDATED 2026-08-27 (v974): the BFK/gluing resummation named here is now EXECUTED (probe seam_bfk_conical_det_probe 21/21, promoted as v974): C_glue = 2^-N exactly (N = 1..4 symbolic, general interval lengths, Laurent-polynomial identity), massless reassembly det'_zeta(Delta_S1) = l^2 = 2^-N prod(2 a_j) (l/N) det'(R_0) with det'(R_0) = N l / prod a_j, jump matrix = inverse mark Green matrix R(m) = (G_m|_marks)^-1 exactly, and the diagonal/scale channel reproduced exactly through log det'Delta(l) - log det'Delta(1/(4 c3)) = (1/(4 c3)) G_reg(0; l). The keystone typing is unchanged: pillowcase moduli, the 8 b1 index budget and a |D|-level BFK as an operator identity stay open with the keystone.

Recorded status: Completion of the computable side [E]/[C]/[O] (diagonal zero at the KMS scale; closed-form resummation; multiplicity dichotomy; residual = the SEAM.EQUIV.01 face alone)