The published foundations of TFPT

SEAM.EQUIV.EDGE.VIRASORO.01Lattice Virasoro/current algebra c=8 [E]/[C]: Casimir c_Dirac=1.000 (1/2 per Majorana), lattice U(1) current Kac-Moody level 1 (<J_n J_-n>=n exact), Sugawara c=dim G/(1+h^v)=8 for BOTH SO(16)_1 120/15 and (E8)_1 248/31; 16*(1/2)=8=g_car+N_fam; SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.BW.UNIFORM.01Uniform-in-N BW tower [E]/[C]: ||Theta K Theta+K||=0 at every N (m=2..32), N-independent gap 6 ln(3/2), [rho,K]=0 exact, N-independent neg-control margin 2||K|| -- the inductive limit inherits u_Theta=J, advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]

SEAM.S3.FROM-P1.01S3 chirality from P1 [E]/[C]: c3's 8=|Z2|*(intK/pi)=2*4 is the one-sided count; a reflection sends Chern C->-C so a two-sided boundary forces C=0, the one-sided seam (no reflection, the |Z2| of c3's 8pi) forces C!=0 -- chirality is a CONSEQUENCE of P1, magnitude c_-=8; SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.KILLTEST.01(E8)_1 character/sector kill-test [E]/[X]: chi=E4/eta^8={1,248,4124,34752}; the discriminators 248=120+128 (weight-1) and |det Cartan E8|=1 vs |det D8|=4 (sectors) -- a measured 120 or 4 sectors would FALSIFY (E8)_1, the data give 248/1 (PASSED); SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.MMST.AUDIT.01MMST citation audit [C]/[O]: hypothesis-by-hypothesis (arXiv:2107.13834) -- H1 CAR class, H2 massless limit, H3 per-copy c=1/2 (Thm 4.11), H4 decoupled additivity 16*1/2=8, H5 WZW range 8<=8<=16 + 120 bilinear currents all MATCH; the SINGLE residual H6 is the 128-spinor extension SO(16)_1->(E8)_1 (det K 4->1), outside MMST's bilinear method -> v459; SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.LATTICEVOA.01Lattice-VOA route to (E8)_1 [E]/[C]: E8 even unimodular (det 1, holomorphic); 248=8+240 currents, the 240 roots split 112 integer + 128 spinor (248=120+128) -- A_{E8} (Adamo-Giorgetti-Tanimoto) constructs the 128-spinor extension MMST (v458) misses; two complementary routes cover (E8)_1, residual = the single realization input (v456); SEAM.EQUIV.01 stays [O]

SEAM.S3.LTO.01S3 commuting-projector realisation [E]/[C]: v367's gapped collar has an exact flat-band parent (Q^2=I, P^2=P=P_gs) that is gap-protected quasi-local (xi~1.14 at M=1, power-law |R|^{-2.1} at the gapless point) and whose flattened modular Hamiltonian inherits Theta K_flat Theta=-K_flat and [rho,K_flat]=0 exactly; the chiral c_-=8 obstructs STRICT commuting-projector locality (Kapustin-Fidkowski), so the realisation is the quasi-local LTO net of NPW26 -- advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]

FORM.SEAM.RESIDUAL.01Formal [F] (Lean kernel): the G-block arithmetic (one-sided 8=|Z2|*intK/pi, reflection C->-C, c_-=8, MMST range 8<=8<=16, decoupled additivity, 248=8+240=120+128, 240=112+128, det K 1 vs 4, + v461 winding=|C|=1!=0 obstruction + v462 Fock counts 120/128/256 + v463 holomorphy selector forced dimV1=248=E8) all kernel-proved by decide; the two cited theorems MERGED into ONE combined axiom seam_realisation_theorem so seamResidualClosed depends on FOUR axioms (down from six); v464 one-particle rigidity is numerical; residual reduced to ONE realization axiom + ONE combined cited theorem; PLUS the v469 parallel peer-reviewed route seamResidualClosed' (invariant-level collar_invariants + crossedproduct_route_theorem; new kernel facts lr_locality_integer, glue_locality_integers, klm_holomorphic, sixteenfold_pin, all [propext only]); SEAM.EQUIV.01 stays [O]

SEAM.S3.LOCALITY.01Strict-locality obstruction exhibited [E]/[C]: the chiral collar's Wilson-loop/Wannier-centre winding = Chern = 1!=0 (vs 0 for the gapped trivial control), the exact obstruction to exp-localised Wannier (Brouder et al./Thouless); so no strictly finite-range commuting projector exists for c_-=8!=0 (Kapustin-Fidkowski) -- v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net, sharpening v460 from citation to exhibited obstruction; SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.SPINOR.01128-spinor extension exhibited [E]/[C]: the Jacobi/E8 identity theta2^8+theta3^8+theta4^8=2E4 makes chi_{(E8)_1}=E4/eta^8=chi_o+chi_s (vacuum 120 + spinor 128 = 248); the finite 16-Majorana Fock space carries C(16,2)=120 currents and 2^8=256=128+128 Ramond states; the critical ring converges c_- -> 8 and x_1 -> 1 as L grows; AGT/AMT (v459) supply the rigorous extension; SEAM.EQUIV.01 existence stays [O]

SEAM.EQUIV.UNIQ.01c=8 holomorphic-uniqueness pin [E]/[C]: c=8 has three level-1 candidates (A8/80, D8/120, E8/248) so c=8 is not sufficient; holomorphy forces the character E4/eta^8 => dim V_1=248 => E8 uniquely; Dong-Mason/Schellekens give holomorphic c=8 VOA unique = V_{E8}, so the identification half of SEAM.EQUIV.01 is [C] (classification-forced); SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.ONEPARTICLE.01one-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]

FLAV.THIRDGEN.PATTERN.01Numerical [N]/[C]/[O] (candidate pattern; record of v84 untouched)

FLAV.DM2RATIO.01Numerical [N]/[C]/[O] (candidate relation; informal pi phi0 table heuristic superseded as comparator)

SEAM.EQUIV.CROSSEDPRODUCT.01Net-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]

ALPHA.QUILLEN.INFLOW.01Inflow 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]

SEAM.EHMODEL.04Numerical [I] (lattice determinants vs exact conical constants) + [A/O] (continuum leg + anchor stay open)

ALPHA.QUILLEN.DETLINE.01Numerical [N] (finite Quillen shadow; gate unchanged)

GRAV.ENTROPIC.ACTION.01Formal/Numerical [E] (five exact identities + quantified gap) + [C] (compression conjecture, Q-target) + [O] (variational completion open)

GRAV.ENTROPIC.HODGE.01Formal/Identity [E] (operator-level Clifford/weights/enumeration, exact) + [C] (physical Q choice) + [O] (AP2 + R^2 unchanged)