The published foundations of TFPT

SEAM.EQUIV.EDGE.CHERN.01Independent topological c_-=8 [E]/[C]: the M=1 collar has integer Chern C=+1 (FHS, no fit; C=0 trivial, C=-1 opposite chirality), the open strip carries |C|=1 chiral edge branch per boundary, and the Li-Haldane entanglement spectrum tracks the edge; bulk-edge gives c_-=16*(1/2)*1=8=g_car+N_fam, corroborating v444 from a different observable. SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.E8.MODULAR.01(E8)_1 modular data [E]/[C]: the character chi=E4/eta^8 is S-invariant (one primary), T-multiplies by e^{-2pi i/3} (=e^{-2pi i c/24}, c=8) and has leading power q^{-1/3} (c=8=0 mod 8, holomorphic) -- the full torus modular signature of the seam (E8)_1 net; SEAM.EQUIV.01 stays [O]

FORM.SEAM.EDGE.CHERN.01Formal [F] (Lean kernel; the Chern integers, the bulk-edge count, the c_-=8=g_car+N_fam assembly, the holomorphic c mod 8 and the order-3 modular T-phase all kernel-proved axiom-free; the cited MMST continuum theorem stays the named residual)

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.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]

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.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]

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

GRAV.ENTROPIC.LEGS.01Numerical [N] (high-precision lattice exhibits) + Formal [E] (single-scale corollary) + [C] (meeting-point reading) + [O] (both legs open)

SEAM.KRONHEIMER.01Reduction/citation [E]/[C]/[O] (graph->geometry bridge discharged to Kronheimer 1989; finite input data compiler-exact; premise transforms, stays open)

FLAV.NUSCALE.04Enumeration/decision [E]/[O] (the 3/5 Clebsch door closed; negative result; window candidate stands)

SEAM.CCC.01Lattice [E] (modular-commutator ground-state measurement + controls + exact additivity) + [O] (continuum limit; SEAM.EQUIV.01 unchanged)

SEAM.MU.01Lattice [E] (parity census double-determined + exact 16-fold-way arithmetic + TEE fence) + [C] (KLM condensation step cited) + [O] (continuum; SEAM.EQUIV.01 unchanged)