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.EDGE.FOURPOINT.01External MMST fact pinned at 4-point [E]/[C]: the edge four-point is quasi-free (G^2=G) and Cauchy-convergent to the free-fermion conformal cross-ratio 2+sqrt(3) (vs an off-scale exponential bulk control), extending v444's two-point scaling limit to 4-point order. The one external MMST existence fact is empirically supported at both orders; SEAM.EQUIV.01 stays [O]

FORM.SEAM.RIGIDITY.FORCING.01Formal [F] (Lean kernel; the entrywise forcing iff (uniform in N), the exact commutant dimension and the order discriminator kernel-proved axiom-free, the operator-level clock-invariance named)

SEAM.EQUIV.MMST.UNIFORM.01Uniform-in-N edge convergence [E]/[C]: two independent edge four-point cross-ratios converge to distinct conformal values (2+sqrt(3) and 2) at one 1/N rate (|Q-Q_inf|*N bounded by a single constant), vs an off-scale bulk control -- the empirical content of the cited MMST uniform-convergence theorem; SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.EDGE.ENTANGLEMENT.01Entanglement c_-=8 [E]/[C]: the Calabrese-Cardy law gives c_Dirac=1=2*(1/2) for the critical free-fermion chain (vs a saturating gapped control), so c_Majorana=1/2 and c_-=16*(1/2)=8=g_car+N_fam -- a third independent reading of the edge central charge after v444/v447; SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.EDGE.CARDY.01Cardy tower {1/16,1/2} [E]/[C]: the edge finite-size spectrum reproduces the exact Ising chiral primary dimensions h_sigma=1/16 (R-NS split) and h_epsilon=1/2 (NS pair gap), L-independent (conformal 1/L scaling); with c=1/2 (v450) the edge is the Ising/Majorana CFT, 16 copies => c_-=8=g_car+N_fam; 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]

SEAM.RIGIDITY.MU4FROMMARKS.01mu4 from the marks [E]/[C]: the four marks ARE mu4 (roots of z^4-1), the order-4 clock z->iz is a 4-cycle in their D4 Moebius stabiliser (cross-ratio 2 fixed), and the form basis omega_k is mu4-graded (eigenvalue i^k) -- so QGEO.SYM.01 follows from marks=mu4 + the existing QGEO.REALIZE.01, no new premise; the rigidity chain closes modulo realisation, 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.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]