The published foundations of TFPT

SEAM.EQUIV.BW.HSMI.01Half-sided modular inclusion [E]/[C]: a Longo standard pair (P>=0, [K,P]=P, J:JKJ=-K, JPJ=P_-) is a +half-sided modular inclusion, so by Borchers the seam modular flow is geometric INTRINSICALLY (no covariance presupposed); face (ii) becomes a structure consequence. Finite witnesses: ladder algebra exact, sl(2,R) positivity, BW reflection; neg control indefinite boost. SEAM.EQUIV.01 stays [O] (continuum standard pair = continuum existence, same wall as MMST/NPW26)

SEAM.EQUIV.CONTINUUM.LR.01Lieb-Robinson motor [E]/[C]: the gapped p+ip collar (v367) has a uniform-in-N gap, exponential clustering with N-independent xi, and a finite light cone, so by Nachtergaele-Sims its bulk thermodynamic limit EXISTS -- grounding the bulk half of the cited MMST existence in OUR model. SEAM.EQUIV.01 stays [O], the open step narrowed to the chiral massless edge scaling limit

SEAM.EQUIV.LTORP.BETA.01beta-interpolation [E]/[C]: the (LTO-RP)/BW data (Theta K Theta=-K, [rho,K]=0) are beta-independent; along the whole path C_beta=1/(1+e^{beta K}) from the NPW26 trace (beta=0) to the seam KMS (beta=1) every covariance is reflection-related, mu4-invariant and valid, norm-continuous, no obstruction -- a PATH argument for v424 (ii), stronger than v426. SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.APPLICABILITY.01Applicability ledger [E]/[C]: a hypothesis-by-hypothesis audit of MMST/NPW26/Adamo finds 11 of 12 hypotheses are TFPT-internal computed facts and exactly 1 is external -- the continuum existence of the chiral massless scaling limit; covariance is an Adamo consequence (defuses v215), not an input. SEAM.EQUIV.01's external dependence is pinned to ONE analytic fact; stays [O]

SEAM.RIGIDITY.UNIFORM.01Uniform-in-N rigidity [E]/[C]: the band-limited mu4-character rigidity of v398 is uniform in N -- exact 4-sector commutation to N=256, a uniform gap, and an N-independent clock-breaking lower bound ||[rho,V]||=2; not a small-N artifact. SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O] (the full-L^2 forcing is the cited rigidity/MMST step)

SEAM.E8.DISCRIMINATOR.01E8 vs SO(16) kill tests [E]/[X]: mode D of v347 realised as discriminators -- torus GSD 1 vs 4, level-1 currents 248 vs 120, lattice theta 240 vs 112 norm-2 roots; c=8 alone does NOT decide (v277), only det K=1 (genus-1, v90) selects E8. Falsifiable: a seam with GSD>1 falsifies (E8)_1. SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.CONTINUUM.EDGE.01Chiral edge scaling motor [E]/[C]: the v367/v368 lattice edge carries the conformal data of a chiral c=1/2 Majorana CFT (linear chiral dispersion v~0.98 with opposite per-edge velocities; algebraic edge correlator eta~1 vs exponential bulk; trivial-phase neg control; N-stable Cauchy scaling-collapse), so the 16-copy edge is a chiral c_-=8 (E8)_1 CFT -- grounding the EDGE half of the cited MMST scaling limit the way v439 grounds the bulk. SEAM.EQUIV.01 stays [O]

FORM.SEAM.BW.HSMI.01Formal [F] (Lean kernel; the standard-pair relations kernel-proved axiom-free, the Borchers step + continuum input named axioms)

SEAM.RIGIDITY.FORCING.01Rigidity FORCING [E]/[C]: commuting with the order-4 mu4 clock is EXACTLY mu4-block-diagonality (entrywise iff swept N=4..128; exact commutant dim sum n_s^2 < N^2; off-block slack zero; the order-2 commutant is strictly larger; 4 distinct eigenvalues), upgrading v442's 'permit' to 'force'. The single residual is the physical transfer lying in the commutant (clock-invariance=QGEO.SYM.01); SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O]

FORM.SEAM.APPLICABILITY.01Formal [F] (Lean kernel; the 11-internal/1-external audit count + MMST arithmetic kernel-proved axiom-free, the one external fact named)

SEAM.RIGIDITY.CLOCKINV.01Clock-invariance derived [E]/[C]: [rho,Lambda]=0 is a CONSEQUENCE of mu4-symmetry of the RP data (the four marks) via OS reconstruction (v54) + the modular formula (v258) -- the OS canonical transfer of mu4-symmetric data inherits the clock (swept N, iff-sharp neg control), is a contraction, and its modular Hamiltonian commutes with the clock. With v445 this reduces the rigidity residual from an operator commutation to 'the RP data is mu4-symmetric' (QGEO.SYM.01); SEAM.EQUIV.01 stays [O]

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)