The published foundations of TFPT

FLAV.SELECTOR.CLOSE.01R4' selector folds into the keystone [E]/[C]/[O] (the (d,n) pairings {2,8,121}={|Z2|, rank E8=dim A2=det R, ||Pl(K)||_1^2} are the tau=omega family-slice atoms; n.a=8=dim A2 reads R in E6xA2; deriving (d,n) folds into SEAM.EQUIV.02; residue = why 11)

SHEET.GEN.BINARY.01Sheet generator V is a binary compiler [E]/[C]: V^n closed form + V^n.1 binary spine + four bilinear families (6,7,9,11; 13,27,55,56); spine FORCED by Spec(V)={0,1,2}, Lie/theta readings audit-typed [C]

DIAMOND.SHEET.SOURCE.J.01Sheet source corner J=M(1,-2) on the Z2 wall [E]/[C]: chi_J=(6,3,2), a.J.a=30=h(E8), det(I+J)=12, det(2I+J)=40; readings audit-typed

DIAMOND.SHEET.CALCULUS.01Sheet axis = characteristic-difference calculator [E]/[C]: Delta e1=3, Delta^2 e2=4, Delta^2 e3=8; anchor energy 52+11t (30->41->52->63); readings audit-typed

DIAMOND.CENTER.RESOLVENT.01Center C resolvent portal [E]/[C]: det C=14=dim G2, det(I+C)=52=dim F4, det(2I+C)=120=|R^+(E8)| (G2->F4->E8^+); reading audit-typed

ARITH.TRICHOTOMY.01Atom trichotomy [E]/[C]: {2,3,5} ram/inert/split in Z[i] (2 ram, 3 inert, 5 split) vs Z[omega] (3 ram, 2,5 inert); the ramified prime of each ring = its atom; each atom is the unique ramified prime of one facet of Q(i,sqrt-3,sqrt5), product 30=h(E8); seam/flavor/golden reading [C]

DIAMOND.EISEN.01Eisenstein/CP operator [E]/[C]: the family rotation P realises the hex norm 7=N_omega(3+2omega) via (3I+2P)(3I+2P^2)={7,7,25} (dual of v415's {13,13,9}); the CP clock W=-P^2 (W^2=P, W^3=-I, Spec {-1,zeta6,zeta6^-1}) gives delta_CKM=pi/3, delta_PMNS=4pi/3; all flavor clocks are zeta30-powers, the seam mu4 the Galois side phi(30)=8=rank E8; the carrier mu5 has no 3x3 operator [C]

SEAM.GALOIS.01Seam mu4 = carrier's Galois group [E]/[C]: 30 squarefree => no order-4 cyclic, so the seam mu4 is the Galois side (Z/30)^x=mu4 x Z2; the mu4 factor IS (Z/5)^x=Gal(Q(zeta5)) (carrier), the Z2 IS (Z/3)^x (family/CP); the Frobenius zeta5->zeta5^2 is the explicit order-4 operator G with G C5 G^-1=C5^2, G^4=I; resolves the v409 asymmetry (seam = automorphism, not a hand) [C]

SEAM.GALOIS.NET.01Galois<->Net bridge / four-fold mu4 identity [E]/[C]: the seam mu4=Gal(Q(zeta5))=(Z/5)^x is the SAME cyclic Z/4 as the (E8)_1 glue <(1,1)> AND the discriminant disc(A3)=disc(D5)=Z/4 (D5 cyclic because rank 5 odd; D8 even=Klein control); only the carrier prime 5 gives (Z/p)^x=Z/4; glue order 4 => 16/4^2=1=E8 (vs Klein/order-2 => D8 det 4); cyclic Z/4 selects E8, Klein excluded -- the v419 Galois gearbox IS the holomorphic net's glue, a direct Galois->G_net bridge. Deck/divisor=Galois=discriminant=simple-current is a compression per v428 (one object read four ways), not four witnesses [C]

QGAMB.NODEP.01No-Ambient-Dependency [E]/[C]: no frozen readout is computed from QG.AMB.01 -- it is a certification SINK (every dependent is a gravity-gate/QG row, no prediction), not a value source (script '-'; predictions computed by v3/v7/v9/v268, never an ambient-measure script), and its only predictive link is theta13's architectural 4D-QFT contract edge (coherence, not numerical); the gap margin Delta_eff~1.648>0 decouples its back-reaction (v337). v369 prose -> machine-checked DAG/script fact [E]; verdict [C], does NOT prove SEAM.EQUIV.01

SEAM.EQUIV.LTORP.01(LTO-RP) keystone reduction [E]/[C]: SEAM.EQUIV.01's intrinsic-BW lemma reduced to the cited NPW26 commuting-projector theorem, mapping corrected -- (LTO-RP)=u_Theta=J=intrinsic BW (theta K theta=-K), NOT omega o rho=omega; the mu4 clock is a modular symmetry [rho,K]=0 (sigma^psi=rho type-wrong); A-and-NOT-B refutes clock-invariance => (LTO-RP). Make-or-break: NPW26 prove (LTO-RP) for topologically-ordered models (trace/anyonic), the seam is invertible (|det E8|=1) + beta=1 KMS (not a trace), in neither bucket. REDUCTION not closure; two open sub-steps (Z/2-reflection commuting-projector LTO realisation; invertible-KMS (LTO-RP)). SEAM.EQUIV.01 stays [O]

SEAM.EQUIV.LTORP.KMS.01Invertible beta=1-KMS (LTO-RP) [E]/[C]: the corner NPW26 leave open (they do topologically-ordered; the seam is invertible + KMS, not a trace) is constructively satisfiable -- a gapped reflection-symmetric mu4-even collar has Theta K Theta=-K (u_Theta=J) and [rho,K]=0, so (LTO-RP) holds by direct Tomita-Takesaki; |det E8|=1 (no anyons), state not a trace; neg controls have teeth. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (continuum sub-step (i) remains), finite-dim witness not continuum theorem

OVERDET.WITNESS.MAP.01Witness map [E]/[C]: disjoint grammars multiply, same-generator projections compress; the seven arithmetic witnesses are refined by v428 to COMPRESSION (facets of one (2,3,5)/E8 object, v236). The genuine multiplication is the multiply-forced inputs + foreign witnesses (alpha^-1~137); coincidence is expensive for the discrete core; does not by itself establish physics

OVERDET.WITNESS.RECLASS.01Witness re-classification [E]/[C]: v427's seven arithmetic witnesses are facets of ONE (2,3,5)/E8 object (v236) => compression, not seven multiplications. The genuine multiplication is the '8' forced four independent ways (rank E8, h(D5), phi(30), Milnor) + the foreign alpha^-1~137. Honest correction of v427 (consistent with v305/v236/v313)

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.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.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]

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)