The published foundations of TFPT

FORM.MU4COMM.01Formal

SEAM.EQUIV.UNIFY.01Synthesis/reduction [E]/[O] (one keystone theorem; QGEO.SYM corollary; QG.AMB decoupled)

FIRSTPRINCIPLES.BOUNDARY.01Boundary audit [E]/[X]/[C] (what is derived / external / tensioned / forbidden)

DYNAMICS.UNIVERSAL.01Universal synthesis [E]/[C] (every sector = a gapped operator with a unique attractor + spectral gap => parameter-free; extends v303 theory-wide; 3 gaps computed, the cross-sector identification [C])

COXETER.PRIME2.01Prime-2 facet of the Coxeter clock [E]/[C] (h(E8)=30=2*3*5; one quadratic facet field per prime, ramified at it; the prime-2/Gaussian/Z4-seam facet completes v383's golden and family facets)

COXETER.ATOMS.01Clockwork coherence [E]/[C] (Coxeter primes = anchor atoms = facet fields; chi=3^6/(3^6-2^6); both dynamic rates carry prime-2; 30=5x6) -- a bird's-eye re-reading, no new number

FR.CLOCK.PROBE.01Clock probe of the frontier [E]/[C] (3 of 4 'external' scheme-numbers -- A_Lambda, Koide rate, axion spine -- land EXACTLY on the {2,3,5} clock; only the proton C_p is genuinely external; a falsifiable anti-numerology test)

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]

ARITH.NORMTRIPLE.01Cyclotomic norm triple [E]/[C]: the carrier split (3,2) over the three atom-rings Z[omega],Z[i],Z[zeta5] gives det(3I+2 Comp(Phi_{3,4,5}))=(7,13,55)=(scalaron,Delta_Q,quark numerator); the carrier-5 clock is the 4x4 Phi5-companion C5 (golden in its 2cos values, N(3+2 zeta5)=55=quark numerator); NEG control anchor (5,4)->(21,41,461) fails in the carrier ring; (3,2) FORCED (v14) not unique, but the UNIQUE CROSS-SECTOR seed -- only (3,2)->(7,13,55) spans gravity/flavor/masses while (2,1)->(3,5,11) is all-core [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]

FR.POLE.QED.01F_pole external [E]/[C]/[O]: the Koide source->pole transfer is a one-loop EM effect; Koide is rescaling-invariant (Q(c*m)=Q(m)) so the shift is flavor-split; the tree->pole gap 0.0022 is one-loop-EM-sized (=phi0/24 to 0.6%, =alpha/pi to ~5%), pole at 2/3 (IR attractor); a 1-loop QED per-lepton-log estimate shifts Koide by EM-loop order toward 2/3; the exact flow time t~2.84 is scheme-dependent external QED, only sized; F_pole stays [C], firewalled (v187) -- rate forced (v327/v82), flow external

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.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.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.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]

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]

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.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]