The published foundations of TFPT

RESIDUAL.CERTIFICATION.01Historical compiler-residual audit [E]; explicitly not a TOE-completeness audit. TFPT.TOE.COMPLETE.01 and all T1-T8 remain Open [O].

UVBRANCH.KILLTEST.01UV-branch kill-test surface [E]/[X]/[C] (carrier-PS on the all-order footing; proton-SAFE hierarchy computed; plain-SM GUT killed; no fake tau_p)

GRAV.AMPLITUDE.01Perturbative graviton amplitude [E]/[C] (entire-form-factor exchange: finite, ghost-free, UV-softened, tree-unitary; perturbative-only)

CORRECTIONS.GAP.01Correction calculus [E]/[C] (each sector's gap sets its first-correction size; (2/3)^6 family vs golden family; the harvest of v383)

CORRECTIONS.BUDGET.01Typed correction budget [E]/[C] (the gap correction is keyed to the mechanism class, not a uniform band; seam-gapped=(2/3)^6/golden, exact-identity=0, external-rate=external, fixed-point=explicit texture)

GRAV.LOOP.FINITE.01Perturbative graviton loop UV-finiteness [E] (power counting; super-polynomial falloff, degree -> -infinity at every loop order, seam-regulated) + loop unitarity [C]/[O] (the honest open point, not claimed solved)

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)

FORM.SPECTRALGAP.01Formal [F] (Lean kernel; no extra axioms) -- the scalar core + the finite-dimensional diagonalisable case of the spectral-gap meta-theorem (v383): gap => unique attractor + geometric decay => parameter-free

ALPHA.QUILLEN.PROGRESS.01Honest attempt / reduction [E]/[O] (a solvable-model zeta-det mechanism + the reduction of ALPHA.QUILLEN.EXACT.01 to the named sub-target 'seam a_2 = 8 b1 c3^6'; the headline target stays OPEN, not closed)

ALPHA.QUILLEN.PROGRESS.02Second honest step / grounding [E]/[O] (a solvable 4D model exhibiting the nonzero a_4 conformal-anomaly order + the v391<->v342 connection; the headline target stays OPEN, not closed)

ALPHA.QUILLEN.PROGRESS.03Third honest step / grounding+reduction [E]/[C]/[O] (b1 = the U(1) a_4 heat-kernel coefficient via beta=a_4; residuals 1&3 collapse to one seam F-normalisation [C]; residual 2 the alpha^3 Chern level located but [O]; the headline target stays OPEN)

ALPHA.QUILLEN.PROGRESS.04Fourth honest step / metric-independence test + conditional level typing [E]/[C]/[O] (the pi-power test isolates a^3 as the unique pi^0/c3^0 metric-independent topological rung; conditional on the CS/eta reading the a^3 coefficient is an integer level, =1 the unit; the EM1 CS-boundary proof and the headline target stay OPEN)

SEAM.S3.SCALINGLIMIT.01Scaling-limit convergence evidence [E] (the finite-size central charge converges to c=8 and Richardson-extrapolates there -- the existence-relevant observable) + the abstract existence stays [O] (cited MMST v336); an honest attack that strengthens, does not close

CORRECTIONS.NUMERIC.01Numeric correction budget [E]/[C] (the typed budget v388 as actual first-correction magnitudes: fixed-point 0.227%, seam-gapped 8.78%/QG-capped 9.62%, exact-identity 0, external quoted -- a computed theoretical-uncertainty surface)

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

HYP.COUPLED.01Coupled hypergraph rewrite [E]/[O] (one local rule unifies carrier x family; joint attractor marks x e0; recovery (2/3)^6 after one hand; v324 fiber emerges as cusp readout; v299 growth unchanged) -- mechanism unified, phi0 + seed remain

HYP.PHI0.01phi0 leading term [E]/[C]/[O] (phibase = F/(g_car N_fam) c3 = 1/(6 pi); F/h = |Z2|/N_fam; Gauss-Bonnet consistent; puncture 48 c3^4 still analytic) -- combinatorial reinterpretation, not full phi0 closure

HYP.PHI0.PUNCTURE.01Named target [E]/[O] (the phi0 puncture 48 c3^4 as the order-4 boundary Seeley-DeWitt contact term of the mu4 puncture, summed over Omega_adm=N_fam dim S+=48; a face of the seam boundary analysis, elevates v396's puncture residual to a citable target; no new gate)

RES.COXETER.SYMMETRY.01Structural lemma [E]/[C] (prime-2 necessary, not autonomous: the sheet involution has spectrum {+-1} and projector {0,1}, none in (0,1), so no prime-2-only attractor; reuses established atoms, no new number) -- closes the v394 open-question corollary

FORM.COXETER.PRIME2.01Lean formalisation [E] (no prime-2-only attractor: involution eigenvalues +-1, projector {0,1}, none in (0,1), while 2/3 is a contraction rate; axiom-pure) -- machine-proves the v409 corollary