The published foundations of TFPT

CELEST.WP1.01Formal/Identity [E] (exact Z4-orbifold/character/spin arithmetic, sympy, negative controls) + [C] (the named clock^2 = deck identification, verdict B) + [O] (celestial programme; SEAM.EQUIV.01 unchanged)

CELEST.SEAM.01Open (research contract; WP1-WP5d executed [E]/[C]/[L] via v492/v493/v495/v496/v497/v498/v500/v501/v504 -- WP5d: alpha (mu-index) + beta (split + strong additivity) executed via v501/v504, all three KLM ingredients witnessed on the lattice, continuum uplift stays [O] (Xu's theorem cited); WP5e alpha stage [E] via v502 + beta stage [E] via v505 (equivariant twistor skeleton; kill not fired) + gamma stage [E] via v508 (exchange no-go with certificate (0, 32, 72); negative result, kill not fired) + epsilon-2 stage [E] via v509 (level-from-flux: the one-level statement is now a THEOREM of clock invariance, the sector-counter dial #prim((E8)_k) = 1 <=> k = 1 pins the level; the PT/Z4 back-reaction stays [O]); + delta-2 stage [E] via v511 (the full-tensor ledger: collapse confirmed by innerness, the cubic d-symbol channel opens T3 with the weaker certificate psi = 64, relaxed solution c_d = 1920 = |W(D5)|); + epsilon-1 stage [E] via v514 (epsilon-1 executed via v514: the O(-2) bulk-axion slot is a construction (equivariant Penrose ledger), lambda-tilde = 6 pinned by three exact ledgers (Okubo / measure cancellation excluding 3 and 12 / flux), and the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from geometry -- the quantised BCOV coefficient and the twisted channels stay [O] (M1-M3 preregistered)); WP5e-M1 DONE (v515: M1 executed via v515: the back-reacted Omega_N is closed-form on the A3 twistor family, all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector and clock covariance, and the lens geometry FORCES the source charge 4 = |mu4|; honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden family)); WP5e-M2+M3 DONE (v516/v517: M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01); the global BCOV quantisation [O] (single remaining milestone -- the exchange sub-branch closed by v508, the epsilon-2 flux dial executed by v509, the delta-2 tensor ledger executed by v511, the epsilon-1 axion slot executed by v514, the M1-M3 back-reaction milestones executed by v515/v516/v517 (the v514 fence fully worked off); named remaining target: the BCOV MEASURE QUESTION -- the delta-1 chain is DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, all three testers fail; tension with v516 stated), so what remains is which of the two exact measures (the declared v516 completion reading vs the v518 derived chiral measure) is the true BCOV measure; the cubic GS term stays dissolved (psi = 64 delivered by v516, no branch fires under the derived measure, v518)))

CELEST.WP2.01Formal/Identity [E] (exact deformation geometry/period/algebra arithmetic, sympy, negative controls) + [C] (three named identifications, verdict B; v216 precision typed) + [O] (algebra/contract programme; SEAM.EQUIV.01 and v216 unchanged)

CELEST.WP3.01Formal/Identity [E] (exact Okubo/Killing/lambda arithmetic for all 8 algebras, Fraction/sympy, sl4 negative control, look-elsewhere battery) + [C] (the convention typing of the 8 pi and sqrt 3; the alignment reading with the look-elsewhere caveat, verdict B) + [O] (celestial programme WP5; SEAM.EQUIV.01 unchanged)

CELEST.WP4.01Formal/Identity [E] (exact counting/character/theta arithmetic, integer/Fraction/sympy; mismatch localisation with exact cross-multiplication) + [C] (literature gradings CP Table 1 + Sym^2(248) decomposition; the boundary-limit reading, verdict B(ii)) + [O] (the constructive limit -> WP5; SEAM.EQUIV.01 unchanged)

CELEST.WP5A.01Formal/Identity [E] (exact stabilisation/derivation/control arithmetic, integer/Fraction; Freudenthal + Weyl + peeling machinery unit-tested) + [C] (the named identifications: radial decoupling = collar limit; Kac singular vector via counting consequence) + [O] (the operator question WP5b-e; SEAM.EQUIV.01 unchanged)

CELEST.WP5B.01Formal/Identity [E] (exact construction/singularity/control arithmetic, integer/Fraction, deterministic; Chevalley basis + affine PBW engine unit-tested on all 61504 pairs) + [C] (the named identifications: Weyl complete reducibility beyond BFS depth 4; Kac mechanics beyond level 2; per-period quarter dictionary) + [O] (the GNS/limit-state question WP5c-e -- the twisted-slot moding tension is the precise WP5c question; SEAM.EQUIV.01 unchanged)

CELEST.WP5C.01Formal/Identity [E] (exact construction/Gram/control arithmetic, integer/Fraction, deterministic; the compact anti-involution machine-determined; the full 9361-block level-2 Gram exact) + [C] (the named identifications: quasi-free extension; Kac positivity beyond level 2; module identification in the component route) + [O] (local normality, net structure/Haag duality, Costello-Li -- WP5d/e; SEAM.EQUIV.01 unchanged)

CELEST.WP5D.01Formal [E] (the exact arithmetic: Cartan dets, KLM/Longo-Rehren quotients, Sigma d^2, theta_v at nu = 16, machine identities) + Lattice/Numerical [L]/[N] (the Gaussian lattice witnesses, ED-validated to 1e-15: extensivity, the ln 2 plateau, the duality-asymmetry witness) + [C] (the continuum dictionary offset = ln index -- KLM/Longo-Xu/CHMP -- and the interacting condensation step) + [O] (continuum Haag duality / local normality / split of the quotient net = WP5d-beta; Costello-Li = WP5e; SEAM.EQUIV.01 unchanged)

CELEST.WP5E.ALPHA.01Formal/Identity [E] (exact prefactor/level arithmetic: Hurwitz-zeta vacuum energies, inner/shift twist distribution, discriminant-form = Casimir two-route identity, current/embedding/central-charge k-fixing, singular-vector dial -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Kac generation for k >= 3) + [O] (the BCOV/Costello-Li twistor inflow = WP5e proper; SEAM.EQUIV.01 unchanged)

CELEST.WP5DB.01Formal [E] (the exact algebra spans/indices, the PP identities with integer attainment, the Lambda^2-compound and parity machine identities, the index consistency) + Lattice/Numerical [L]/[N] (the deficit tables, the nome ladder, the power laws, the Klich-Levitov slope; Gaussian-exact, ED-validated) + [C] (the KLM dictionary: Longo-Xu bounded-deficit dichotomy, Longo split heredity, CHMP order parameters) + [O] (the continuum proof for the seam quotient net and the condensed (E8)_1 net -- Xu's orbifold theorem cited, not claimed; WP5e/Costello-Li; SEAM.EQUIV.01 unchanged)

CELEST.WP5E.BETA.01Formal/Identity [E] (exact equivariant arithmetic: Atiyah-Bott denominators/characters, per-sector Okubo decomposition with the rigid 32 T3 residual, index bridge f(m) = ch2(T_m) with the -78 defect, geometric level dials, axion-slot weight arithmetic -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the dictionaries above) + [O] (the global BCOV quantisation = WP5e proper; SEAM.EQUIV.01 unchanged)

CELEST.WP5E.GAMMA.01Formal/Identity [E] (exact vertex-space/selection/certificate arithmetic: invariant quadratics dim 2 and quartics dim 5 by Weyl nullspace, product theorem, rank-3 certificate (0, 32, 72), naturalness resolution, SO(16)/D8 controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the number fences 32 = 20 + 12, 72 = 2 lambda~^2_e8) + [O] (the twisted BCOV contact terms delta-1 (binary) + the full-tensor ledger delta-2; SEAM.EQUIV.01 unchanged; CELEST.SEAM.01 stays Open)

CELEST.WP5E.EPS2.01Formal/Identity [E] (exact CPS-skeleton/pairing/lockstep/sector arithmetic: S^3 period (2 pi i)^2 N, exceptional flux 2 pi N, contraction 2N, pairing enumeration 48 -> 2, fluxes (64, 60, 64), current count (240, 0, 0, 0), #primaries (1, 3, 5, 10, 15, 27), lockstep theorem det(A-1) = -4 with the 0/24 negative probe, k = 2 / SO(16) / A2 controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the CPS dictionary itself on PT/Z4: their geometry is the C^2 blow-up with one (-1)-sphere and so(8) at negative level, not the A3 ALE -- typed as an analogy until the backreaction is computed) + [O] (the type-I B-model backreaction on PT/Z4 -- twisted Kodaira-Spencer measure, the A3 Omega_N; the same fence as v502 S5.7/v505)

CELEST.WP5E.DELTA2.01Formal/Identity [E] (exact Hom-table/certificate arithmetic: g0 = d5 (+) a3 without u(1), sector factorisation into minuscule orbits, innerness theorem at arities 2 and 3, bilinear table (2, 1, 1, 1), trilinear neutral dims (3, 2, 2, 1, 1), unique symmetric survivor = su(4) d-symbol, Q_dd = (1/60)(T3 - P3/4) two routes, rank-3/4 certificate psi(A_fix) = 64, relaxed solution (-1, 8, 2; 1920), SO(16)/false-g0/false-sector controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the number fences 64 = dim g1 = 2^6, 1920 = |W(D5)| = 8 x 240; the |W(D5)| reading typed look-elsewhere-loaded and convention-contingent by v513) + [O] (the arity >= 4 ledger, the physical justification of the cubic GS term, ch2-naturalness for cubic vertices UNDECIDED; delta-1 UNDECIDED at exploration level at promotion time; since DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, tension with v516 stated); SEAM.EQUIV.01 unchanged; CELEST.SEAM.01 stays Open)

CELEST.DTERM.NONDERIV.01Formal/Identity [E] (exact convention/enumeration/lattice/index arithmetic -- sympy/Fraction, no floats, Wolfram-mirrored); the v511 [C] fence '1920 = |W(D5)|' TEXT-SHARPENED to look-elsewhere-loaded and convention-contingent (fence stays [C]); [O] the physical justification of the cubic GS term and the generation of the 32

CELEST.WP5E.EPS1.01Formal/Identity [E] (exact slot/residue/GH arithmetic: equivariant Penrose ledger block by block d <= 6, character series P_m, Molien hypersurface relation degree 8, Okubo perfect square (6<x,x>)^2, measure chain mu-exact excluding 3 and 12, flux single-channel iff k = 1, GH centre dichotomy, period lockstep 4 pi t0 (i-1)(1, i, i^2), Coxeter clock from geometry, EH log ledger with asymptotic coefficient exactly 0, multipole selection rule m = 0 mod 4 -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Costello's flat-PT matching lambda_g^2 E = A: the measure chain is CLASSICAL bookkeeping) + [O] (the quantised BCOV one-loop coefficient on PT/Z4; the twisted channels 32 T3 not covered by the bulk axion -- M1-M3 preregistered with success + kill criteria)

CELEST.WP5E.M2.01Formal/Identity [E] (exact weight/contact/table/control arithmetic: completion-weight identity w = 4h = -4 ch2 with no free scale, parameter-free locks (T5 = 0 any scale, 4:3 ratio, T3 budget => c = |mu4|), contact term two bases, per-sector Okubo squares 36<x,x>^2/det_j, total (5/4) x 36, certificate kills 32 -> 0 / 72 -> 0 / psi 64 -> 0, exchange-span remainder, mu-blind chain, all five controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the DECLARED completion reading: loop = unphased sector trace with the same zero-mode measure -- supported by the delta-1 modular-completion finding, not derived) + [O] (the BCOV-integral derivation of the completion reading, delta-1b/1c/1d; SEAM.EQUIV.TWISTOR.01 unchanged)

CELEST.WP5E.DELTA1.01DECIDED: kill under the derived measure, tension with v516 stated ([E] exact Weil/Q(zeta_8) algebra, loop words + koboundary defects + 1/960 phase arithmetic, ledgers, G = f1 f3, the decision table; [C] the f1f3/KS-weight identification + kernel-family convention + chi_k choice; [O] the BCOV-integral derivation that decides between the declared (v516) and derived (v518) measures -- since decided at probe level by v520 for the declared reading, residual [O] = the constructive w_m derivation; NO marker moves)

SEAM.EQUIV.MMST.01Conditional [C] (closed modulo cited theorems: the MMST/OS continuum existence, v336; every computable level pinned [E]; Lean FORM.SEAM.MMST.01)