The published foundations of TFPT

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)

COSMO.RUNNING.01Identity [E] (LO relations + exact slow roll) + Physical [C] (data confrontation; N_star input)

COSMO.MUDIST.01Physical [C] (Chluba-window bridge, data-limited) + Identity [E] (comb-blindness attenuation)

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)

QGEO.EMERGE.LIGHT.01Formal [E] (exact structure: pillowcase = half torus pair structure, permutation identities R^2 = Z2, clock-forcing a3 = a2 = a1 = 0 with j = 1728 identically, Coxeter charpoly) + Numerical [N] (eta/E2*/j at 40 digits; v280-style lattice DtN curves) + [C] (clock = carrier input) + [O] (the dynamical realisation of the clock); NO marker moves -- QGEO.REALIZE.01 stays [C]/[O]

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)

SEAM.CLOCK.RIGIDITY.01Formal/Identity [E] (Theorem A: complete PGL2(C) square-root classification, D4 centrality criterion, counterexample, n-arithmetic, stabiliser trichotomy -- sympy exact; Theorem B: U^2 = (-1)^F in exact Cl(16) Fractions, nonsplit Z4 extension, Z8 tower, R-sector split control, 16-fold-way arithmetic -- Wolfram-mirrored) + [C] (the one alignment bit: deck central in the mark-D4); QGEO.REALIZE.01 stays [C]/[O], P2.TYPING.01 R1 mapped onto the same bit

SEAM.BIT.FREEDOM.01Formal/Identity [E] (dihedral theorem: unique free involution = antipode, Aut(C_16) = D_16 complete census, quotient topology chi 0 vs 1; complete Cl(16) Fock census NONSPLIT <=> FREE over all 17 involutions; real-structure classification: two deck-commuting families, per-configuration uniqueness table; harmonic + free => b = +-i exactly; counterwitnesses -- sympy exact / exact Cl(16) Fractions, Wolfram-mirrored) + [C] (the square-modulus datum tau = i); SEAM.CLOCK.RIGIDITY.01, SEAM.BIT.ORIGIN.01 and QGEO.REALIZE.01 stay as they are (text precised only), P2.TYPING.01 R1 halved

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)

SEAM.TAU.FLAG.01Formal/Identity [E] (complete V4/D4 symmetry census, flag/arc orbit arithmetic, odd-split and K3-indicator closed forms, twist mode identity, H4 Schur rigidity, negative controls -- sympy exact + mpmath 30 digits, Wolfram-mirrored) + [C] (the symmetry-lift bit itself: flag transitivity stays a postulate; the gain is the FORM, not a derivation)

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