The published foundations of TFPT

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.ORIGIN.01Formal/Identity [E] (origin theorem: half-period fixed points, rational descent identity, free-involution census in all three stabiliser types; core solve lambda in {-1, 2, 1/2}; CM dictionary; equivalence tetrad + circularity audit; fermion dichotomy: Utilde^2 = 256 gamma-monomial vs +2^7 scalar in exact Cl(16) Fractions, root counts 2 vs 0, arrangement census 28/21 -- sympy exact, Wolfram-mirrored) + [C] (the one alignment bit: which half-period, i.e. the deck position); SEAM.CLOCK.RIGIDITY.01 and QGEO.REALIZE.01 stay as they are, P2.TYPING.01 R1 unchanged

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

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.M1.01Formal/Identity [E] (exact residue/family/period/kernel arithmetic: Poincare residue three routes, cover factor 4 = |mu4|, closed twistor family XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4(1 - la^4)^2, CY-compatible clock lift Omega -> +Omega, period lockstep 2 pi i t0 (i-1)(1, i, i^2) with covariance Pi -> i Pi, BM kernel (2 pi i)^2 anchor + dK = 0 + scale degree 0, lens forcing N in 4Z, 12 conifold nodes on Z8 -- sympy exact, no floats, Wolfram-mirrored) + [C] (global patching of the local BM kernels on the resolved 3-fold; Hitchin small resolution; the CPS 'exceptional-curve branes = glue currents' dictionary) + [O] (M2: the twisted KS measure per sector; M3: the a0 uplift; the quantised BCOV coefficient on PT/Z4)

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.M3.01Formal/Identity [E] (exact kernel/expansion/period/monodromy arithmetic: null-coordinate harmonicity for arbitrary kernels, residue V-ledger identity 1/(2r), kernel log coefficient 4 = |mu4| with first correction a0/eta^4 and exact m-grading, GLT tower p_{4k} = 4(-a0)^k with the n = 0 mod 4 selection rule, exceptional-locus log 4 log t0 + clock phase 4i, period response 1/4 integrating to monodromy i, a0-rigid node support and (2 pi i)^2 fluxes, all four controls -- sympy exact, Wolfram-mirrored) + [C] (the GLT dictionary eta_p log eta_p = the GH Kahler-potential kernel; Lindstrom-Rocek / Ivanov-Rocek) + [O] (the full nonlinear Kahler potential of the resolved A3 ALE; 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)

SEAM.EQUIV.TWISTOR.01Open (research route [O]: the Costello-Li construction; preparation [E]/[C] via the CELEST.SEAM.01 work packages v492-v518)

WOIT.OS.TWISTOR.01Open (research contract)

WOIT.THETA.FREE.01Identity/Lattice (exact sympy + exact Cl(16); 40-digit gap-certified RP inertia certificates)