Original ledger status
CELEST.WP1.01
WP1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 1, verdict B): the E8 mu4-glue grading (v128) IS the flat Z4 monodromy of the glue-equivariant celestial chiral algebra on the A3 ALE space C^2/Z4 -- the Bittleston-Homans-Sharma CCA construction (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case) refined to Z4 with a Kronheimer-Nakajima flat connection at infinity. Sympy exact throughout. [E] INNER GRADING / FLAT MONODROMY: h = (2,2,2,2,2;0^4) gives <alpha,h> = glue class mod 4 for ALL 240 roots => glue = Ad(exp(2 pi i h/4)), an order-4 torus element = a flat Z4 monodromy; the A3-side detector h' = (0^5;1,1,1,-3) reads the SAME class -- the glue charge is the diagonal (1,1) of the discriminant group Z4 x Z4 (the v92/v125 Lagrangian glue); Z2 halfway: 120 = SO(16), 128 = spinor (v125 sub-Q {0,2}). [E] SPATIAL SIDE: C^2/Z4 under (z1,z2) -> (i z1, i^-1 z2) IS the A3 singularity XY = Z^4 (invariant ring exact; Hilbert series (1+t^4)/((1-t^2)(1-t^4)); Molien table verified; conjugation symmetry N[d][1] = N[d][3]). [E] EQUIVARIANT SECTOR CLOSES: the glue-equivariant SDYM(E8) modes (j + m = 0 mod 4) form a Lie subalgebra with D[d] = 60(d+1) even / 64(d+1) odd -- uniform per parity ONLY because dim g_j = (60,64,60,64) (the v128 dims are the consistency condition); zero modes (d=0) = g0 = carrier D5+A3+Cartan = 60; asymptotic density 62/248 = 1/4 = 1/|Z4|; EH/Z2 halfway towers 120/128. [E] CHARACTER: Theta_E8 = sum of the 4 glue-coset thetas (shell 1 = (52,64,60,64) from the LATTICE, independent of the root construction); the 4 sector characters Theta_Cj/eta^8 sum EXACTLY to chi_(E8)_1 = 1+248q+4124q^2+34752q^3 (v377 series); sector weights h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8) = the v92/v125 discriminant form (5x^2+3y^2)/8 mod 1; glue diagonal h = (0,1,1,1) INTEGER = v125 isotropy = the extension to (E8)_1 is local. [E] CRITICAL CORRECTION (why verdict B): the A3 deck diag(i,i^-1) induces z -> -z on the celestial sphere (projective order 2, the sheet flip) -- NOT the order-4 clock z -> iz; any SU(2) lift of the clock has order 8 (spin group Z8, would quotient to A7); exact bridge (spin clock)^2 = deck, |Z8| = 8 = 2|mu4| (the c3 = 1/(8 pi) winding integer); the order-4 clock IS realised as the U(2) phase diag(i,1) (order 4, fixed {0,inf}, free mu4 orbit -- all v180 properties; normalises the deck, det = i rotates the holomorphic symplectic form by the mu4 generator -- Kaehler, not triholomorphic). [E] GRAVITY SIDE: clock-invariance selects from the versal A3 family XY = Z^4 + a2 Z^2 + a1 Z + a0 EXACTLY the 1-parameter family XY = Z^4 + a0; its 4 branch points are ONE mu4 orbit with cross-ratio 2 -- the seam marks (v179 clock orbit; v168/v214 pillowcase j = 1728). [E] NEGATIVE CONTROLS: false spatial action diag(i,i) killed 3 ways (det = -1 not in SL(2,C); Poisson closure fails 14/14; grades unsymmetric); false glue (60/64 swap) breaks additivity on 3112/6720 root pairs; rigidity: exactly {id, j -> -j} = Aut(Z4) of 24 relabelings are additive. [C] the clock-deck identification is a NAMED extra identification (clock^2 = deck on the sphere, sheet-Z2 spin bookkeeping) -- verdict B, not A. [O] structural arithmetic ONLY: no celestial duality is constructed; CELEST.SEAM.01 (WP2-WP5) stays a research contract; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP1 [E]/[C]/[O]: E8 mu4-glue = inner order-4 torus element = flat Z4 monodromy on the A3 ALE; equivariant SDYM(E8) sector closes (60/64 towers, zero modes = carrier, density 1/4); (E8)_1 character = sum of the 4 glue sectors, weights = discriminant form; clock = U(2) diag(i,1), clock^2 = deck (Z8, 8 = 2|mu4|); clock-invariant deformation XY = Z^4 + a0, branch points = one mu4 orbit, cross-ratio 2; negative controls + Aut(Z4) rigidity; verdict B; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; SEAM.EQUIV.01; E8.GLU.01; v92; v125; v128; v168; v179; v180; v214; v377
Scripts: v492_celestial_z4_orbifold.py
Location: tfpt_research_contracts
Original CSV, line 555Original ledger status
CELEST.SEAM.01
THE CELESTIAL-HOLOGRAPHIC RESEARCH CONTRACT (2026-07-20; the fourth research contract alongside CONTRACT.U.01, CONTRACT.G.01 and CONTRACT.F.01): HYPOTHESIS (as corrected by WP1) -- the seam is the glue-equivariant Z4-orbifold sector of the celestial chiral algebra on the A3 ALE space C^2/Z4; the mu4 clock is the Kaehler U(2) phase diag(i,1) whose square is the deck group (spin bridge Z8, 8 = 2|mu4| = the c3 = 1/(8 pi) seam winding integer); the associated twistorial bulk is the SELF-DUAL sector of TFPT gravity. TYPING / NON-CIRCULARITY: admissible inputs = the finite seam/glue data (v92/v125 glue, v128 graded hull, v168/v214 pillowcase, v180 clock) + the WP1/WP2 [E] arithmetic (v492/v493); NOT admissible as input: the continuum existence of the holomorphic (E8)_1 net on the seam (exactly the SEAM.EQUIV.01 target -- assuming it would be circular); scope = the self-dual sector only, never full Einstein gravity (the v358/v359 entanglement-equilibrium route and GRAV.ENTROPIC.ACTION.01 are separate statements about the full equation). WORK PACKAGES: WP1 DONE (v492, verdict B -- match under the one named identification clock^2 = deck); WP2 DONE (v493, verdict B -- both targets landed and SHARPENED: a0 = pure seam SCALE, the tau = i pillowcase shape frozen (I = 12 a0, J = 0 identically, j = 1728 for all a0); the 3 spheres carry the 3 nontrivial mu4 characters, the clock = Coxeter element of W(A3) = the Picard-Lefschetz monodromy of the family, surviving direction = the chi_1-Fourier diagonal; BHS Z2 -> Z4 transfer with corrections linear in a0, grade-conserving, a0 <-> the c^2 slot); WP3 DONE (v495, verdict B -- exact arithmetic alignment EXISTS: lambda_e8 = 6 (unit-trace) resp. 1/10 (adjoint-trace), (kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300, the Okubo coefficient 5/(2(dim+2)) DERIVED for all 8 algebras with a sharp sl4 negative control and the closed form lambda~^2 = h_vee + 6 across the Deligne series; BUT the look-elsewhere battery shows the alignment format passes 8/8 -- ZERO selectivity; only g_car = 5 | h_vee is e8-selective (1/8), lambda-integrality 2/8 shared with sl3, the isolating conjunction POST HOC: the c3-connection survives as convention-level compatibility + genuine lambda-arithmetic, NOT as e8-selective evidence -- alignment survives, selectivity does not); WP4 DONE (v496, verdict B(ii) -- the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading, the mismatch localised THREE ways (spin unbounded below; growth n^{2/3} vs n^{1/2}, Meinardus poles s = 2 vs 1; the level-2 null ideal deletes exactly 27000 = 30^3 = h_vee^3 with no jet analogue), BUT the boundary/period reading holds exactly at the current stratum (zero modes = 60 = C0 currents, jet slice cycles (60,64,60,64), one mu4 period sums to 248, glue weights (0,1,1,1) integer) -- the SAME MMST scaling-limit structure as SEAM.EQUIV.01 (v336/v449): the character is a boundary/limit SHADOW of the S-algebra, the constructive limit question goes to WP5); WP5a DONE (v497, 2026-07-21 -- 'the null ideal from the limit': the boundary limit made a PRECISE coefficientwise limit via the chi_w family with stabilisation threshold w = n+1 (w = 2 = the chiral jet grading); the null ideal DERIVED from root data, Sym^2(248) = 27000 + 3875 + 1 with 27000 = h_vee^3 by Freudenthal + Weyl + peeling; quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control: four blocks, 5304 != 14^3, h = 1/2 breaks one-block fusion; the celestial route now has the SAME two-step shape as MMST -- limit + maximal ideal -- with the ideal quantitatively fixed); WP5b DONE (v498, 2026-07-21 -- 'the singular vector as an operator', SUCCESS on the preregistered criteria: |s> = (E^theta_{-1})^2|0> constructed EXPLICITLY in a machine-built Chevalley/Frenkel-Kac basis (cocycle asymmetry on all 57600 pairs, SGN = -1 forced by Jacobi, kappa derived), J^a_1|s> = 0 for all 248 generators with case tally 190/57/1 (exactly one case sees k), level dial 2(1-k): the deletion object exists ONLY at k = 1 and not in the centerless loop algebra; mu4: j(theta) = 1 (h-adapted chamber), clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 with the direct g_0-orbit BFS matching Freudenthal through depth 4, quotient 4124 = sector sum; controls: the 248-current state and generic level-2 states NOT singular, the cube at k = 2 (generic Kac mechanics, honestly typed), D8 has the singular vector too but h = 1/2 breaks one-block fusion -- one-block closure stays E8/mu4-specific; HANDOVER to WP5c: in the twisted quarter-slot moding two C_1 modes never sum to 8 quarters (minimum 6 = q^{3/2}) -- the per-period dictionary (v496), not the per-slot identification, carries |s> to q^2); WP5c DONE (v500, 2026-07-21 -- 'the GNS limit state', SUCCESS on the preregistered criteria: the quasi-free family omega_w (affine k=1 vacuum on the currents via the machine-determined compact anti-involution theta(e_a) = -e_{-a}, unique on all 61504 basis pairs, + x^{wr}-contracted radial oscillators) is positive for every finite w and stabilises EXACTLY at the WP5a threshold (omega_w = omega_inf mod x^{N+1} for w >= N+1, sharp at w = N); GNS(omega_inf) levels (1, 248, 4124) via the FULL exact 9361-block level-2 Gram: rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight (Freudenthal on all 9361 weights), every block PSD, rank table per Weyl orbit (0,0,1,8,44), h.w. norms (3844,49,2,0); the clock descends to GNS with level-2 rank split (1036,1024,1040,1024) = the per-period lattice theta-split -- the two-routes identity at the STATE level; |s> IS the zero vector of GNS(omega_inf), resolving the WP5b S4.10 twisted-slot tension; CCR obstruction: NO w-uniform damping state exists -- the family formulation is necessary AND the family exists, KILL not triggered; controls: k=2 keeps everything, k=0 has no current layer, D8 gives 2076 = one block of four, wrong family erases the 248 layer 710955 != 248, no damping keeps 897266 != 248); WP5d-alpha DONE (v501, 2026-07-21 -- 'the two-interval index from the lattice', the mu = 1 indicator chain PASSES: fermionic two-interval MI extensive (c fit 0.5000, max |F| = 6e-5 at N = 512, falling with N -- the mu = 1 reference), the sector-summed orbifold prescription pays exactly one classical bit with the 16-layer seam carrier saturating Delta_2 = ln 2 to machine precision (1.1e-15 at N = 512), so [F : F_even] = 2 and mu_gauged = 4 = the v490 parity census (two independent lattice witnesses); complementary-pair budget <= ln 4 = ln mu(SO(16)_1) as a double-limit statement + the orbifold duality-failure witness D(E) != D(E'); condensation arithmetic anchored at both ends: det Cartan(D5) x det Cartan(A3) = 16, KLM/Longo-Rehren 16/4^2 = 4/2^2 = 1, Sigma d^2 = (4,4,1), theta_v = 1 exactly at nu = 16 (rivals != 1) => mu = 1 after condensation, preregistered KILL 'mu-offset != 0 after condensation' NOT triggered; controls: nu = 1 offset non-removable, trivial phase all zero, wrong sector sum loses the full ln 2; WP5d-beta DONE (v504, 2026-07-21 -- 'split + strong additivity from the lattice', the two remaining KLM legs: strong additivity ALGEBRAICALLY EXACT with the shared boundary Majorana (Even(A) v Even(B) = Even(A u B), GF2 spans full 64/64..1024/1024, matrix rank 32/32; disjoint exactly index 2 = the v501 ln-2 bit, localised at the split point), the entropic touching defect BOUNDED < ln 2 with the Ising 1/4-exponent approach (p = 0.2444 vs 2 Delta_mu = 1/4) while the preregistered U(1)/Dirac control DIVERGES ((1/2) ln Var Q, Klich-Levitov pinned, infinite index) -- bounded-vs-divergent is the lattice discriminator and the quotient net sits on the bounded side; split via the elliptic-nome sigma_k ladder (rate pi K(1-x)/K(x) to 1.3-2.0%, sigma_1 ~ x^{1/2}, trace norm summable) with EXACT orbifold inheritance (C -> -C machine identity + the Lambda^2 C compound -- Longo heredity); Pimsner-Popa E(a) - a/2 = PaP/2 identically, lambda = 1/2 = 1/[F:F_even] with exact integer attainment (16384 + 2048 monomials, 0 violations), lambda_E4 = 1/4 = 1/mu, index consistency exp(Delta) = 2 = 1/lambda_PP over two independent routes: WITH v501 ALL THREE KLM INGREDIENTS of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice; the continuum uplift stays [O]: 'finite-group orbifolds of completely rational nets are completely rational' is XU'S THEOREM, cited not claimed -- the proof for the concrete seam quotient net and the interacting condensed (E8)_1 net remains WP5e/Costello-Li); WP5e: alpha stage (CFT-side prefactor + level pinning) executed via v502 (2026-07-21, CELEST.WP5E.ALPHA.01 -- the q^{-1/3} prefactor derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions) and k = 1 forced three independent ways (current condition h(J) = k = 1, conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0, c = 8 <=> 240(k-1) = 0, plus the WP5b singular-vector dial 31124 - 27000 = 4124), with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing); beta stage (equivariant twistor skeleton) executed via v505 (2026-07-21, CELEST.WP5E.BETA.01 -- the Atiyah-Bott/Lefschetz fixed-point skeleton of the inflow exact: denominators (2,4,2), characters (248,0,-8,0) two routes, per-sector Okubo with the RIGID 32 T3 twisted residual, index bridge f(m) = ch2(T_m) with glue defect -78 both routes, level dials k = 1 geometric (current count 240/0/0/0), and the E3 refutation a0 = graviton modulus O(2) NOT the axion O(-2) with the three sphere classes filling the three twisted axion slots; the preregistered kill 'inflow demands a level != 1' does NOT fire on the equivariant skeleton); gamma stage (the sphere-axion pairing check) executed via v508 (2026-07-21, CELEST.WP5E.GAMMA.01 -- an honest rigid NEGATIVE result with certificate: the W(D5) x W(A3)-invariant vertex spaces are exactly dim 2 (quadratics) and dim 5 (quartics), the product of any two invariant quadratics has zero T5/T3 content, so the exchange matrix is rank 2 with rank([M | A_fix]) = 3 and annihilator certificate (Phi_T5, Phi_T3, Phi_P)(A_fix) = (0, 32, 72) -- the three sphere axions CANNOT cancel the rigid 32 T3 residual by quadratic-vertex exchange, rule- and coupling-independently; the slot bijection is untouched and the preregistered level-kill still does not fire); WP5e: alpha (CFT side, v502) + beta (equivariant twistor skeleton, v505) + gamma (exchange no-go, v508) executed; the global BCOV quantisation stays [O] -- the exchange sub-branch is closed by v508; epsilon-2 executed via v509 (2026-07-21, CELEST.WP5E.EPS2.01, 'the CPS level-from-flux dial', verdict B): the one-level statement is now a THEOREM (lockstep from clock invariance -- one mu4 orbit of branch points, |Pi_j| = sqrt(2) t equal on all three spheres, det(A-1) = -4 no invariant flux vector; falsifier Z^4 - Z: 0/24 orderings lockstep), level = 1 carries the new sector-counter dial #prim((E8)_k) = 1 <=> k = 1 (affine marks: (1, 3, 5, 10, 15, 27) for k = 1..6), fluxes (64, 60, 64) with uniformly one quantum per charged current and the naive 'level = total flux' killed by the lockstep test itself; the PT/Z4 back-reaction stays [O]; delta-2 executed via v511 (2026-07-21, CELEST.WP5E.DELTA2.01, intermediate verdict, exact -- the full-tensor ledger: the v508 collapse is confirmed full-tensorially and arity-crossing by the innerness theorem (h in the g0 Cartan, all 15 non-neutral trilinear triples Hom = 0), but the su(4) d-symbol -- the unique symmetric trilinear on e8 -- opens the T3 channel (Q_dd = (1/60)(T3 - P3/4), Phi_T3(Q_dd) = 1/60 != 0), leaving the pairing obstructed only by the weaker certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 64, and exactly solvable relaxed with c_d = 1920 = |W(D5)|); delta-2 executed via v511 -- the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T3 with the weaker certificate psi = 64; delta-1 since DECIDED NEGATIVE under the DERIVED measure by v518 (2026-07-22, CELEST.WP5E.DELTA1.01: the derived chiral measure -- blockwise SL(2,Z) covariance solved for, the mu4 multiplier obstruction a CHARACTER (koboundary defects (1,1,1) on all 15 pairs, lambda(gamma) = i^{2B + C/4} on Gamma_1(4)) cancelled exactly by the twisted fibre block f1 f3 = G, NOT by the three sphere axions -- fails all three preregistered testers under both derived solutions chi_4/chi_10, with no (N1, N2) rescue in the positive cone: a genuine KILL on the derived surface, in stated TENSION with the DECLARED completion reading of v516 which delivers the psi = 64 slice and cancels 32 T3 -- both are exact, and which of the two is the true BCOV measure is the named [O]); the pre-v518 state was: the strict holomorphic reading is refuted at the Z2 anchor, the contact term is the modular completion (Harvey-Moore tau-integral) with the forced leading (T5,T3) ratio 4:3 -- the named remaining targets are the tau-integral and the physical justification of the cubic GS term (c_d = 1920; the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513; the convention-stable [E] core is c_d = 32 x 60), plus the epsilon-1 O(-2) bulk-axion construction (success: partition function = E4/eta^8 including the q^{-1/3} prefactor FROM THE TWISTOR SIDE; kill: anomaly mismatch at the Costello-Li grade -- neither the CFT-side dials of v502 nor the equivariant skeleton of v505 trigger it). KILL TESTS (pre-registered): (K1) the equivariant structure could have contradicted 52+64+60+64 -- SURVIVED by WP1 AND WP2 (false-glue control: 3112/6720 violations show the test has teeth; WP2 deformation side: no shape modulus survived clock-invariance, no sector leak -- the clock-variant controls fail two independent ways each); (K2) one-loop associativity demanding extra fields TFPT lacks -> contract dies (Costello's axionic mechanism passes E8 at the algebra level; the TFPT field content is the open half); (K3) EVALUATED by WP3 (v495): does NOT fire -- exact alignment exists ((kappa/c3)^2 = 12 resp. 1/300); but its SCOPE IS DEMOTED on the look-elsewhere evidence (the alignment format passes 8/8 across Costello's whole list): compatibility, not e8-selective evidence; (K4) EVALUATED by WP4 (v496): fires only against the exact-block reading -- the sector arithmetic holds exactly, so the contract does NOT degrade to 'E8 is an admissible celestial gauge algebra'; WP4 lands at the boundary-limit reading (verdict B(ii), the SEAM.EQUIV.01 scaling-limit shape). HONEST SUCCESS ESTIMATE: full success (WP5e, constructive holography at the Costello-Li grade) low probability; the partial-findings expectation is now REALISED AND EXCEEDED: WP1-WP4 + WP5a-WP5d all executed with sharp structural results (verdicts B, B, B, B(ii); ALL FIVE constructive WP5 milestones landed -- null ideal, singular vector, GNS limit state, two-interval index, split + strong additivity (both WP5d stages) -- plus the WP5e alpha stage (CFT-side prefactor + level pinning, v502) AND the WP5e beta stage (equivariant twistor skeleton with the honest a0 refutation, v505) AND the WP5e gamma stage (the exchange no-go with certificate, v508) AND the WP5e epsilon-2 stage (the CPS level-from-flux dial with the lockstep theorem and the sector-counter dial, v509); AND the WP5e delta-2 stage (the full-tensor ledger with the innerness theorem and the cubic d-symbol channel, v511); WP5e proper, the global BCOV quantisation, is the single remaining open piece -- narrowed by v508/v509/v511 to the delta-1 tau-integral (since DECIDED by v518, CELEST.WP5E.DELTA1.01: kill under the derived measure, all three preregistered testers fail; the measure tension with v516 is the named [O]), the physical justification of the cubic GS term (c_d = 1920; the |W(D5)| reading look-elsewhere-loaded and convention-contingent -- v513) and the epsilon-1 bulk-axion construction (executed via v514) plus the PT/Z4 back-reaction (M1 executed via v515: the back-reacted Omega_N closed-form with (2 pi i)^2-integral lockstep flux vector and forced source charge 4 = |mu4|; 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 clock^2 = deck correction, the Coxeter/Picard-Lefschetz monodromy, the non-selective GS alignment, the boundary-limit localisation of the character mismatch, the explicit singular vector with its level dial 2(1-k), the limit state with the ideal in its GNS kernel, the entropically measured mu-chain 16 -> 4 -> 1, and the lattice KLM completeness witness (split + strong additivity + finite mu, v501/v504). SEAM.EQUIV.01 stays [O] throughout; nothing here moves it.
Recorded status: Open (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)))
Dependencies, status detail and source
Canonical status: Open (research contract): seam = glue-equivariant Z4-orbifold sector of the CCA on the A3 ALE; clock^2 = deck (Z8 spin bridge); twistorial bulk = self-dual sector only; WP1-WP4 + WP5a-WP5d executed (verdicts B, B, B, B(ii); WP5a: null ideal derived, 27000 = h_vee^3, quotient = sector sum = 4124, stabilisation threshold w = n+1; WP5b: the singular vector explicit, J^a_1|s> = 0 on all 248 generators with case tally 190/57/1, level dial 2(1-k), j(theta) = 1, clock phase i^2 = -1), WP5c DONE (v500: GNS limit state omega_inf with the null ideal in its kernel -- level-2 rank 4124, kernel 27000 = V(2 theta) weight by weight, clock split (1036,1024,1040,1024), |s> = the GNS zero vector, CCR necessity of the family), WP5d-alpha DONE (v501: two-interval index -- 16-layer ln 2 plateau at 1e-15, [F : F_even] = 2, mu_gauged = 4 = v490, KLM chain 16 -> 4 -> 1 anchored at both measured ends, theta_v = 1 at nu = 16), WP5e-alpha DONE (v502: q^{-1/3} prefactor = exact mu4 vacuum-energy bookkeeping (shift orbifold, discriminant form = Casimir two ways) + k = 1 forced three ways, glue-h integrality honestly demoted), WP5e-beta DONE (v505: the equivariant anomaly ledger exact -- AB characters (248,0,-8,0), per-sector Okubo with the rigid 32 T3 residual, index bridge f(m) = ch2(T_m) with defect -78 two routes, geometric k = 1 dials, a0 refuted as GS axion (graviton slot O(2)), the three sphere classes = the three twisted axion slots; kill not fired), WP5d-beta DONE (v504: split + strong additivity from the lattice -- Even(A) v Even(B) = Even(AuB) EXACT with the shared boundary point (disjoint index exactly 2 = the v501 ln-2 bit), bounded-vs-divergent entropic discriminator (Z2 deficit < ln 2 with the Ising 1/4-exponent approach vs U(1) divergence, Klich-Levitov pinned), elliptic-nome split ladder with exact orbifold inheritance (C -> -C + Lambda^2 C, Longo heredity), Pimsner-Popa lambda = 1/2 = 1/[F:F_even] and lambda_E4 = 1/4 = 1/mu with integer attainment, index consistency exp(Delta) = 2 = 1/lambda_PP two routes -- all three KLM ingredients witnessed on the lattice; continuum uplift stays [O], Xu's theorem cited), WP5e-gamma DONE (v508: the sphere-axion pairing check -- exchange no-go with certificate: invariant vertex spaces exactly dim 2/5 by Weyl nullspace, product theorem kills every exchange image in the T3 direction, rank([M | A_fix]) = 3 with certificate (Phi_T5, Phi_T3, Phi_P) = (0, 32, 72), naturalness dissolves scale-independently, SO(16)/D8 controls; slot bijection untouched, level-kill still not fired), WP5e-epsilon-2 DONE (v509: the CPS level-from-flux dial, verdict B -- the one-level statement is now a THEOREM (lockstep from clock invariance, falsifier Z^4 - Z: 0/24), level = 1 carries the new sector-counter dial #prim((E8)_k) = 1 <=> k = 1, fluxes (64, 60, 64) with one quantum per current, the naive 'level = total flux' killed; the PT/Z4 back-reaction stays [O]), WP5e-delta-2 DONE (v511: the full-tensor ledger -- delta-2 executed via v511 -- the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T3 with the weaker certificate psi = 64; delta-1 since DECIDED NEGATIVE under the DERIVED measure by v518 (2026-07-22, CELEST.WP5E.DELTA1.01: the derived chiral measure -- blockwise SL(2,Z) covariance solved for, the mu4 multiplier obstruction a CHARACTER (koboundary defects (1,1,1) on all 15 pairs, lambda(gamma) = i^{2B + C/4} on Gamma_1(4)) cancelled exactly by the twisted fibre block f1 f3 = G, NOT by the three sphere axions -- fails all three preregistered testers under both derived solutions chi_4/chi_10, with no (N1, N2) rescue in the positive cone: a genuine KILL on the derived surface, in stated TENSION with the DECLARED completion reading of v516 which delivers the psi = 64 slice and cancels 32 T3 -- both are exact, and which of the two is the true BCOV measure is the named [O]); the pre-v518 state was: the strict holomorphic reading is refuted at the Z2 anchor, the contact term is the modular completion (Harvey-Moore tau-integral) with the forced leading (T5,T3) ratio 4:3 -- the named remaining targets are the tau-integral and the physical justification of the cubic GS term (c_d = 1920; the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513)), WP5e-epsilon-1 DONE (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] (the single remaining milestone: Costello-Li uplift -- the exchange sub-branch closed by v508, the flux dial executed by v509, the 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 named remaining target is the BCOV measure question (the delta-1 chain is DECIDED by v518: kill under the derived measure, tension with v516 stated; the cubic GS term stays dissolved)); kill tests evaluated: K1 survived, K3 not fired (scope demoted -- alignment non-selective, compatibility not evidence), K4 lands at the boundary-limit reading; SEAM.EQUIV.01 untouched [O]
Dependencies: CELEST.WP1.01; CELEST.WP2.01; CELEST.WP3.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; CELEST.WP5DB.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; CELEST.WP5E.M3.01; CELEST.WP5E.DELTA1.01; SEAM.EQUIV.01; CONTRACT.G.01; GATE.METRIC.01; v92; v125; v128; v168; v180; v214; v492; v493; v495; v496; v497; v498; v500; v501; v502; v504; v505; v508; v509; v511; v514; v515; v516; v517; v518
Scripts: v492_celestial_z4_orbifold.py
Location: tfpt_research_contracts
Original CSV, line 556Original ledger status
CELEST.WP2.01
WP2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 2, verdict B; companion of CELEST.WP1.01/v492): the clock-invariant deformation XY = Z^4 + a0 of the A3 seam orbifold -- geometry, period/monodromy household, and the Bittleston-Homans-Sharma deformed-algebra pattern (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case, fibre XY - Z^2 = -c^2) transferred to k = 4. Sympy exact throughout (47 checks). [E] GEOMETRY: P(iZ) = P(Z) forces a3 = a2 = a1 = 0 SHARPLY and two-sided (prod_k (Z - i^k w) = Z^4 - w^4: mu4-stable root set <=> clock-invariant quartic); XY = Z^4 + a0 smooth iff a0 != 0; disc = 256 a0^3; branch points Z^4 = -a0 = ONE free mu4 orbit in every fibre; cross-ratio 2 / j = 1728 for ALL a0 (binary-quartic invariants I = 12 a0, J = 0 IDENTICALLY) => a0 is a PURE SCALE, the tau = i pillowcase (v168/v214) is FROZEN -- no shape modulus survives clock-invariance. [E] PERIODS / MONODROMY: Pi = t(i-1)*(1, i, i^2) -- all 3 vanishing spheres carry the SAME holomorphic volume sqrt(2) t (lockstep, no relative modulus); the clock acts on H2 as a COXETER ELEMENT of W(A3) (char x^3+x^2+x+1, eigenvalues {i,-1,-i} = the A3 exponents 1,2,3; order = Coxeter number h(A3) = 4 = |mu4| = N_fam + 1); NO invariant cycle (det(A-1) = -4); the surviving deformation direction is the weight-1 chi_1-Fourier diagonal (1, i, i^2), NOT the naive invariant diagonal; the clock IS the Picard-Lefschetz monodromy of the family (a0 -> e^{2 pi i} a0 = the 4-cycle); versal weights (a3,a2,a1,a0) = (1,2,3,0) = the REGULAR mu4 representation, invariant line = a0 alone. [E] DEFORMED ALGEBRA (BHS Z2 -> Z4): mode dictionary {w[m,n]: m = n mod 4} <-> {X^p Y^q Z^r: r <= 3}; flat deformation (counting level); bracket {f,g} = -4*Nambu(XY - Z^4 - a0) ANCHORED at the a0 = 0 orbifold w-sector ({X,Y} = 16 Z^3, {Y,Z} = -4Y, {Z,X} = -4X), Jacobi + Casimir exact, uniform clock-degree -1; S-algebra corrections exactly LINEAR in a0 at the Z4 wrap r+r' >= 4, conserving the mu4 grade -- NO sector leak, so the v492 glue-equivariant SDYM(E8) sector deforms consistently; a0 <-> the BHS c^2 slot (weight 0, the order-c^2 splitting-function correction slot); matrix factorisations (McKay modules) deform along every 1+3 and 2+2 orbit splitting; the clock fixes exactly the antipodal 2+2 sector = the EH/Z2 middle node (4 = 2 x 2). [E] NEGATIVE CONTROLS (doubly sharp): a1 Z => j = 0 (hexagonal CM point, maximally non-square); pure a2 Z^2 stays SINGULAR (A1 node); a2 != 0 => j = 1556068/81 != 1728 (exact instance, cross-ratio and binary-quartic routes agree); grading leak explicit (Z^2 * Z^2 picks up grades {0,1,2} in the variant ring). [C] THREE NAMED IDENTIFICATIONS (why verdict B, not A): (1) the fibrewise -4*Nambu bracket is THE k = 4 CCA bracket (BHS prove k = 2; anchored here at a0 = 0, twistorial derivation not redone); (2) period = root difference (standard A-series residue normalisation); (3) the seam-scale reading of a0 presupposes the WP1 clock-deck bridge (clock^2 = deck, Z8, 8 = 2|mu4|). [C] v216 PRECISION (typed, NO marker move): GIVEN the order-4 clock, the clock-invariant deformation freezes the (2,2,2,2) modulus to the square AUTOMATICALLY (only the scale a0 stays free) -- a RELOCATION of the same order-4 carrier input to the celestial side (the clock IS the input), not a new derivation; v216 keeps its [O]-residual. [O] OPEN: full deformed twisted-sector OPEs; the W(mu)-type scaling-limit identification for k = 4 (BHS announce the ADE case); the spacetime splitting-function check; WP3 (GS coefficient alignment), WP4 (character vs S-algebra grading), WP5 (constructive holography); SEAM.EQUIV.01 (continuum (E8)_1 net on the seam) stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP2 [E]/[C]/[O]: clock-invariance selects XY = Z^4 + a0 sharply; a0 = pure seam scale, tau = i pillowcase frozen (j = 1728 for all a0); clock on H2 = Coxeter element of W(A3) = Picard-Lefschetz monodromy; surviving direction = chi_1-Fourier diagonal, spheres in lockstep; BHS Z2 -> Z4: -4*Nambu bracket anchored at the orbifold, corrections linear in a0, grade-conserving (no sector leak), a0 <-> c^2 slot; negative controls j = 0 / j != 1728 / grading leak; verdict B; SEAM.EQUIV.01 and v216 [O] untouched
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v168; v179; v180; v214; v216; v492
Scripts: v493_celestial_wp2_clock_deformation.py
Location: tfpt_research_contracts
Original CSV, line 557Original ledger status
CELEST.WP3.01
WP3 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 3, verdict B; the pre-registered 'alignment test only'): the Green-Schwarz axion coefficient lambda_g of celestial/twistorial SDYM (Costello, arXiv:2111.08879, eq. 1.2.4: kappa = lambda_g/(8 pi sqrt 3)) against the TFPT seam constant c3 = 1/(8 pi). Exact arithmetic throughout (Fraction/sympy, 25 checks). [E] OKUBO IDENTITY DERIVED, NOT QUOTED: Sum_alpha <alpha,x>^4 = [5/(2(dim g+2))] (Sum_alpha <alpha,x>^2)^2 as an exact polynomial identity for ALL 8 algebras on Costello's list (sl2, sl3, g2, so8, f4, e6, e7, e8; e6/e7 carved out of the v492 E8 root set) -- 'no independent quartic Casimir'; NEGATIVE CONTROL: sl4 (which HAS one) fails the proportionality (5/32 on s0^4 vs 3/32 on s0^2 s2^2) -- the list is sharp. [E] Killing normalisation Sum <alpha,x>^2 = 2 h_vee <x,x> exact (h_vee = 2,3,4,6,9,12,18,30); Deligne dimension formula dim g = 2(5h_vee-6)(h_vee+1)/(h_vee+6); CLOSED FORM lambda~^2 = 10 h_vee^2/(dim+2) = h_vee + 6 EXACTLY (8,9,10,12,15,18,24,36) -- fixes Costello-Paquette eq. (1.5); fundamental Dynkin indices i_R = (1,1,2,2,6,6,12,60) from explicit weight systems. E8: lambda~ = 6 (unit-trace) and lambda_fund = 1/10 (adjoint-trace) EXACT. Cross-checks: Costello's sl2 = 8, sl3 = 9 reproduced; HIS PRINTED lambda^2(so8) = 3/2 is a factor-2 slip (his own weight content gives Sum w^4 = 192, not 96: lambda^2 = 3, Okubo-consistent 12/4). [E] ALIGNMENT: (kappa/c3)^2 = lambda^2/3 = 12 = |mu4| N_fam (unit-trace) resp. 1/300 (adjoint-trace), both EXACT anchor rationals; kappa/c3 ITSELF is IRRATIONAL (2 sqrt 3 resp. 1/(10 sqrt 3)) -- the pre-registered 'lambda/sqrt3 rational?' reading FAILS, only the SQUARED alignment holds; anchor decomposition h_vee = 30 = |Z2| N_fam g_car (all three TFPT atoms once, squarefree), phi(30) = 8 = rank(E8), dim+2 = 250 = 2 * 5^3, lambda~ = 6 = |Z2| N_fam, lambda_fund = 1/10 = 1/(|Z2| g_car). [E] LOOK-ELSEWHERE (the HARD FENCE -- alignment != selection): the alignment test AS SUCH passes 8/8 (ZERO selective power for e8), h_vee {2,3,5}-smoothness 8/8, lambda~-integrality 2/8 (sl3 + e8), lambda_fund-rationality the same 2/8, phi(h_vee) = rank 4/8, phi(h) = rank 5/8; the ONLY e8-selective single test is g_car = 5 | h_vee (1/8); the conjunction that isolates e8 (1/8) was assembled POST HOC from correlated sub-tests -- it quantifies where e8 sits, it is NOT a preregistered discriminator. [E] KILL-TEST K3 EVALUATED: does NOT fire (exact alignment EXISTS) but its scope is DEMOTED -- the c3-connection of the celestial route survives as convention-level compatibility + genuine lambda-arithmetic, NOT as e8-selective evidence. [C] TYPED: the 8 pi in Costello's (1.2.4) is the 2 pi convention of the anomaly polynomial (his A.0.10 constant descended to spacetime), NOT an independent occurrence of the seam constant; the sqrt 3 is the anomaly's own (sqrt 48 = 4 sqrt 3 from the twistor box diagram, 48 = 2*4!); WP3 is an alignment yes/no ONLY -- it can never derive c3 from celestial data (contract wording, honoured). [O] WP5 (constructive holography) and SEAM.EQUIV.01 stay [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP3 [E]/[C]/[O]: Okubo coefficient 5/(2(dim+2)) derived for all 8 algebras (sl4 control); lambda~^2 = h_vee + 6 closed form; lambda_e8 = 6 resp. 1/10 exact; (kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300 exact, kappa/c3 irrational; so8 slip flagged; look-elsewhere: alignment 8/8 (zero selectivity), g_car | h_vee 1/8 (e8 only), joint test post hoc; K3 not fired, scope demoted -- compatibility, not e8-selective evidence; verdict B; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v492; v493
Scripts: v495_celestial_wp3_gs_alignment.py
Location: tfpt_research_contracts
Original CSV, line 558Original ledger status
CELEST.WP4.01
WP4 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 4, verdict B(ii); the 'type mismatch, head-on'): the (E8)_1 character E4/eta^8 against the glue-equivariant E8[C^2] S-algebra. Exact integer/Fraction/sympy arithmetic (25 checks). [E] BASE: v492 replication (240 = 52+64+60+64, dims with Cartan (60,64,60,64)); equivariant tower D[d] = 60(d+1) even / 64(d+1) odd; cumulative generator count QUADRATIC -- 31 s^2 + 92 s + 60 (even) / 31 s^2 + 94 s + 63 (odd) exactly, leading 31 = k + h_vee (arithmetic note); [C] CP Table 1 (arXiv:2201.02595): J[m,n] carries 2d spin 1 - (m+n)/2, UNBOUNDED BELOW along the tower => no rational VOA (L0 bounded below) contains the towers as graded pieces. [E] EQUIVARIANT HILBERT SERIES: chi_sp(t) = (60 + 128t + 60t^2)/(1-t^2)^2 (palindromic numerator, value 248 at t = 1); ALL mu4 collapses w in {1,-1,i,-i} and the SU(2)_+ primary slice (60+64t)/(1-t^2) fail to start (1, 248, ...) -- the 'row sum' route fails at the first coefficient. [E] CHARACTER SIDE: chi = E4/eta^8 = (1, 248, 4124, 34752, 213126, 1057504, 4530744) exact through q^6; machinery cross-check q j(tau) = E4^3/eta^24 = (1, 744, 196884, 21493760); theta split (shell 1 = (52,64,60,64)), sector currents (60,64,60,64), glue-diagonal h = (0,1,1,1) INTEGER (v125 isotropy = locality); Sugawara c = 248/31 = 8 = rank = 2|mu4| = g_car + N_fam. [E] BRIDGES: zero-mode slice D[0] = 60 = the C0-sector current count EXACTLY; the single-line jet slice z1^d cycles (60,64,60,64) sector by sector; BOUNDARY/PERIOD LIMIT: one full mu4 period of loop energies carries dim g_(n mod 4) = (64,60,64,60), sum = 248 EXACTLY -- the single-particle level of the 248q stage; 248 is NOT a jet-tower dimension (d <= 100) -- only the period collapse produces it; BUT the loop Fock at integer level 1 counts 897266 >> 248: the rational truncation (null ideal) is NOT in the jet/loop algebra -- it must come from a limit. [E] MISMATCH LOCALISED THREE WAYS: (a) GRADING (spin unbounded below; 248 no tower dimension); (b) GROWTH: jet Fock f = (1, 128, 8436, 381056) vs chi = (1, 248, 4124, 34752) -- TWO-SIDED failure (f1 < chi1 undercount, f2 > chi2 overcount), f_n/chi_n strictly increasing n = 1..12 (exact cross-multiplication); Meinardus poles s = 2 vs s = 1 => growth exponents n^{2/3} vs n^{1/2}, asymptotically DISJOINT counting types; (c) TRUNCATION: free-current level-2 count 31124 = 248 + Sym^2(248); the character keeps 4124 = 1 + 248 + 3875; the deficit is EXACTLY 27000 = 30^3 = h_vee^3 -- the (E8)_1 null ideal deletes the 27000 in Sym^2(248) = 1 + 3875 + 27000 ([C] standard) with NO analogue in the jet algebra (level 1 agrees: 248 = 248). [E] KILL-TEST K4 EVALUATED, three-way: (i) exact realisation NO; (ii) boundary-limit realisation PARTIAL YES -- current stratum, period sum 248, integer glue weights and theta split hold exactly; the missing piece is precisely the rational truncation, which must come from a scaling/compactification limit -- the SAME structure as the MMST scaling-limit route of SEAM.EQUIV.01 (v336/v449: the rational net exists in the LIMIT of the lattice collar, not in the pre-limit algebra); (iii) the hard mismatch is exactly located; K4 does NOT degrade the contract to 'E8 is an admissible celestial gauge algebra' (the sector arithmetic holds exactly). [C] VERDICT B(ii): the (E8)_1 character is a BOUNDARY/LIMIT SHADOW of the equivariant S-algebra, not a conformal block of it in the bulk (jet) grading -- WP4 sharpens into a statement of SEAM.EQUIV.01 type (rational structure emerges only in a limit). [O] the constructive boundary limit itself (which limit, which topology, convergence) stays OPEN and belongs to WP5/SEAM.EQUIV.01; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP4 [E]/[C]/[O]: the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading -- mismatch localised three ways (spin unbounded; growth n^{2/3} vs n^{1/2}; level-2 null ideal 27000 = h_vee^3 absent from the jets); BUT the boundary/period reading holds exactly at the current stratum (zero modes 60, jet slice (60,64,60,64), mu4 period sum 248, glue weights (0,1,1,1) integer) -- the MMST scaling-limit shape of SEAM.EQUIV.01; K4 fires only against the exact reading; verdict B(ii); constructive limit -> WP5; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492
Scripts: v496_celestial_wp4_character_shadow.py
Location: tfpt_research_contracts
Original CSV, line 559Original ledger status
CELEST.WP5A.01
WP5a OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5a, the first constructive milestone of WP5 -- 'the null ideal from the limit'; the v496 boundary-limit shadow made precise): exact integer/Fraction arithmetic throughout (34 checks), no floats. [E] STABILISATION THEOREM: a one-parameter family chi_w of graded Fock characters on the CHIRAL jet generators (loop energy m = p-q >= 1, radial number r = min(p,q); the m = 0, r >= 1 invariant-ring tower carries g_0), generator energy E_w = m + w*r in quarter units (u = q^{1/4}). (i) w = 2 IS the chiral jet grading (E_2 = p + q identically; generator counts 64,120,128,180,192,240,256,300 per degree); (ii) the u^n coefficient of chi_w equals the quarter-moded loop Fock for ALL w >= n+1 and is STRICTLY larger for every w <= n (verified n <= 8, w <= 10) -- the v496 'boundary-limit shadow' is the precise coefficientwise limit epsilon = 1/w -> 0 of a family CONTAINING the jet grading, with explicit threshold w = n+1; (iii) the limit does NOT produce the character (897266 >> 248 at integer level 1): limit and quotient are two separate, honestly typed steps; convention clarified: 897266 = quarter-moded loop Fock at u^4 = INTEGER level 1 (v496 S4.5). [E] NULL IDEAL DERIVED FROM ROOT DATA (not cited): Freudenthal multiplicity machinery (unit-tested on the adjoint: zero-weight mult 8, dim 248 by Weyl = orbit sum), dominant-weight tally of Sym^2(248) (30876 pairs, zero-weight mult 156) and generic character peeling DERIVE Sym^2(248) = V(2 theta) + V(omega) + V(0) = 27000 + 3875 + 1 (peeling residual exactly zero; independent Weyl-orbit cross-check); 27000 = 30^3 = h_vee^3 EXACTLY; C2 = (124, 96, 0) with 124 = C2(27000), Dynkin indices 6750/750 integers; level-1 integrability <2 theta,theta_vee> = 4 > 1 and <omega,theta_vee> = 2 > 1 => the only level-1 primary is the vacuum ([C] Kac); the maximal submodule of the level-1 vacuum Fock is generated by the level-2 singular vector of weight 2 theta ([C] Kac, used only through its counting consequence): level-2 quotient 31124 - 27000 = 4124 = 1 + 248 + 3875 = chi_2 EXACTLY. [E] TWO ROUTES, ONE NUMBER: the mu4 theta-split sector characters Theta_Cj/eta^8 (v492 S4 machinery, the boundary/twisted presentation) summed at q^2 = (1036,1024,1040,1024) give the SAME 4124 -- the untwisted null-ideal quotient and the mu4-sector lattice sum count the same level-2 space; at q^3 the sum is 34752 = chi_3; ideal bookkeeping beyond level 2 typed [C] (ideal series (0,0,27000,2600000,...) nonnegative, strictly increasing through q^12). [E] NEGATIVE CONTROLS: (a) SO(16)_1 through the SAME pipeline (D8 frame): Sym^2(120) = 5304 + 1820 + 135 + 1 derived (FOUR components vs E8's three); 5304 != h_vee(so16)^3 = 14^3 = 2744 -- the h_vee^3 identity is NOT generic; quotient 7380 - 5304 = 2076 = Theta_D8/eta^8 at q^2 (independent lattice route: the pipeline is the correct general level-1 vacuum recipe, validated on a second algebra); BUT the D8 block weights are (0, 1/2, 1, 1) -- h = 1/2 non-integer, the four blocks can NOT fuse into one local character (four-block theory) vs E8's integer (0,1,1,1) fusing into E4/eta^8: the procedure is not trivially projective; (b) false periodisation: period sums over P = 1, 2, 8 quarter-steps give 64, 124, 496 -- only P = 4 = |mu4| gives 248; (c) swapped glue (60/64 exchanged): the zero-mode anchor breaks (D[0] = 64 != 60 = lattice C0 currents). [C] NAMED IDENTIFICATIONS: 'radial decoupling = MMST collar thickness -> 0' (the v336/v449 lattice-collar route has the SAME two-step shape: pre-limit overcounting algebra, rational net only in the limit) is a named identification, not a computed isomorphism; the Kac singular-vector statement enters only through its counting consequence. [O] HONEST LIMIT (= the rest of WP5): the limit does NOT generate the truncation -- the singular vector as an OPERATOR in the limit net stays open (WP5b); the GNS limit state (kernel containing the ideal) and local normality / Haag duality of the limit net stay open (WP5c/WP5d); convergence topology beyond coefficientwise stays open; the celestial route now has the SAME two-step shape as MMST (limit + maximal ideal), with the ideal quantitatively fixed. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5a [E]/[C]/[O]: the boundary limit is a precise coefficientwise limit -- chi_w family (E_w = m + w r, quarter units) contains the chiral jet grading at w = 2 and stabilises to the quarter-moded loop Fock at threshold w = n+1 (exact, n <= 8, w <= 10); the limit does NOT produce the truncation (897266 >> 248) -- but the ideal SIZE is DERIVED from root data (Sym^2(248) = 27000 + 3875 + 1, 27000 = h_vee^3, Freudenthal + Weyl + peeling) and the quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control (four blocks, 5304 != 14^3, h = 1/2 breaks fusion); only P = |mu4| = 4 gives 248; open: singular vector as operator (WP5b), GNS limit state (WP5c), local normality/Haag duality (WP5d), Costello-Li uplift (WP5e); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492; v496
Scripts: v497_celestial_wp5a_null_ideal.py
Location: tfpt_research_contracts
Original CSV, line 562Original ledger status
CELEST.WP5B.01
WP5b OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5b, the second constructive milestone of WP5 -- 'the singular vector as an operator'; WP5a/v497 fixed the ideal's size and location, WP5b constructs the deleting object): exact integer/Fraction arithmetic throughout (53 checks), no floats, deterministic. [E] CONSTRUCTION: |s> = (E^theta_{-1})^2|0>, theta the highest root, weight 2 theta (the top of the 27000), level 2 = 8 quarter units = q^2 (an INTEGER level); the Chevalley/Frenkel-Kac basis is MACHINE-built from the v497 root data (bimultiplicative 2-cocycle eps(a,b) = (-1)^(m_a^T B m_b), asymmetry verified on all 57600 ordered root pairs; the [e_a,e_{-a}] = SGN*H(a) sign FORCED by the Jacobi identity, SGN = -1, not chosen; the invariant form kappa DERIVED from invariance, kappa(theta_vee,theta_vee) = 2; Jacobi exact on ALL 57600 sl2-slice triples + 6000 seeded triples + all 496 theta-adjacent triples; all root-sum structure constants N_ab = +-1). [E] SINGULARITY (main theorem, machine): J^a_1|s> = 0 for ALL 248 basis elements, 0 violations; case classification of the J_1 condition: 190 vanish at the first bracket, 57 at the second, exactly 1 (a = F^theta) by the central-term cancellation -- the ONLY case that sees the level k; plus J^a_2|s> = 0 (all 248), E^a_0|s> = 0 (all 120 positive roots), h_i-eigenvalues = <2 theta, a_i_vee> exactly, Shapovalov norm 0 at k = 1 (4 at k = 2); the affine PBW engine ([J^a_m,J^b_n] = [a,b]_{m+n} + m k kappa(a,b) delta_{m+n,0}) unit-tested on all 61504 basis pairs. [E] LEVEL DIAL: the F^theta_1 coefficient is 2(1-k) -- 2 at k = 0, 0 EXACTLY at k = 1, -2 at k = 2: in the centerless loop algebra (k = 0) the deletion object does NOT exist -- the central extension is load-bearing. [E] MU4 DATA (glue frame, v492/v496): theta-grading dims (1,56,134,56,1), g_0 = e7 + u(1); the h-adapted Weyl chamber MACHINE-constructed (8 simple roots re-derived, Cartan det 1, E8 Dynkin diagram with branch lengths [1,2,4]); theta_glue the unique dominant root, height 29 = h_vee - 1, <theta,h> = 5 => glue class j(theta) = 1; clock phase on |s> = i^(2j) = -1; class(2 theta) = 2 sheet-EVEN (consistent with WP5a S6.8); the deleting theta-sl2 crosses the sheet-ODD classes (1,3) while its square is sheet-even; E^theta expressible in the sector-C_1 equivariant variables (spatial degree m = -j mod 4, 2 monomial slots) and the square is mu4-INVARIANT; quarter bookkeeping (v496 per-period dictionary): E^theta_{-1} = 4 quarter units = one full mu4 period (248 currents), |s> = 8 quarters = q^2; cross-table theta-grading x glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)). [E] LEVEL-2 MATCH: weight 2 theta has multiplicity 1 in the level-2 Fock => U(g)|s> is THE 27000; dim V(2 theta) = 27000 = 30^3 = h_vee^3 (Freudenthal + Weyl-orbit sums, v497 replication); the direct g_0-lowering orbit BFS from |s> reproduces the Freudenthal weight multiplicities of V(2 theta) EXACTLY, weight by weight, through depth 4; quotient 31124 - 27000 = 4124 = theta-split sector sum (1036,1024,1040,1024); LATTICE KERNEL READING: (2theta,2theta)/2 = 4 > 2, so the quotient (E8)_1 theory has NO momentum-2theta state at q^2 -- |s> IS the kernel generator of Fock -> E4/eta^8. [E] NEGATIVE CONTROLS (honest separation): (a) E^theta_{-1}|0> (level 1) NOT singular -- exactly one witness (a = F^theta, -> -k|0>), the 248-current state that must REMAIN (chi_1 = 248); (b) 4 seeded generic level-2 states all non-singular (14-33 witnesses each); (c) at k = 2 the square is not singular but the CUBE (E^theta_{-1})^3|0> IS -- the (E^theta)^{k+1} pattern is GENERIC Kac level mechanics, not E8-specific; (d) D8/SO(16)_1 through the SAME pipeline (Chevalley + Jacobi on all 12544 sl2-slice triples): (E^theta_{-1})^2|0> IS singular there too (the singular-vector mechanism is level-1 GENERIC), BUT dim V(2 theta_D8) = 5304 != 14^3 and D8 keeps 3 extra level-1 primaries (comark-1 weights, h = (1/2,1,1), h = 1/2 NON-integer breaking one-block fusion) vs E8 (comarks all >= 2, vacuum the ONLY level-1 primary, glue weights (0,1,1,1) integer) -- the ONE-BLOCK closure is the E8/mu4-specific part; the singular vector itself is generic level-1 Kac mechanics, honestly separated. [C] NAMED IDENTIFICATIONS: Weyl complete reducibility of the finite-dim level-2 Fock extends the depth-4 orbit agreement to the full dim U(g)|s> = 27000; 'the singular vector generates the maximal submodule' (Kac) used only beyond level 2; the per-period quarter dictionary (v496 S4) for the u-unit bookkeeping. [O] HONEST LIMIT (= the precise WP5c question): in the TWISTED quarter-SLOT moding of the chi_w limit Fock two sector-C_1 modes never sum to 8 quarters (0 solutions, minimum total 6 = q^{3/2}) -- the per-PERIOD dictionary (v496), not the per-slot identification, carries |s> to q^2; exactly the GNS/limit-state seam (kernel containing the ideal) that WP5c must construct; local normality/Haag duality (WP5d) and Costello-Li (WP5e) stay OPEN. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5b [E]/[C]/[O]: the deletion operator EXISTS and is explicit -- |s> = (E^theta_{-1})^2|0> in a machine-built Chevalley/Frenkel-Kac basis, J^a_1|s> = 0 on all 248 generators (case tally 190/57/1, exactly one case sees k), level dial 2(1-k): only k = 1 deletes, no deletion object in the centerless loop algebra; mu4-compatible: j(theta) = 1, clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 (orbit BFS = Freudenthal through depth 4), quotient 4124 = sector sum; controls: the 248-current state and generic states not singular, cube at k = 2 (generic Kac mechanics), D8 contrast (5304 != 14^3, h = 1/2 breaks fusion) -- one-block closure E8/mu4-specific; open: the twisted-slot moding carries |s> only via the per-period dictionary -- the precise WP5c GNS/limit-state question; WP5c-e stay [O]; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP4.01; SEAM.EQUIV.01; v128; v492; v496; v497
Scripts: v498_celestial_wp5b_singular_vector.py
Location: tfpt_research_contracts
Original CSV, line 563Original ledger status
CELEST.WP5C.01
WP5c OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5c, the third constructive milestone of WP5 -- 'the GNS limit state'; WP5a/v497 made the boundary limit precise, WP5b/v498 built the deleting operator, WP5c constructs the STATE whose GNS kernel contains the null ideal): exact integer/Fraction arithmetic throughout (35 checks), no floats, deterministic. [E] CONSTRUCTION: the quasi-free family omega_w -- loop sector (r = 0) = the affine k=1 vacuum n-point functions via the v498 PBW engine and the MACHINE-DETERMINED compact anti-involution theta(e_a) = -e_{-a}, theta(h) = h (the UNIQUE sign that is an anti-automorphism on ALL 61504 basis pairs; contravariance <J^x_{-1} u, v> = <u, J^{theta x}_1 v> exact); radial sector (r >= 1) = oscillator pairings x^{w r} (the exact Gibbs regulator e^{-beta E_w}, kept x-adic; rational sample x = 1/2). [E] CONVERGENCE SHARP: loop entries w-independent; radial entries x^{w r} have x-adic order w r >= w, so omega_w = omega_inf mod x^{N+1} for ALL w >= N+1 (N = 8) and NOT yet at w = N (the r = 1 modes sit at order N exactly) -- the WP5a stabilisation threshold w = n+1 holds IDENTICALLY at the state level; positivity of every omega_w checked at the one-particle level (all 36 radial modes). [E] THE KERNEL (core result, fully machine): level-2 Fock = 31124 monomials in 9361 weight blocks (dim profile 164:1, 37:240, 7:2160, 1:6960); the FULL exact Gram computed block by block: level-1 rank 248 POSITIVE DEFINITE (240 root modes at norm k, Cartan block = the E8 Cartan matrix, det 1); level-2 rank 4124 EXACTLY (the preregistered target check), kernel 27000, every one of the 9361 blocks PSD; rank per Weyl orbit (norm 8/6/4/2/0) = (0,0,1,8,44), and (dim, rank, inertia) CONSTANT along every Weyl orbit (verified on all blocks, no orbit assumption); kernel = V(2 theta) WEIGHT BY WEIGHT (rank_lambda = dim_lambda - mult_lambda(V(2 theta)) for all 9361 weights, Freudenthal cross-check); component route: unique highest-weight vectors at (0, omega, theta, 2theta) with norms (3844, 49, 2, 0); direct ideal checks F_i|s> and F_j F_i|s> norm 0 and orthogonal to their full blocks; GNS graded dims (1, 248, 4124) = E4/eta^8 through q^2. [E] MU4 EQUIVARIANCE: frame map glue -> standard via the UNIQUE E8 diagram isomorphism; omega_w o clock = omega_w for ALL w (nonzero vacuum values only at total weight 0 = clock class 0); the clock unitary DESCENDS to GNS(omega_inf); GNS level-2 rank split by clock class = (1036, 1024, 1040, 1024) = Theta_Cj/eta^8 at q^2, level-1 split (60,64,60,64) -- the WP5a two-routes identity now holds at the STATE level. RESOLUTION OF WP5B S4.10: |s> IS the ZERO VECTOR of GNS(omega_inf) -- nothing needs to 'carry' it to q^2; the twisted two-C1-mode floor (6 quarters = q^{3/2}) concerns only pre-limit objects (GNS has no state at 6 quarters: 6 mod 4 != 0). [E] NEGATIVE CONTROLS: (a) k = 2: <s|s> = +4 (WP5b sign confirmed), all 240 norm-8 and all 6720 norm-6 monomials nonzero, dominant blocks FULL rank -- the GNS kernel contains NO level-2 ideal (the deletion is a k = 1 state-level fact); (b) k = 0: level-1 Gram identically 0 -- no current layer without the central extension; (c) D8/SO(16)_1 same pipeline: level-1 rank 120 PD, FULL level-2 Gram over 7380 monomials in 2705 blocks: rank 2076 = 7380 - 5304, kernel = V(2 theta_D8) weight by weight, PSD, lattice route Theta_D8/eta^8 = (1, 120, 2076) agrees; but h = (0, 1/2, 1, 1): the vacuum GNS is 1 of 4 blocks vs E8 = the FULL character -- ONE-BLOCK completeness of the limit state is E8/mu4-specific; (d) wrong family E'_w = m w + r: level-1 limit rank 0 (the 248 layer ERASED), u^4 survivor content 710955 != 248 -- the kill has teeth; (e) no damping: the w-uniform undamped vacuum keeps the full jet Fock, 897266 != 248; (f) CCR OBSTRUCTION: on a FIXED radial pairing c > 0 every state has omega(b b*) >= c (positivity) -- NO w-uniform state damps the radial modes: the contract's family formulation is NECESSARY, and the family EXISTS => the preregistered KILL is probed and NOT triggered. [C] NAMED IDENTIFICATIONS: the quasi-free/Gaussian extension of the positive one-particle data to the full *-algebra (standard); Kac positivity beyond level 2 (levels <= 2 machine-checked here); complete reducibility / multiplicity-freeness in the component route (the rank is [E], the module identification is the [C] part); k = 2 non-dominant blocks by Weyl-orbit rank constancy. [O] HONEST LIMIT: local normality of omega_inf, the net structure / Haag duality (WP5d), Costello-Li (WP5e), and the operator-algebraic closure of the radial contraction beyond the graded/x-adic topology stay OPEN. VERDICT: SUCCESS on the preregistered criteria at the graded-state level (target check dim GNS level-2 = 4124 met exactly). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5c [E]/[C]/[O]: the GNS limit state EXISTS and is explicit -- the quasi-free family omega_w (affine k=1 vacuum + x^{wr}-contracted radial oscillators; compact adjoint machine-determined on all 61504 pairs) is positive for every finite w, stabilises exactly at the WP5a threshold w = N+1, and its limit has GNS levels (1, 248, 4124): full 9361-block exact Gram, level-2 rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight, every block PSD, rank table (0,0,1,8,44); the 248 layer survives (level-1 rank 248 PD); the clock descends to GNS with rank split (1036,1024,1040,1024) = the per-period theta-split; |s> is the GNS zero vector (WP5b S4.10 resolved at the state level); CCR obstruction: no w-uniform damping state -- the family is necessary AND exists (KILL not triggered); controls: k = 2 keeps everything, k = 0 has no current layer, D8 = one block of four, wrong family/no damping caught (710955/897266 != 248); open: local normality, net/Haag duality (WP5d), Costello-Li (WP5e); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP5B.01; SEAM.EQUIV.01; v128; v492; v496; v497; v498
Scripts: v500_celestial_wp5c_gns_limit_state.py
Location: tfpt_research_contracts
Original CSV, line 564Original ledger status
CELEST.WP5D.01
WP5d-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, alpha stage -- 'the two-interval index from the lattice'; the local-net question): numpy/scipy Gaussian machinery + exact Fractions (39 checks). Roadmap discriminator: (E8)_1 => mu = 1 (Haag-dual two-interval net, ONE sector); SO(16)_1 => mu = 4; preregistered KILL = 'the two-interval mu-offset does NOT vanish after the condensation'. [E-machine]/[L] SETUP: the v367 seam edge as 16 chiral Majorana layers (c_- = 8 = g_car + N_fam, v489), faithful 1d reduction = the critical Majorana ring (v450 style, NS closure, c = 1/2 per layer); Peschel-exact; three two-interval prescriptions (fermionic rho / orbifold sector sum {1, P_A, P_B, P_AB} / dual algebra {1, P_AB}); the superselection identity [rho, P_AB] = 0 machine-exact; EVERY Gaussian formula (covariance round trip, Peschel entropies, trace formula, interval parity = Pfaffian, disjoint Renyi-2 deficit) validated against exact diagonalisation to <= 1.5e-15. [L/N] FERMIONIC MI EXTENSIVE (the mu = 1 reference): I(x) = -(c/3) ln(1-x) with c fitted 0.5000; max |F(x)| = 5.97e-5 at N = 512 over the full x grid, |F| falling monotonically to 1.3e-6 at N = 1024 -- the lattice witness of Casini-Huerta extensivity. [L/N] THE MU-OFFSET (core): Renyi-2 exactly Delta_2 = -ln[(1+R)/2] with R = Tr(rho P rho P)/Tr rho^2; the 16-layer seam carrier (diagonal Z2) saturates ln 2 to MACHINE PRECISION (|Delta_2 - ln 2| = 1.1e-15 at N = 512, 0 at N = 1024/2048; single-layer R falls as N^-0.500, Ising disorder decay); ED (N = 12..24) confirms vN and Renyi-3/4 deficits track it (not a Renyi-2 artefact); index reading [F : F_even] = exp(Delta_inf) = 2.000000 => mu_gauged = exp(2 Delta_inf) = 4.000000 = v490's mu_pre (TWO independent lattice witnesses: parity census there, entropic two-interval offset here); gauging every layer separately costs 16 ln 2 -- the offset measures ln[index of the subnet], not a universal blob. [L/N] HAAG-DUALITY INDICATOR: fermionic complementarity S(E) = S(E') to 8.2e-12 (graded-duality proxy); the orbifold BREAKS it (D(E) - D(E') = 0.427 at a = 64 -- the direct lattice witness that Even(E)-prime != Even(E-prime)); the complementary-pair sum is exchange-symmetric (dev 4.3e-14) and bounded by ln 4 = ln mu(SO(16)_1), max 1.2637 at the self-dual x = 1/2, -> ln 2 at the squeezed ends: the ln 4 KLM budget is a DOUBLE-LIMIT statement (sum of the two one-sided ln 2 saturations), NOT a fixed-x identity -- the first naive fixed-x check was WRONG and was replaced by the honest budget + asymmetry pair; the dual-algebra prescription {1, P_AB} is machine-identically defect-free. [E] CONDENSATION ARITHMETIC, LATTICE-ANCHORED: det Cartan(D5) * det Cartan(A3) = 4 * 4 = 16 = mu(carrier); KLM/Longo-Rehren quotients mu/|L|^2: 16/4^2 = 1 (one-step mu4 glue) and 16/2^2 = 4 -> 4/2^2 = 1 (via SO(16)_1), both routes exact Fractions; Sigma d_i^2 cross-checks (Ising 1+1+2 = 4, SO(16)_1 four abelian sectors = 4, (E8)_1 one sector = 1); theta_v = e^{2 pi i nu/16} = 1 EXACTLY at nu = 2 c_- = 16 (bosonic, condensable, v490), rivals nu = 1, 2, 8 give theta_v != 1. THE ANCHORED CHAIN: mu = 1 (graded fermion, MEASURED offset 0) -> 4 (gauged, MEASURED ln 2 offset + v490 parity census) -> 1 (theta_v = 1 extension, ARITHMETIC [C]: the condensed (E8)_1 net is interacting, not free-fermion computable -- v490's honest fence): the lattice pins BOTH measurable ends, the arithmetic closes the middle. [E/N] NEGATIVE CONTROLS: (a) nu = 1 single Majorana: offset non-trivial (Delta_2 = 0.6364 at N = 2048, rising toward ln 2) and theta_v = e^{i pi/8} != 1 -- NO bosonic condensation channel, the offset NOT removable: the KILL discriminator has teeth; (b) trivial dimerised phase: c = 0.00000, MI <= 1.3e-7, 16-layer deficit <= 2.6e-8 -- the ln 2 is a property of the CRITICAL seam edge, not of the prescription; (c) wrong sector sum ({1, P_AB} only, 2 of 4 structures): offset EXACTLY 0 instead of ln 2 -- measurably wrong bookkeeping, as preregistered. [C] the continuum dictionary 'offset = ln index' (KLM two-interval index; Longo-Xu; Casini-Huerta-Magan-Pontello entropic order parameters) and the condensation step itself (theta_v = 1 simple-current extension). [O] continuum Haag duality, local normality, split/strong additivity of the quotient net (WP5d-beta, running as exploration), Costello-Li (WP5e). VERDICT: the mu = 1 indicator chain PASSES; the preregistered KILL is NOT triggered. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal [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)
Dependencies, status detail and source
Canonical status: WP5d-alpha [E]/[L]/[N]/[C]/[O]: the two-interval mu-index measured entropically on the seam lattice edge -- fermionic MI extensive (mu = 1 reference, c fit 0.5000, |F| -> 0), the sector-summed orbifold pays exactly one classical bit (16-layer carrier ln 2 plateau to 1.1e-15 at N = 512), [F : F_even] = 2, mu_gauged = 4 = the v490 census (two independent lattice witnesses); complementary-pair budget <= ln 4 (double-limit statement) + the orbifold duality-failure witness; condensation arithmetic anchored at both measured ends: 16 -> 4 -> 1 (KLM/Longo-Rehren, Sigma d^2, theta_v = 1 exactly at nu = 16, rivals != 1) => mu = 1 after condensation, preregistered KILL not triggered; controls: nu = 1 non-removable, trivial phase zero, wrong sector sum loses the full ln 2; open: WP5d-beta (continuum Haag duality / local normality / split, exploration running) + WP5e (Costello-Li); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5C.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v378; v450; v489; v490
Scripts: v501_celestial_wp5d_two_interval_index.py
Location: tfpt_research_contracts
Original CSV, line 565Original ledger status
CELEST.WP5E.ALPHA.01
WP5e-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, alpha stage -- 'prefactor and level bookkeeping', the CFT-side anchor of the BCOV/Costello-Li twistor uplift): exact sympy/Fraction arithmetic, no floats (33 checks). Roadmap target: boundary partition function = E4/eta^8 INCLUDING the q^{-1/3} prefactor; preregistered KILL = 'the anomaly inflow demands a level != 1'. [E] PART A -- THE PREFACTOR IS EXACT mu4 VACUUM-ENERGY BOOKKEEPING: Hurwitz-zeta derivation (zeta(-1,theta) = -B2(theta)/2 symbolic; E_b(theta) = -1/24 + theta(1-theta)/4 exact; fermion counterpart E_f = -E_b, Ising h_sigma = 1/16 per Majorana); the clock is INNER -- h = (2,2,2,2,2;0^4) and the A3 detector h' = (0^5;1,1,1,-3) BOTH read the glue class mod 4 on all 240 roots ((h,h) = 20, (h',h') = 12, sum 32 = 2*4^2) -- so ALL 8 Cartan/torus-boson directions sit at clock eigenvalue 1: twist distribution theta = 0^8 in EVERY sector C_j, a SHIFT orbifold (lattice cosets, integer-moded oscillators), NOT a rotation orbifold; E_osc(C_j) = 8*(-1/24) = -1/3 = -c/24 SECTOR-INDEPENDENT (c = 8 = rank = Sugawara 248/31 -- the central-charge route and the boson-count route agree); sector ground exponents -1/3 + (0,1,1,1). [E] DISCRIMINANT FORM = CASIMIR ENERGY, TWO ROUTES: coset minima h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8), glue diagonal (0,1,1,1) INTEGER; route 1 = SPECTRAL FLOW (s_j = j h/4 costs j^2(h,h)/32: 5j^2/8, 3j^2/8, sum j^2 = 0 mod 1 = the isotropy/Lagrangian condition); route 2 = FREE FERMIONS, exact not just mod 1 (D5 = 10 Majorana -> 5/8, A3 = D3 = 6 -> 3/8, glue diagonal = 16 = 2^(g_car-1) = the WP5d seam carrier -> 1; R-NS shift 1/16 per Majorana); HONEST SHARPENING (i): the ROTATION reading fails -- the geometric A3 deck diag(i,i^-1) gives twisted Casimir 3/16 != 3/8 (the v492 'clock != deck' lesson at the Casimir level, factor 2). [E] CHARACTER RECONSTRUCTION: all glue-coset norms even (0 odd cross-terms: isotropic diagonal <=> even extended lattice = E8); sum of the four Theta_Cj = E4 = (1,240,2160,6720); sector characters have leading exponents -1/3 + h_j, leading coefficients (1,64,60,64), currents (60,64,60,64), and sum EXACTLY to E4/eta^8 = q^{-1/3}(1,248,4124,34752). [E] PART B -- k = 1 FORCED THREE INDEPENDENT WAYS: (i) the CURRENT CONDITION -- simple-current weights are exact closed forms (D5: h(k om_v) = k/2, h(k om_{s,c}) = 5k/8; A3: h(k om_j) = k j(4-j)/8), the Z4 extension closes onto the E8 ADJOINT (60+64+60+64 = 248 spin-1 currents) only if h(J) = 1, and h(J; k) = k => k = 1 uniquely; HONEST SHARPENING (ii): glue-h INTEGRALITY h(J^j; k) = k(0,1,1,1) holds for ALL k = 1..8 -- the originally conjectured integrality route fixes NOTHING (the naive dial is dead; the three other dials carry); (ii) CONFORMAL EMBEDDING: D5+A3 in E8 reduces to 47k^2 + 219k - 266 = 47(k-1)(k+266/47) = 0 (unique positive root k = 1), D8 in E8 to 128k(1-k) = 0 (128 = spinor dim); (iii) CENTRAL CHARGE = THE PREFACTOR ITSELF: 248k/(k+30) = 8 <=> 240(k-1) = 0 (240 = #roots), -c/24 = -1/3 only at k = 1; PLUS the WP5b singular-vector dial: PBW engine ||(E^theta_{-1})^m 0||^2 = m! k(k-1)...(k-m+1), singular vector EXACTLY at grade k+1, Weyl dims from root data (248, 3875, 27000; Sym^2 248 = 30876), k = 1 deletes V(2 theta): 31124 - 27000 = 4124 = chi_2 (singular-vector consistency); k = 2: <s|s> = +4 = the WP5b/WP5c value. [E] NEGATIVE CONTROLS: (a) D8/SO(16)_1 -- same rank 8, same c = 8, SAME q^{-1/3} prefactor, but coset weights (0,1/2,1,1): the prefactor does NOT discriminate E8 vs SO(16), the h-INTEGRALITY of the glue diagonal does; Theta_D8 + Theta_s = E4 (the {1,s} Lagrangian extension IS E8); (b) WRONG TWIST: Z2 rotation gives E = +1/6 (h = 1/2 -- no mu4-shift sector there); a mu4 rotation makes E_j = (-1/3, 1/24, 1/6, 1/24) sector-DEPENDENT (the common prefactor and the single eta^8 break); (c) WRONG k in {2,3,4}: ALL FIVE dials fail coherently (h(J) = k != 1; embedding residuals 360/814/1362; c != 8; prefactor != q^{-1/3}; <s|s> = 4/12/24 so 27000 survives, level-2 dim stays 31124 != 4124); at k = 1 all five pass. [C] Kac generation of the maximal submodule by the grade-(k+1) singular vector for k >= 3 (k = 1, 2 machine-anchored by WP5b/WP5c). [O] HONEST FENCE -- WP5e is NOT closed: the BCOV/Costello-Li construction itself (the boundary partition function from the twistor side, the actual anomaly-inflow computation on PT, Costello's one-loop axion coupling lambda_g/(8 pi sqrt 3), the Costello-Paquette-Sharma level-from-flux mechanism) stays OPEN; v495 anchors the Lie-algebra arithmetic of the axion coefficient only; the KILL branch is NOT triggered on the CFT side -- all computable dials say k = 1 -- but its twistor side is untested. v502 anchors the CFT side of WP5e (alpha stage). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5e-alpha [E]/[C]/[O]: the q^{-1/3} prefactor of E4/eta^8 derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3 at c = 8; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions; rotation reading fails 3/16 != 3/8) and k = 1 forced three independent ways (current condition h(J) = k = 1; conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0; c = 8 <=> 240(k-1) = 0; plus the WP5b singular-vector dial 31124 - 27000 = 4124) with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing; controls: D8 same prefactor (discriminator = h-integrality), rotation twists break the common prefactor, wrong k fails all five dials; the BCOV/twistor inflow itself stays [O] -- WP5e not closed, SEAM.EQUIV.01 untouched
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; SEAM.EQUIV.01; v492; v496; v497; v498; v500; v501
Scripts: v502_celestial_wp5e_prefactor_level.py
Location: tfpt_research_contracts
Original CSV, line 566Original ledger status
CELEST.WP5DB.01
WP5d-BETA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, beta stage -- 'split + strong additivity from the lattice'; the two remaining Kawahigashi-Longo-Mueger legs of complete rationality = split property + strong additivity + finite two-interval mu-index, the index being WP5d-alpha/v501): numpy/scipy Gaussian machinery + exact GF2/integer algebra (37 checks, deterministic, ~12 s); the U(1)/Dirac number-conserving subnet is the preregistered negative control -- the current-net analogue where strong additivity famously FAILS in the continuum (Buchholz-Schulz-Mirbach). [E] STRONG ADDITIVITY, ALGEBRAICALLY EXACT: 'touching' = shared boundary Majorana => Even(A) v Even(B) = Even(A u B) EXACTLY (GF2 spans full: 64/64, 256/256, 512/512, 1024/1024 on four sizes; literal matrix rank 32/32); DISJOINT intervals give exactly HALF (rank 16/32, index 2 -- the missing sector is odd(x)odd, the single ln-2 bit of v501, localised at the split point); the U(1)/neutral algebras do NOT generate the union even WITH the shared site (gaps 2/10/52, GROWING -- the charge-transfer tower is missing): the exact lattice shadow of the current-net strong-additivity failure (infinite index). [N] ENTROPIC WITNESS: the fermionic touching MI diverges with exact (c/3) ln 2 increments per doubling (0.115525, dev 1.2e-6 -- pure UV point-gap, present for EVERY net, hence NOT the discriminator); the honest discriminator is BOUNDED vs DIVERGENT sector-sum deficit: the Z2/orbifold deficit rises 0.4045 -> 0.5899 (N = 64 -> 4096), stays < ln 2, with (ln 2 - Delta_2) ~ N^(-p), p = 0.2444 vs 2 Delta_mu = 1/4 (the Ising disorder-operator dimension h_mu = 1/16 per chirality); split-point scan exactly reflection symmetric (3.9e-14); ED vN deficits same direction; gap continuity to the v501 x -> 1 endpoint. HONEST NOTE: the naive 'defect -> 0' expectation is FALSE at the sharp lattice split -- bounded <=> finite index (Longo-Xu) is the correct statement. U(1) CONTRAST: D_U1(touch) BURSTS the finite-group ceiling ln 2 from L = 128 on and keeps growing as (1/2) ln Var Q_A with Var Q_A = (1/pi^2) ln L + const (Klich-Levitov slope 0.10134 vs 1/pi^2 = 0.10132): infinite index, no saturation. [N] SPLIT PROPERTY (nuclearity witness): the singular values of the off-diagonal covariance block for DISJOINT intervals fall exponentially in k with rate = the elliptic nome pi K(1-x)/K(x) to 1.3-2.0% (d = 16/64/256/512 at N = 4096: rates 2.29/3.24/4.82/5.81); sigma_1 ~ x^0.5044 vs 1/2 (the dimension-1/2 Majorana two-point law); trace norm summable and falling (Buchholz-Wichmann nuclearity proxy); the dimerised control decays exponentially per site (a gapped seam splits trivially). [E] ORBIFOLD INHERITANCE, EXACT AT MACHINE LEVEL: P_A maps the coupling block C -> -C (sigma_k identical, 8.3e-17) and the even-bilinear coupling Gram is the second compound Lambda^2 C with singular values {sigma_a sigma_b} (2.8e-14) -- the SAME ladder at doubled rate; the orbifold Renyi-2 MI is < 3% of the fermionic one and falls with x (split heredity under finite index, Longo, [C]). [E] PIMSNER-POPA / LONGO-REHREN, EXACT: E(a) - a/2 = PaP/2 holds IDENTICALLY (the PP inequality with lambda = 1/2 = 1/[F : F_even], one line, exact); ATTAINMENT in exact integers: all 16384 (even, odd) monomial pairs at N = 8 Majoranas -- 8192 extremal with M^2 = 4, Tr M = 0, PMP = -M exactly, 0 violations; all 2048 hermitianised odd monomials at N = 12 (2048/2048); random pencil sweep min lambda = 0.5000000000; the two-interval group {1, P_A, P_B, P_AB} gives lambda = 1/4 = 1/mu exactly; INDEX CONSISTENCY, TWO ROUTES: exp(Delta_inf) = 2.000000000 = 1/lambda_PP (the v501 entropic route, |Delta - ln 2| = 1.1e-15 at N = 512) and exp(2 Delta_inf) = 4 = 1/lambda_E4 = mu -- the Longo-Rehren index witnessed entropically AND operator-algebraically, independently; U(1) control: lambda = 1/(m+1) -> 0 (infinite index, the operator-algebraic twin of the entropic divergence). [E/N] NEGATIVE CONTROLS + FINITE-SIZE HONESTY: dimerised/trivial phase -- every indicator trivial (touching deficit N-independent to 3.6e-15, MI bounded, sigma_1 exponential); U(1) small-L honesty (non-monotone below L ~ 16, documented); Z2 finite-size honesty: at N = 4096 the touching deficit still sits 14.9% below ln 2 (the N^{-1/4} Ising approach is SLOW -- the verdict rests on the bound + fitted exponent, stated as witness, not proof); wrong prescription {1, P_AB} on the touching pair: deficit EXACTLY 0 (machine identity) -- the bookkeeping is falsifiable. [C] the lattice -> continuum KLM dictionary: bounded deficit = finite index = complete rationality (Longo-Xu); split heredity under finite index (Longo); the entropic order parameters (CHMP); the nome ladder = the free-fermion two-interval modular package. [O] the continuum proof: the theorem 'finite-group orbifolds of completely rational nets are completely rational (hence split + strongly additive)' is XU'S THEOREM (algebraic orbifold CFTs) -- CITED, NOT CLAIMED; the continuum proof for THE seam quotient net itself and for the interacting condensed (E8)_1 net stays open (WP5e / Costello-Li); SEAM.EQUIV.01 stays [O], untouched; no marker moves. VERDICT: with v501, all THREE KLM ingredients of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice for the orbifold prescription; the U(1) control fails both new legs exactly as the continuum current net does.
Recorded status: Formal [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)
Dependencies, status detail and source
Canonical status: WP5d-beta [E]/[L]/[N]/[C]/[O]: split + strong additivity witnessed on the seam lattice edge for the orbifold prescription -- strong additivity EXACT with the shared boundary Majorana (GF2 spans full, rank 32/32; disjoint index exactly 2 = the v501 ln-2 bit), the entropic touching defect BOUNDED < ln 2 with the Ising 1/4-exponent approach (p = 0.2444) while the preregistered U(1) control diverges ((1/2) ln Var Q, Klich-Levitov pinned): bounded-vs-divergent is the lattice discriminator; split via the elliptic-nome sigma_k ladder (1.3-2.0%, sigma_1 ~ x^{1/2}, trace norm summable) with EXACT orbifold inheritance (C -> -C + Lambda^2 C compound, Longo heredity); Pimsner-Popa E(a) - a/2 = PaP/2 identically (lambda = 1/2 = 1/[F:F_even], integer attainment 16384 + 2048 with 0 violations), lambda_E4 = 1/4 = 1/mu, index consistency exp(Delta) = 2 = 1/lambda_PP two routes; controls: dimerised trivial, wrong prescription exactly 0, finite-size honesty documented; with v501 all THREE KLM ingredients of complete rationality witnessed on the lattice; continuum uplift stays [O] (Xu's theorem cited, not claimed; WP5e/Costello-Li); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5D.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v450; v489; v490; v501
Scripts: v504_celestial_wp5d_klm_completeness.py
Location: tfpt_research_contracts
Original CSV, line 568Original ledger status
CELEST.WP5E.BETA.01
WP5e-BETA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, beta stage -- 'the equivariant anomaly ledger on twistor space', the twistor-side milestone of the BCOV/Costello-Li uplift; WP5e-alpha = v502 anchored the CFT side): exact sympy/Fraction arithmetic, no floats (47 checks). Roadmap target: the anomaly-inflow side of the boundary partition function; preregistered KILL = 'the anomaly inflow demands a level != 1'. [E] ATIYAH-BOTT LEDGER: denominators det(1 - g_j^{-1}|_{T0 C^2}) = (2, 4, 2) for j = 1,2,3; sum of reciprocals 5/4 = (|Z4|^2 - 1)/12 (the A-series Dedekind identity); the origin is the only fixed point; equivariant characters tr_{g_j}(Ad E8) = (248, 0, -8, 0) by TWO independent routes (graded dims / root sum); invariant average 60 = the D5+A3 carrier; Frobenius 61568; tr_{g_2} = -8 = 120 - 128 (so16 adjoint-minus-spinor supertrace). [E] OKUBO PER SECTOR (the honest sharpening): the INVARIANT sector obeys Okubo exactly -- Q^(0) = 36<x,x>^2 with 36 = lambda~^2_e8 (the v495 identity re-derived through the mu4-glue grading); the TWISTED sectors do NOT reduce to (tr_{g_j} X^2)^2 (independent-quartic content T5: (-8, 32, -8), T3: (48, -64, 48)); in the AB-weighted fixed-point sum the D5 quartic CANCELS exactly and the A3 quartic leaves the RIGID residual 32 T3 (32 = (h,h) + (h',h'), a [C] coincidence); no nonzero reweighting of the twisted sectors alone is quartic-free, and the only quartic-free weighting on the class blocks is the uniform Z4 average (= the invariant projector); the graded GS-exchange ansatz cancels sector 0 and is exactly rank-obstructed in sectors 1, 2, 3 -- twisted-sector closed strings must carry the rest [O]; heterotic identities exact (Tr_120 = 8tr^4 + 3(tr^2)^2, Tr_128 = -8tr^4 + 6(tr^2)^2, Tr_E8 = 9(tr^2)^2). [E] INDEX BRIDGE (the centrepiece): the per-character AB sum f(m) = (1/4) sum_j (i^{jm} - 1)/det_j = (0, -3/8, -1/2, -3/8) equals ch2(T_m) = -(C^{-1})_mm/2 EXACTLY for m = 1, 2, 3 -- the E1 fixed-point ledger and the E2 McKay/Kronheimer intersection ledger are the SAME arithmetic; the glue-bundle ch2 defect is -78 by BOTH routes (integer; per-class (-24, -30, -24), the middle -30 = -h_vee a [C] coincidence); bridge identity sum_m h_A3(m) = 5/4 = sum_j 1/det_j; sphere pairing <alpha,h> integer on all 240 roots, per-class Killing sums (320, 320, 240, 320), total 1200 = 2 h_vee (h,h). [E] k = 1 GEOMETRIC: the lattice current count (norm-2/k vectors in the glue cosets) = 240 at k = 1 and EXACTLY 0 at k = 2, 3, 4 (no adjoint closure); embedding residual 47k^2 + 219k - 266 = (0, 360, 814, 1362); h(J;k) = k integer for ALL k -- integrality alone fixes nothing (honest, exactly as v502); v493 lockstep periods => ONE scale => ONE level (det(A-1) = -4, no invariant flux vector). [E] E3 REFUTATION (prominent): the clock-invariant modulus a0 in O(8) generates a Hamiltonian in O(2) -- a0 fills the BSS GRAVITON slot (s = 2), NOT the axion slot O(-2) (s = 0): Hamiltonian weights X = -2 != +2 = Y, mismatch 4 = -deg K_PT = |mu4| (a [C] coincidence) -- 'the theory brings its GS axion as a0' is REFUTED; what DOES match: the three H2(ALE) classes carry exactly the Coxeter eigencharacters {i, -1, -i} = the three twisted ledger sectors (bijection, no invariant class: det(A-1) = -4); slot count 1 tower slot O(-2) + 3 sphere classes = 4 = |mu4| (soft count, not a construction). [E] NEGATIVE CONTROLS: diag(i,i) breaks the ledger FOUR independent ways (complex denominators, reciprocal sum 1/4 != 5/4, omega-weight -1 not Calabi-Yau, invariant ring needs 5 generators); SO(16) glue: a mere Z2 monodromy (classes {0,2}), characters (120, 0, +120, 0) wrong sign AND magnitude at j = 2, odd McKay nodes empty, ch2 defect -30 != -78, and bulk Okubo FAILS (T5 = 16, T3 = -32: independent quartic Casimir, no single-axion GS -- the v495 sl4-style control transplanted); k = 2: current count 0, residual 360, c-prefactor -31/48 != -1/3, while naive integrality dials stay integer (the kill is the closure dial). [C] dictionaries: CS(rho) mod 1 = discriminant weight; 32 = (h,h) + (h',h'); -30 = -h_vee; the Omega-contraction identification of the axion slot. [O] HONEST FENCE: the global BCOV/Kodaira-Spencer theory on PT/Z4 (quantisation, non-fixed-point contributions, the twisted closed-string measure), the O(-2) bulk-axion construction, the Costello-Paquette-Sharma level-from-flux mechanism, and the pairing of the rigid 32 T3 twisted residual with the three sphere axions all stay OPEN. VERDICT B: the preregistered kill ('inflow demands a level != 1') does NOT fire on the equivariant skeleton -- all computable twistor-side pieces are consistent with k = 1 and kill k = 2, 3, 4 via the closure dial; what the skeleton cannot decide is the actual BCOV one-loop coefficient on PT/Z4. WP5e: alpha (v502) + beta (v505) executed; SEAM.EQUIV.01 untouched; no gate closes, no marker moves.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5e-beta [E]/[C]/[O]: the Atiyah-Bott/Lefschetz skeleton of the twistor inflow exact -- denominators (2,4,2) with Dedekind sum 5/4, characters (248,0,-8,0) two routes, invariant Okubo 36<x,x>^2 with the twisted sectors NOT reducing and the AB sum leaving the rigid 32 T3 residual; index bridge f(m) = ch2(T_m) exact with glue defect -78 both routes; level dials k = 1 geometric (current count 240/0/0/0, residual 0/360/814/1362, one scale => one level -- since v509 (WP5e-epsilon-2) a THEOREM of clock invariance, no longer a heuristic); E3 refuted: a0 = graviton modulus O(2), not the axion O(-2) (mismatch 4 = |mu4|), the three sphere classes = the three twisted axion slots; controls diag(i,i)/SO(16)/k = 2 quantified; kill branch does not fire on the equivariant skeleton; the global BCOV quantisation stays [O] -- with the rigid 32 T3 twisted residual and the sphere-axion pairing as the named remaining targets; SEAM.EQUIV.01 untouched
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP2.01; CELEST.WP3.01; CELEST.WP4.01; CELEST.WP5E.ALPHA.01; SEAM.EQUIV.01; v492; v493; v495; v496; v502
Scripts: v505_celestial_wp5e_beta_equivariant_ledger.py
Location: tfpt_research_contracts
Original CSV, line 569Original ledger status
CELEST.WP5E.GAMMA.01
WP5e-GAMMA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, gamma stage -- 'the sphere-axion pairing check', an honest rigid NEGATIVE result in the v505 style; exact sympy/Fraction arithmetic, no floats, 27 checks, ~4 s): can the three twisted sphere axions eta_m (m = 1, 2, 3; Coxeter eigencharacters {i, -1, -i} on H^2(ALE), v505 S5.8) cancel the rigid 32 T3 residual of the Atiyah-Bott anomaly ledger (v505 S3.7: A_fix = Q^(1)/2 + Q^(2)/4 + Q^(3)/2 = 9P1 - 30P2 - 15P3 + 32 T3) by Green-Schwarz-type EXCHANGE with sector-compatible quadratic vertices? NO, with a certificate. [E] VERTEX-SPACE THEOREM (structural backbone): the W(D5) x W(A3)-invariant quadratics on the 8-dim glue Cartan are EXACTLY span{s5, s3} (dim 2, bidegree dims (1, 0, 1)) and the invariant quartics EXACTLY span{P1, P2, P3, T5, T3} (dim 5, bidegree dims (2, 0, 1, 0, 2)) -- computed by exact Weyl nullspace arithmetic over ALL bidegrees, the vertex basis is COMPLETE; PRODUCT THEOREM (the master kill): (a s5 + b s3)(c s5 + d s3) = ac P1 + (ad+bc) P2 + bd P3 identically -- the product of ANY two invariant quadratics has identically ZERO T5- and T3-content, hence EVERY scalar-exchange image (any number of axions, any selection rule, any coupling values) is annihilated by Phi_T3 while Phi_T3(A_fix) = 32 != 0: the pairing is decided before enumeration. [E] SELECTION-RULE ENUMERATION (Z4 charge m + j + j' = 0 mod 4): the strict two-index reading COLLAPSES -- the class bilinears C_{j,j'} = sum_{alpha+beta=0} <alpha,x><beta,x> are nonzero ONLY for j' = -j (C_{j,-j} = -K_j), so the rule forces m = 0: the sphere axions have NO invariant bilinear Cartan vertex at all (16 raw triples, 4 nonzero, all m = 0); twist-insertion reading (the generous route): eta_1 <-> K^(3), eta_2 <-> K^(2), eta_3 <-> K^(1), eta_0 <-> K^(0); exchange channels E13 = K^(3)K^(1) = (16, -96, 144, 0, 0), E22 = (K^(2))^2 = (16, 32, 16, 0, 0), bulk E00 = 225 E22; SIDE DISCOVERY: K^(0) = -15 K^(2) -- the two even-sector quadratics are PARALLEL (this collapse drives the Phi_P obstruction). [E] UNSOLVABILITY WITH CERTIFICATE (the core result): exchange matrix M is 5 x 2, rank 2, kernel 0 (a solution, if it existed, would be unique/rigid); rank([M | A_fix]) = 3 > 2: UNSOLVABLE; annihilator certificate: im(M) is annihilated by Phi_T5, Phi_T3, Phi_P = 3[P1] - [P2] - [P3] with values (0, 32, 72) on A_fix = (9, -30, -15, 0, 32) -- TWO independent obstructions: 32 T3 (structural, rule-independent) and Phi_P = 72 (selection-rule-driven: even the factorised part 9P1 - 30P2 - 15P3 is unreachable through charge-0 channels because K^(0) || K^(2) collapses the charge-0 product span to 2 dims); Phi_P(Q^(0)) = 0 (the bulk Okubo square stays inside the reachable slice -- the v505 sector-0 mechanism is undisturbed); ker M = 0 forces the unique non-disturbing coupling c = 0: the sphere axions are RIGIDLY SILENT in the exchange ansatz; relaxed control: dropping charge conservation makes the P-block fully reachable (rank 3, det -64) but T5 = T3 = 0 persists -- the deep kill is T3. [E] NATURALNESS RESOLUTION: ch2-natural couplings (f(1)f(3), f(2)^2) = (9/64, 1/4) give image (25/4, -11/2, 97/4, 0, 0) and AB-weight couplings (1/2, 1/4) give (12, -40, 76, 0, 0) -- BOTH have certificate values (0, 0) instead of the required (32, 72): the failure is scale- and coupling-INDEPENDENT (no 'unnatural factor' -- the ansatz space misses the target identically; the naturalness question dissolves). [E] NEGATIVE CONTROLS: (a) invariant axion only = v505 S3.9 replicated + sharpened (A_fix not in span{(K^(0))^2}); (b) flipped rule m + j - j': twist-insertion channel set unchanged (the Phi_T3 kill is rule-flip invariant), two-index reading changes (only eta_2 couples, the bulk axion gets span{K_0, K_2}, det 224, the P-part becomes reachable) -- 32 T3 stays dead: Phi_P is rule-dependent, Phi_T3 is not; (c) SO(16) glue: odd McKay classes EMPTY (no sphere partners at all) and the AB sum (15, -18, -15, -4, 40) does not even cancel T5 -- the mechanism is structurally BROKEN, not merely obstructed (E8 is doubly special); (d) D8 native: quartic 16 T8 + 12 s8^2 (2 h_vee(D8) = 28 Killing), no A3 block, no T3 structure -- the pairing question cannot even be posed. [C] NUMBER FENCES (typed [C] only, look-elsewhere discipline): 32 = (h,h) + (h',h') = 20 + 12; 72 = 2 * 36 = 2 lambda~^2_e8. [O] SHARPENED O-FENCE: the exchange sub-branch is DEAD; the 32 T3 residual MUST be carried by twisted BCOV contact terms -- binary criterion WP5e-delta-1: the one-loop coefficient on C^2/Z4 must come out exactly (9, -30, -15, 0, 32), COMPUTED (orbifold elliptic-genus / twisted-character arithmetic), not fitted; named escape WP5e-delta-2: the full-tensor ledger with non-Cartan external legs (the collapse is a Cartan-restricted statement); plus WP5e-epsilon-1 (the O(-2) bulk-axion construction, lambda~ = 6) and WP5e-epsilon-2 (CPS level-from-flux). The slot bijection (v505 S5.8) is UNTOUCHED; the preregistered level-kill still does NOT fire; SEAM.EQUIV.01 stays [O]; NO marker moves anywhere -- CELEST.SEAM.01 stays Open, the O-fence is only SHARPENED.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5e-gamma [E]/[C]/[O]: the exchange route for the sphere-axion pairing is KILLED with a three-functional certificate -- invariant quadratics on the glue Cartan exactly span{s5, s3} (dim 2), quartics exactly span{P1, P2, P3, T5, T3} (dim 5), the product of any two invariant quadratics has zero T5/T3 content (master kill: Phi_T3 annihilates every exchange image, Phi_T3(A_fix) = 32 != 0); strict two-index rule collapses the sphere couplings entirely (m = 0 only), twist-insertion channels give a rank-2 exchange matrix with rank([M | A_fix]) = 3 and certificate (Phi_T5, Phi_T3, Phi_P) = (0, 32, 72); K^(0) = -15 K^(2) side discovery; ch2-natural and AB-weight couplings miss scale-independently (naturalness dissolves); SO(16) has no sphere partners AND uncancelled T5, D8 has no T3 structure; the 32 T3 burden moves to the twisted BCOV contact terms (delta-1, binary: coefficient exactly (9, -30, -15, 0, 32) computed not fitted) and the full-tensor ledger (delta-2); slot bijection untouched, level-kill still not fired, SEAM.EQUIV.01 stays [O], no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; SEAM.EQUIV.01; v492; v493; v495; v502; v505
Scripts: v508_celestial_wp5e_gamma_pairing.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 573Original ledger status
CELEST.WP5E.EPS2.01
WP5e-EPSILON-2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, epsilon-2 stage -- 'the CPS level-from-flux dial', verdict B; exact sympy/Fraction arithmetic, no floats, 28 checks, <1 s): derive the boundary level k = 1 from flux quantisation on the lockstep spheres, following the Costello-Paquette-Sharma (CPS) Burns-holography mechanism (PRL 130, 061602; arXiv:2306.00940). [E] CPS SKELETON REPLICATED EXACTLY: the S^3 period of the brane backreaction 3-form int Omega_N = (2 pi i)^2 N (pullback density -sin(2 theta), phases cancel exactly; their eq. 3.33 -- large-gauge invariance of hCS quantises N in Z); the exceptional-sphere Kahler flux int_{P^1} omega = 2 pi N exactly (K = |u|^2 + N log|u|^2, the flat part dies at t -> 0; their eq. 3.37); the boundary level magnitude 2N via the symplectic-boson double contraction (exact for N = 1, 2, 3; their eq. 9.4), Dynkin normalisation T(so8 vector) = 1, T(adj) = 6 -- the CPS formula 'level = flux quantum x Dynkin index of the boundary matter', level LINEAR in the flux. [E] PAIRING MATRIX PINNED, NOT POSTULATED: the v505 ch2 ledger (f(m) = -(C^-1)_mm/2 = (-3/8, -1/2, -3/8)) + integrality + unimodularity + effectivity pin P_mi = c1(T_m).[Sigma_i] = delta_mi by COMPLETE enumeration (row candidates 8/6/8; 48 unimodular solutions of the full form condition P C^-1 P^T = C^-1; with P >= 0 exactly 2 = identity + A3 diagram flip); full c1 matrix -(C^-1)_mn; glue FLUXES F_i = sum_m dim(g_m) P_mi = (64, 60, 64) = dim g_i, total 188 = 248 - 60, flip-invariant; flux per charged root current uniformly ONE quantum through exactly one sphere. [E] LEVEL FORMULA, honest: 'level = total open-string flux' is KILLED by the lockstep test itself (k_i = (64, 60, 64) unequal -- F_i is the FLAVOUR multiplicity, in CPS language the brane matter content); what works: k = flux per adjoint current = F_i/dim g_i = (1, 1, 1), anchored non-circularly by the lattice current count (240, 0, 0, 0) for k = 1..4 (v505 S4.8) and the embedding index j(D5 in E8) = j(A3 in E8) = 1 (Killing coefficient 60 = 2 h_vee(E8), both factors). ORD-4-VS-LEVEL-1 RESOLVED: the clock order counts FRACTIONAL flux sectors, not the level -- mu = det(D5) det(A3) = 16 = ord(monodromy)^2, the mu4 glue diagonal is isotropic and Lagrangian (order 4 = sqrt 16), KLM condensation 16 -> 16/4^2 = 1 = zero residual fractional flux; NEW DIAL: #primaries((E8)_k) = (1, 3, 5, 10, 15, 27) for k = 1..6 from the affine marks -- exactly ONE sector iff k = 1 (level from sector count, independent of the CPS transplantation). [E] THE LOCKSTEP THEOREM (the lifting of v505 S4.10 'one scale => one level' to a theorem): clock invariance P(iZ) = P(Z) forces a3 = a2 = a1 = 0 sharply, the four branch points are ONE mu4 orbit i^k t, the periods Pi_j = t(i-1) i^{j-1} have |Pi_j| = sqrt(2) t EQUAL on all three spheres, the monodromy acts as the pure phase i on the period vector and det(A - 1) = -4 != 0: no invariant flux vector, no sphere-dependent level assignment possible -- with the CPS flux-level dictionary, k1 = k2 = k3 IS A THEOREM of clock invariance. NEGATIVE PROBE (new falsifier): the clock-forbidden family Z^4 - Z (smooth, disc = -27) has branch points {0, 1, w, w^2}: NOT a mu4 orbit, and over ALL 24 orderings ZERO chains give three equal period moduli (1 vs sqrt 3). HONEST LIMIT: uniform flux doubling keeps the lockstep -- the theorem proves EQUALITY of the three k, not the value; the value 1 comes from the current/sector counters and, CPS-faithfully, from the minimal quantum N = 1. [E] NEGATIVE CONTROLS: k = 2 (current count 0, embedding residual 360, c = 31/2 => prefactor -31/48 != -1/3, #primaries 3 != 1 -- while the lockstep dial honestly does NOT fire); SO(16) glue (classes (60, 0, 60, 0) => fluxes (0, 60, 0): spheres 1/3 FLUX-DARK, ch2 defect -30 != -78, condensation stops at 16/2^2 = 4 = det Cartan(D8): four sectors survive); A2 intersection form (det 3 != 4 = |Z4|: McKay bijection broken, glue diagonal 23/24 not integer, mu = 12 not a perfect square: NO Lagrangian subgroup exists, Dedekind 2/3 != 5/4, Coxeter order 3 != 4)
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5e-epsilon-2 [E]/[C]/[O] (verdict B): k = 1 from flux quantisation -- partially derived. DERIVED exact: one level is a THEOREM (lockstep from clock invariance: one mu4 orbit, |Pi_j| = sqrt(2) t equal, monodromy = pure phase i, det(A-1) = -4 no invariant flux vector; falsifier Z^4 - Z: 0/24 orderings lockstep), flux-per-current = 1 uniformly (pairing matrix = identity up to diagram flip, pinned by complete enumeration from the v505 ch2 ledger; fluxes (64, 60, 64) = flavour multiplicity, NOT the level -- the naive reading killed by the lockstep test itself), the sector-counter dial pins the value (16 = ord^2 fractional sectors condense through the Lagrangian mu4 glue to 1; #primaries((E8)_k) = 1 iff k = 1), and the CPS skeleton is exact (level linear in the flux quantum, minimal quantum N = 1 <-> k = 1). ANALOGY [C]: the CPS dictionary on PT/Z4. OPEN [O]: the type-I B-model backreaction on PT/Z4. The v505 'one scale => one level' sentence is lifted to theorem status (text move, no gate move); WP5e stays [O]; SEAM.EQUIV.01 untouched; no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; SEAM.EQUIV.01; v492; v493; v502; v505; v508
Scripts: v509_celestial_wp5e_eps2_level_from_flux.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 574Original ledger status
CELEST.WP5E.DELTA2.01
WP5e-DELTA-2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, delta-2 stage -- 'the full-tensor ledger', the named escape route of the v508 exchange no-go, intermediate verdict, exact; integer-scaled weight multisets + Kostant/Weyl alternating sums + sympy/Fraction, no floats, 41 checks, ~4 s): does the FULL adjoint tensor structure of e8 (non-Cartan external legs, higher-arity vertices) open a sphere-axion channel to the rigid 32 T3 residual of the Atiyah-Bott ledger (v505 S3.7: A_fix = (9, -30, -15, 0, 32) in (P1, P2, P3, T5, T3))? The collapse is CONFIRMED full-tensorially, but one cubic door opens. [E] STRUCTURE: g0 = d5 (+) a3 is SEMISIMPLE with NO u(1) factor (the 52 class-0 roots split 40 + 12 into pure D5 + pure A3 roots, root span rank 8 = rank E8); the twisted sectors factorise multiplicity-free into minuscule Weyl orbits g1 = (16_s, 4), g2 = (10, 6), g3 = (16_c, 4bar) = (g1)^*, each an irreducible g0-module (64, 60, 64); INNERNESS THEOREM: the grading element h = (2,2,2,2,2;0^4) lies in the CARTAN OF g0, so t = exp(2 pi i h/4) is in the torus of G0 and every g0-invariant tensor automatically carries total sector charge 0 mod 4 -- the v508 collapse is a full-tensor statement AND holds across arities (all 15 non-neutral sector triples: Hom = 0; the sphere axions eta_1,2,3 have no invariant partner operator at any arity <= 3). [E] HOM TABLES: bilinear dim Hom_{g0}(g_j x g_j') nonzero ONLY at j' = -j with dims (2, 1, 1, 1) (Killing blocks; all 12 charged pairs vanish); invariant symmetric bilinear forms on e8 = 4-dim (2 + 1 + 1), the extra blocks B13/B22 restrict to zero on the Cartan (no new quadratic exchange content); trilinear neutral multisets {0,0,0}: 3 (= 1 sym + 2 antisym), {0,1,3}: 2, {0,2,2}: 2, {1,1,2}: 1, {2,3,3}: 1 -- ALL cross-sector trilinears are pure brackets (antisymmetric in the repeated legs, they vanish with two identical external legs); the ONLY totally symmetric survivor on all of e8 is the su(4) d-symbol (so(10) has no cubic Casimir: Inv(S^3 45) = 0 protects T5, Inv(S^3 a3) = 1). [E] THE NEW CHANNEL: the quartic projection of the d-symbol exchange is Q_dd = (0, 0, -1/240, 0, 1/60) = (1/60)(T3 - P3/4), computed by TWO independent routes (Killing-propagator contraction with 2 h_vee = 60 verified from the roots; hand route |x^2 - (tr x^2/4) 1|^2/60); Phi_T3(Q_dd) = 1/60 != 0 -- v508's master-kill functional Phi_T3 does NOT annihilate the trilinear exchange (the product theorem covers quadratic vertices only); bracket vertices, tadpoles and mixed d x f die exactly; no T5 analogue exists (Phi_T5 stays a universal annihilator, exactly as A_fix requires). [E] NEW CERTIFICATE: M = [E13, E22, E00, Q_dd] has rank 3, augmented rank 4 -- the pairing in the charge (generous) reading remains OBSTRUCTED, but with the WEAKER certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 72 - 8 = 64 (the T3 obstruction 32 is traded for a combined (P3, T3) obstruction: the d-channel produces T3 only with the fixed shadow P3 = -T3/4, and the leftover P-block violates the charge-0 span -- the K0 || K2 collapse of v508); the RELAXED reading (charge bookkeeping dropped on the bilinear P-block) becomes EXACTLY solvable: A_fix = -u + 8v + 2w + 1920 Q_dd (residual 0) -- where v508 found T5 = T3 = 0 persisting even relaxed, the d-channel closes the T3 gap with small integer coefficients. [C] NUMBER FENCES (typed [C] only, look-elsewhere discipline): 64 = dim g1 = 2^6; 1920 = |W(D5)| = 2^7 x 3 x 5 = 8 x 240 -- FENCE SHARPENED (2026-07-22, v513/CELEST.DTERM.NONDERIV.01): the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513; the convention-stable [E] content is c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60. [O] OPEN AFTER DELTA-2: the arity >= 4 ledger; the physical justification of the cubic GS term (c_d = 1920 under the e8-Killing propagator); ch2-naturalness for CUBIC vertices UNDECIDED (no canonical geometric datum exists for a cubic vertex); the burden on delta-1 (twisted BCOV contact terms) is SHARPENED: it must supply either the psi = 64 slice or the selection-rule relaxation -- and the delta-1 exploration run (probe, 27/27, exploration-level, v-numbered on demand) stands UNDECIDED: the strict holomorphic q^0 reading is REFUTED at the Z2/Eguchi-Hanson anchor (method boundary, not a kill), the forced leading (T5,T3) ratio is 4:3 (sector 2 : sectors 1+3), the SL(2,Z) orbit decomposition 15 = 12 + 3 identifies the contact term as the MODULAR COMPLETION of the AB data (Harvey-Moore tau-integral, not a free number), and massive coset levels n >= 2 are mandatory. [E] NEGATIVE CONTROLS: (a) Killing control = v508 replication (A_fix = (9, -30, -15, 0, 32), E13 = (16, -96, 144, 0, 0), E22 = (16, 32, 16, 0, 0), E00 = 225 E22); (b) SO(16)/D8 glue: bilinears (1, 0, 1), trilinears (1, 0, 1, 0), Inv(S^3 adj_d8) = 0 -- no symmetric cubic, no odd sectors, no A3 block: the delta-2 channel cannot even be posed (E8/mu4 doubly special); (c) deliberately false g0 = d5 (+) a2 (+) u(1): bilinear entries (5, 2, 2) != (2, 1, 1) and Inv(S^3 g0) = 6 != 1; false sector content (16_s, 4bar) kills every pairing -- the computation SEES the exact structure. SEAM.EQUIV.01 stays [O]; NO marker moves anywhere -- CELEST.SEAM.01 stays Open, the O-fence is only SHARPENED.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5e-delta-2 [E]/[C]/[O] (intermediate verdict): the full-tensor ledger -- the v508 collapse is CONFIRMED full-tensorially and arity-crossing (innerness theorem: h in the g0 Cartan, every invariant tensor charge-0 mod 4; all 15 non-neutral trilinear triples Hom = 0), and the charge-0 ledger contains exactly ONE structure invisible to the quadratic ansatz: the su(4) d-symbol (unique symmetric trilinear on e8, so(10) has no cubic Casimir), whose exchange quartic Q_dd = (1/60)(T3 - P3/4) breaks the Phi_T3 master kill (1/60 != 0) -- the T3 channel OPENS; the pairing stays obstructed in the charge reading with the weaker certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 64 = dim g1 [C], and becomes exactly solvable relaxed: A_fix = -u + 8v + 2w + 1920 Q_dd with c_d = 1920 = |W(D5)| [C] (the |W(D5)| reading look-elsewhere-loaded and convention-contingent -- v513); SO(16)/D8 has no symmetric cubic and no odd sectors (E8/mu4 doubly special), false g0/sector controls separate; the burden on delta-1 is sharpened to the psi = 64 slice or the selection-rule relaxation (delta-1 itself UNDECIDED at exploration level at promotion time, since DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, tension with v516 stated; the pre-v518 state: strict holomorphic reading refuted at the Z2 anchor, contact term = modular completion / Harvey-Moore tau-integral, forced (T5,T3) ratio 4:3, massive coset levels n >= 2 mandatory); SEAM.EQUIV.01 stays [O], no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; SEAM.EQUIV.01; v492; v493; v502; v505; v508; v509
Scripts: v511_celestial_wp5e_delta2_tensor_ledger.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 575Original ledger status
CELEST.DTERM.NONDERIV.01
THE c_d NEGATIVE CERTIFICATE -- CAN GEOMETRY DERIVE c_d = 1920? (2026-07-22; anti-numerology certificate in the v508 style, honest rigid NEGATIVE result; 24 checks, ~2 s; NO marker moves -- the v511 [C] fence stays [C] and is text-sharpened only, the physical generation of the 32 stays [O], CELEST.SEAM.01 stays Open): the v511 full-tensor ledger found exactly ONE legal cubic channel (the su(4) d-symbol, Q_dd = (1/60)(T3 - P3/4)) with the relaxed solution A_fix = -u + 8v + 2w + 1920 Q_dd and the [C] fence 1920 = |W(D5)|; this module stress-tests the fingerprint via four candidate routes with a PRE-DECLARED look-elsewhere ledger. [E] CONVENTION TABLE: c_d in {1920, 120, 256, 32, 69120} across five normalisations (probe tr-fund/e8-Killing; Gell-Mann; su(4)-Killing; plain trace; ch3-normalised d/6) -- only 1 of 5 hits |W(D5)|; the convention-STABLE content is the factorisation c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60, both factors [E] (Killing propagator 60 from sum_alpha <alpha,e>^2 = 120 over the actual roots; T3-slot fixing via det(P-block) = -64 != 0: c_d is pinned by the T3 slot alone). [E] K1 DEAD: the charge-legal flux pairings (1,3)/(2,2) give 2048/1800 -- they BRACKET 1920 and both MISS; the only flux hit 64*60/2 is the FORBIDDEN (1,2) pairing (charge 3 mod 4, killed by the v511 innerness theorem); LOOK-ELSEWHERE LEDGER: 924 expressions from 21 [E] numbers x grammar {a*b, a*b/2, a*b/4, a+b}, 11 hits on 1920; control targets 1800 -> 8, 2048 -> 5, 2016 -> 0 -- the catalog's multiplicative structure makes any smooth number near 2000 easy to hit. [E] K2 (quantisation gives only a lattice): integer couplings in tr-fund units + e8-Killing propagator put c_d in 60 Z -- 1920 is the 32ND multiple, no minimality; strict ch3-unit couplings (6 Z) give 2160 Z and EXCLUDE 1920 outright; the cubic quantum is lattice-dependent (6 on coroots, 3/8 on weights); the residual integer 32 has 12 signed factorisations, nothing forces it. [E] K3 PROVENANCE CLASH (the death blow to the index route): the su(4) cubic indices per class are A_m = (0, 16, 0, -16) with twisted sum sum_m i^m A_m = 32i, NUMERICALLY equal to Phi_T3(A_fix) = 32 -- BUT the quartic class decomposition (10, -4, 30, -4) is dominated by classes 0/2 while the cubic 32 is built 16 + 16 from classes 1/3: same number, disjoint mechanism = coincidence exposed; the Weyl side is WRONG (the d-channel is an A3-side object, |W(A3)| = 24; D5 has no cubic invariant at all, Inv(S^3 45) = 0; |W(E8)|/|W(A8)| = 1920 is a second accidental 1920 inside E8 combinatorics). [E] K4 NEGATIVE CONTROLS: SO(16)/D8 fails every candidate formula consistently (no d-channel, cubic class sums 0, dark spheres); k = 2 flux doubling scales the flux formula x4 (7680) while c_d is root-theoretic and static; false g0 = d5+a2+u(1) has SIX symmetric cubics (the question cannot be posed); so(10) sanity (Tr X^3 = 0, T5 protection). VERDICT: NUMEROLOGY as a derivation claim -- the v511 [C] fence stays a fence, now typed look-elsewhere-loaded; the honest [E] core is c_d = 32 x 60; the physical generation of the 32 stays [O].
Recorded status: Formal/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
Dependencies, status detail and source
Canonical status: c_d non-derivation [E]/[O]: the fingerprint c_d = 1920 = |W(D5)| does NOT survive the honest battery -- convention table {1920, 120, 256, 32, 69120} (only 1 of 5 normalisations hits), charge-legal flux products 2048/1800 bracket and miss (the 64*60/2 hit is charge-forbidden), look-elsewhere 11/924 vs control 1800 -> 8/924, quantisation gives only lattices (60 Z, 32nd multiple; ch3 units exclude), and the twisted cubic index 32i clashes in provenance with the true quartic 32 (classes 1/3 vs 0/2) -- NUMEROLOGY as a derivation claim; the convention-stable [E] core is c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60; the v511 [C] fence text-sharpened only, the physical generation of the 32 stays [O]; no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; v505; v508; v509; v511
Scripts: v513_celestial_dterm_nonderivation.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 577Original ledger status
CELEST.WP5E.EPS1.01
WP5e-EPSILON-1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 work package 5e, epsilon-1 stage -- 'the O(-2) bulk-axion slot', verdict B; exact sympy/Fraction arithmetic, no floats, 34 checks, ~4 s): build the s = 0 tower field explicitly (v505 E3: the BSS axion is, via Omega-contraction, the O(-2) slot; a0 does NOT fill it), pin lambda~ = 6 (v495: lambda~^2 = 36 = h_vee + 6) as a residue with explicit factor bookkeeping on PT/Z4, and take the first honest back-reaction step (the Gibbons-Hawking A3 form of the CPS Omega_N mechanism, arXiv:2306.00940). [E] EQUIVARIANT O(-2) SLOT: pushforward ledger pi_* O(-2) on PT' = tot(O(1)+O(1) -> P^1): per spacetime degree d exactly (d+1)^2 = nullspaces of the wave operator (1, 4, 9, 16, 25, 36 for d = 0..5); the v492 action on the fibre coordinates (mu1, mu2) -> (i mu1, i^-1 mu2), incidence forces column weights (+1, +1, -1, -1); the equivariant bookkeeping closes BLOCK BY BLOCK (all d <= 6, all 4 characters); character series P_0 = 1 + 3t^2 + 15t^4 + 21t^6 + 45t^8 + ..., P_1 = P_3 = 2t + 8t^3 + 18t^5 + ..., P_2 = 6t^2 + 10t^4 + 28t^6 + ...; the d = 0 slot has multiplicity (1, 0, 0, 0) -- THE BULK AXION SURVIVES THE PROJECTION (density 1/|Z4|); Molien invariant ring = (1 - t^8)/((1 - t^4)^2 (1 - t^2)) -- Z in O(2), X, Y in O(4), relation degree 8 = the a0 weight (v492/v493 bridge); E3 bijection sharpened: twisted characters {i, -1, -i} with minimal content (2t, 6t^2, 2t) and NO degree-0 mode = the Coxeter eigenvalues on H^2(ALE), det(A-1) = -4; graviton control: the O(+2) slot starts invariant only at fibre degree 4 with multiplicity 3 = {X, Z^2, Y} (the cohomological form of v505 X = -2 != +2 = Y). [E] LAMBDA~ = 6 TRIPLY PINNED: (1) Okubo: Tr_adj X^4 = 36<x,x>^2 = (6<x,x>)^2 on the 240 glue roots, 36 = h_vee + 6, lambda~ = 6 = |Z2| N_fam; (2) measure cancellation: mu = 1/4 enters three times (vertex^2 mu^2, propagator 1/mu, anomaly mu) and cancels EXACTLY -- lambda~_eff^2 = 36 for ARBITRARY mu; wrong bookings quantified and excluded: anomaly-only => lambda~ = 3 = N_fam, missing propagator renormalisation => lambda~ = 12 = |mu4| N_fam; (3) flux: minimal CPS quantum N = 1, a single exchange channel exactly at k = 1 (#primaries), (kappa/c3)^2 = 36/3 = 12 exact. [E] GH/A3 BACK-REACTION STEP: invariant 4-centre configurations = exactly 2 branches (axis^4 = pure resolution or ONE free mu4 orbit = pure deformation); free orbit => prod(Z - i^p z0) = Z^4 - z0^4 => XY = Z^4 + a0 with a0 = -z0^4 (the v493 family RE-DERIVED from the geometry); monodromy a0 -> e^{2 pi i} a0 = one clock step; the GH point of charge k is EXACTLY R^4/Z_k (k = 4 => seam boundary S^3/Z4); periods int omega = 4 pi (x_q - x_p), free orbit: Pi_j = 4 pi t0 (i-1)(1, i, i^2), |Pi_j| equal (LOCKSTEP, counter-check of v509); the Coxeter clock FROM THE GEOMETRY (A^4 = 1, det(A-1) = -4, Pi.A = i Pi); the CPS 'N log' analogue = source charge 4 = |mu4| (flux -4 pi per centre); HONEST SHARPENING: for the Ricci-flat ALE the asymptotic log coefficient is EXACTLY 0 (EH: u K' = sqrt(u^2 + a^4), det g = 1) -- the log lives at the exceptional locus (K' ~ a^2/u, flux 2 pi a^2); Burns contrast (K = u + N log u => det g != 1, only scalar-flat); bonus: multipole selection rule m = 0 mod 4, first symmetry-breaking multipole (4, +-4) carries -a0 with amplitude (35/16) sin^4 th. [E] NEGATIVE CONTROLS: so8 (lambda~^2 = 12 irrational; perfect squares in the Deligne series only {9, 36} = {sl3, e8}); diag(i, i) (Veronese cone, 5 generators, no hypersurface, the bijection collapses); k = 2 (three channels; honest: lambda~^2 = 36 is k-independent, the kill is the closure dial)
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: WP5e-epsilon-1 [E]/[C]/[O] (verdict B): the O(-2) bulk-axion slot -- executed. CONSTRUCTED exact: the slot is an explicit equivariant mode ledger (Penrose accounting closes block by block, d <= 6, all 4 characters; d = 0 multiplicity (1, 0, 0, 0): the bulk axion survives the projection; Molien = the v492 hypersurface XY = Z^4 with relation degree 8 = the a0 weight; twisted minimal content (2t, 6t^2, 2t) = the Coxeter eigenvalues -- the E3 bijection per character; graviton control O(+2) separates a0); lambda~ = 6 is pinned by three exact ledgers (Okubo (6<x,x>)^2; measure chain mu-exact with the wrong bookings 3 and 12 excluded; flux single-channel iff k = 1); the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from centre geometry (2 branches, period lockstep, source charge 4 = |mu4|, EH asymptotic log exactly 0 -- the CPS log is an exceptional-locus statement, multipole rule m = 0 mod 4 with (4, +-4) carrying -a0). CONDITIONAL [C]: Costello's flat-PT matching. OPEN [O]: the quantised BCOV one-loop coefficient on PT/Z4 and the twisted channels (32 T3) -- M1 (A3 Omega_N), M2 (twisted KS measure), M3 (a0 uplift) preregistered with success + kill criteria; WP5e stays [O]; SEAM.EQUIV.01 untouched; no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; SEAM.EQUIV.01; v492; v493; v495; v505; v508; v509; v511
Scripts: v514_celestial_wp5e_eps1_axion_slot.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 578Original ledger status
CELEST.WP5E.M1.01
MILESTONE M1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M1 -- 'the A3 Omega_N', SUCCESS on the preregistered v514 S8.1 criterion; exact sympy arithmetic, no floats, 30 checks, ~1 s): compute the A3 analogue of CPS eq. 3.5 on PT/Z4 -- the back-reacted holomorphic 3-form Omega_N sourced by the glue currents wrapping the three lockstep spheres -- and check its periods exactly (preregistered SUCCESS = a closed-form Omega-deformation whose S^3/Z4 periods are (2 pi i)^2 x (integer flux vector) reproducing the lockstep phase structure (i-1)(1, i, i^2); KILL = unequal sphere fluxes or a non-integer period). [E] RESIDUE FORM DERIVED: on F = XY - P(Z) all three ambient representatives satisfy dF ^ omega2 = -dX^dY^dZ and pull back to omega2 = dX^dZ/X (Poincare residue, derived not assumed); cover pullback at a0 = 0: omega2 = 4 dz1^dz2 = |mu4| x (flat form) -- the residue normalisation CARRIES the source charge 4; clock diag(i,1): omega2 -> i omega2 (v492 S5 replicated); weight ledger 4+4+2-8 = 2, Omega in O(4) = -deg K_PT; EH anchor 2 = |Z2|; wrong-weight control Z in O(4) breaks three ways (4 != 2, 6 != 4, 5 != 3). [E] THE TWISTOR FAMILY CLOSED: from the four O(2) centre sections q_p(la) = i^p t0 - i^-p t0 la^2 the family is EXACTLY XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2 (e1 = e3 = 0 identically; la = 0 gives exactly the v493 clock family; a2 in O(4) opens only off-seam, a2(0) = a2(inf) = 0); clock lift gamma: (Z, la) -> (iZ, -i la), gamma^4 = 1, Omega -> +Omega (CY-compatible); Omega = omega2 ^ <la dla> in O(4), weight chain 4+4+2-8+2 = 4. [E] OMEGA_N CLOSED + QUANTISED PERIODS: Omega_N = Omega_0 + sum N_p K_p with CPS/Bochner-Martinelli kernels on the centre twistor lines L_p = {X = Y = 0, Z = q_p(la)}; dK = 0 away from the source, scale degree 0; PERIOD TABLE: boundary S^3/Z4 at the orbifold point (2 pi i)^2 x 1 (minimal invariant charge 4 = |mu4| -- the lens fundamental domain FORCES N in 4Z: quotient large-gauge quantisation (2 pi i)^2 N/4 integer only for N = 0 mod 4), deformed phase (2 pi i)^2 x 4, linking S^3 uniformly (2 pi i)^2 x N, per-sphere flux vector (2 pi i)^2 x N (1,1,1) (forced uniform twice over: clock orbit + K4 connectivity of the four centre lines, all 6 pairs meet), seam-fibre phase vector 2 pi i t0 (i-1)(1, i, i^2); the undeformed Omega_0 carries ZERO quantised 3-flux (all flux sourced); (2 pi i)^2 = (fibre residue) x (transverse residue) (density -sin 2 theta); orbifold-limit cover count 16 = |mu4|^2; minimal quantum 1 <-> k = 1 (v509 bridge). [E] CLOCK COVARIANCE: periods 2 pi i (q_{j+1} - q_j) generic-exact (profile and path drop identically), seam vector moduli all 2 sqrt(2) pi t0, Pi_{j+1}(-i la) = i Pi_j(la) for ALL la; the 12 collision nodes sit exactly on the 8 eighth roots of unity (8 = 2|mu4|, the Z8 seam bridge), antipodal double nodes, clock orbits 4+4+4, honest conifold points (Hessian det 512 t0^6, quadric rank 4, exceptional curves = sphere classes). [E] KILL PROBE (honest): the clock-forbidden family Z^4 - Z: 0/24 orderings lockstep (vs 8/24 for the mu4 orbit), la-support 12 twelfth roots != Z8, no antipodal pairing, no clock lift -- BUT the (2 pi i)^2 quantisation itself holds there TOO (local gauge invariance): INTEGRALITY ALONE IS NOT THE DISCRIMINATOR; the discriminator is the lockstep phase/modulus structure + the clock forcing of equal fluxes (unequal N_p allowed on the forbidden family); the v509 dial survives, the KILL does not fire. [E] ANCHORS + CONTROLS: EH/Z2 from the same machinery (XY = Z^2 - t0^2(1 - la^2)^2, exactly 2 nodes at la = +-1, lens forces charge 2 = |Z2|); CPS original = k = 1 flat patch (full S^3: every integer N allowed -- the unconstrained anchor); diag(i,i): Omega_0 -> -Omega_0 (no invariant 3-form, collapse at step zero) while K stays invariant (U(2)); fractional charges N = 1, 2, 3 excluded (lens periods 1/4, 1/2, 3/4)
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: M1 [E]/[C]/[O] (SUCCESS on the preregistered criterion): M1 executed via v515 -- the back-reacted Omega_N is closed-form on the A3 twistor family (Omega_N = Omega_0 + sum N_p K_p, BM kernels on the centre twistor lines), all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector N(1,1,1,1) and clock covariance Pi -> i Pi, and the lens geometry FORCES the source charge 4 = |mu4| (quotient large-gauge quantisation: only N = 0 mod 4 passes); honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden Z^4 - Z family vs 8/24) + the clock forcing of equal fluxes; neither preregistered KILL fires. CONDITIONAL [C]: global kernel patching, Hitchin small resolution, CPS brane dictionary. OPEN [O]: M2 (twisted KS measure), M3 (a0 uplift), the quantised BCOV coefficient on PT/Z4; SEAM.EQUIV.TWISTOR.01 untouched (v515 is preparation, not the Costello-Li construction); no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.BETA.01; SEAM.EQUIV.TWISTOR.01; v492; v493; v505; v509; v514
Scripts: v515_celestial_wp5e_m1_omega_n.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 579Original ledger status
CELEST.WP5E.M2.01
MILESTONE M2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M2 -- 'the twisted Kodaira-Spencer measure', SUCCESS on the preregistered v514 S8.2 criterion ON THE DECLARED COMPLETION MEASURE, verdict B; exact sympy/Fraction arithmetic, no floats, 23 checks, ~1 s): quantise the O(-2) tower on PT/Z4 using the eps1 mode ledger and compute the one-loop box coefficient PER SECTOR (preregistered SUCCESS = the bulk sector reproduces lambda~^2 = 36 with the eps1 measure bookkeeping AND the twisted sectors pair with the three sphere axions cancelling the rigid 32 T3 residual; KILL = a leftover independent quartic (T3-type) in ANY sector). DECLARED MEASURE ANSATZ (stated before computing): v508 killed every exchange realisation; the delta-1 exploration identified the twisted contact term as the MODULAR COMPLETION of the Atiyah-Bott data with forced leading (T5, T3) ratio 4:3; M2 declares contact_j = (Q^(0) - Q^(j))/det_j -- in the channel with the g^j zero-mode normalisation 1/det_j the twisted-sector LOOP contributes the UNPHASED sector trace, of which the AB skeleton kept only the phase-weighted insertion part. [E] THE COMPLETION-WEIGHT IDENTITY (the centrepiece): per glue class w_m = sum_j (1 - i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4 h_m = |mu4| h_m = -4 ch2(T_m) EXACTLY (h_m = m(4-m)/8 the v502 sector Casimir energies, ch2(T_m) the McKay/Kronheimer charges, v505 S4.3 index bridge): the three sphere axions pair through their OWN ch2 charges -- an exact identity, NO free scale, no fit. [E] PARAMETER-FREE LOCKS: T5 = 0 for ANY scale c (the ch2 pattern alone kills T5); ratio h_2 : h_1 = 4 : 3 = the delta-1-forced leading ratio (REPRODUCED, not fitted); the T3 budget fixes the scale c = 4 = |mu4| uniquely; the unique delta-1 leading weights (3/2, 2) reproduced as (4h_1, 4h_2). [E] THE CONTACT TERM: channel form = class form = (36, 120, 60, 0, -32) EXACTLY (one object, two bases); the contacts carry nonzero T5/T3 per channel (genuine contact, not exchange -- the v508 product theorem replicated as control). [E] SUCCESS TABLE: skeleton_j + contact_j = Q^(0)/det_j = 36<x,x>^2/det_j -- a PERFECT Okubo square (T5 = T3 = 0, lambda~^2 = 36) in EVERY twisted channel ((18, 9, 18) x <x,x>^2); total A_fix + contact = 45<x,x>^2 = (5/4) x 36<x,x>^2 = (Dedekind sum) x (Okubo) = the UNIQUE quartic-free weighting of v505 S3.8 (the rigidity theorem is exactly what the measure lands on); both v508 certificates killed (Phi_T3: 32 -> 0, Phi_P: 72 -> 0); the v511 slice psi = 64 SUPPLIED EXACTLY (psi(contact) = -64) -- no cubic d-channel needed, c_d is free to be 0; remainder inside the exchange span (45<x,x>^2 = (45/16) E22 = (1/80) E00, Phi_P = 0): the BULK axion cancels every channel with the SAME lambda~^2 = 36, the twisted propagator carrying the AB zero-mode factor 1/det_j; the v514 S3.4 measure chain stays mu-blind. [E] NEGATIVE CONTROLS: wrong scale c in {1, 2, 3, 5} leaves T3 = (24, 16, 8, -8) -- only c = 4 = |mu4| clears the budget; shuffled weights (h_1 <-> h_2) break T5 = -14 AND T3 = 36; SO(16) glue keeps T5 = 20 / T3 = -40 in the summed trace -- the KILL branch FIRES there (Okubo failure inherited, E8 doubly special); diag(i, i) degenerates to the Z2 target and breaks the conjugation pairing + the index bridge; the Z2/Eguchi-Hanson anchor PASSES with the same mechanism (A_2 + contact = 9<x,x>^2 = (1/4) x 36<x,x>^2, coupling scale 2 = |Z2| = centre count). [E] MULTIPLIER NOTE (cross-reference only, substance belongs to the parallel delta-1d strand): the contact weights 4h are REAL RATIONALS carrying no multiplier system of their own; the delta-1c residual phases are the SQUARES of the A3 discriminant T-phases e(3a^2/8) -- orders (1, 8, 2, 8) squaring to (1, 4, 1, 4), a finite mu4 system; Gauss sums 2e(3/8), 2e(5/8), full Weil sum 4. [C] THE COMPLETION READING (the mandatory fence): the loop of twisted states carries the UNPHASED sector trace with the same zero-mode normalisation -- declared, supported by the delta-1 modular-completion finding, NOT derived from the BCOV integral. [O] the BCOV-integral derivation of the completion reading (delta-1b/1c/1d territory; since executed and DECIDED by v518 -- kill under the derived measure, tension stated -- and the measure question since DECIDED at probe level by v520: the completion reading is now DERIVED at probe level from two independent constructive sources (F-independence + Quillen pairing) under the typed premises TP-1..TP-4 (v520); the residual [O] is the constructive BCOV derivation of the w_m normalisation itself -- since DERIVED at probe level by v523 (CELEST.WP5E.WM.01): 1/det_j is the Atiyah-Bott/zeta-determinant fixed-point factor, computed from three independent sources under the typed premises TP-REG/TP-Q/TP-NUM/TP-CH, and the residual [O] narrows to the GLOBAL BCOV integral beyond the fibre zero-mode factor); SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). VERDICT: SUCCESS wording met on the declared completion measure; the KILL does NOT fire; NO marker moves anywhere.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: M2 [E]/[C]/[O] (SUCCESS on the preregistered criterion, on the DECLARED completion measure): M2 executed via v516 -- the completion contact contact_j = (Q^(0) - Q^(j))/det_j carries the exact weight identity w_m = sum_j (1 - i^{jm})/det_j = 4 h_m = |mu4| h_m = -4 ch2(T_m) (the sphere axions pair through their own McKay ch2 charges, no free scale); parameter-free locks (T5 = 0 for any scale, ratio 4:3 reproduced, T3 budget forces c = 4 = |mu4|); every twisted channel becomes the perfect Okubo square 36<x,x>^2/det_j, total 45<x,x>^2 = (5/4) x 36 = Dedekind x Okubo (the v505 S3.8 unique quartic-free weighting); both v508 certificates killed and the v511 psi = 64 slice supplied exactly -- the cubic d-channel is NOT needed (c_d free = 0); controls: wrong scale/shuffle/so16 (KILL fires there)/diag(i,i)/Z2-EH anchor (scale 2 = |Z2|). FENCE [C]: the completion reading (loop = unphased sector trace with the same zero-mode normalisation) is DECLARED within this module, supported by the delta-1 modular-completion finding, NOT derived from the BCOV integral here -- the completion reading is since DERIVED at probe level from two independent constructive sources (F-independence + Quillen pairing) under the typed premises TP-1..TP-4 (v520); the w_m normalisation is now DERIVED at probe level -- 1/det_j is the Atiyah-Bott/zeta-determinant fixed-point factor, computed from three independent sources (v523); the residual [O] narrows to the global BCOV integral beyond the fibre zero-mode factor; no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.DELTA2.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; SEAM.EQUIV.TWISTOR.01; v495; v502; v505; v508; v511; v513; v514; v515
Scripts: v516_celestial_wp5e_m2_twisted_ks_measure.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 580Original ledger status
CELEST.WP5E.M3.01
MILESTONE M3 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M3 -- 'the a0 uplift', SUCCESS on the preregistered v514 S8.3 criterion; exact sympy arithmetic, no floats, 23 checks, ~7 s): uplift the (l, m) = (4, +-4) GH multipole (= the a0 direction) to the Beltrami differential on PT/Z4 and compute the induced Kahler-potential correction (preregistered SUCCESS = a log-type correction whose coefficient is tied to the source charge 4 = |mu4|, matching the V-ledger; KILL = coefficient decoupled from the centre count). THE UPLIFT OBJECT (declared): the generalized-Legendre-transform kernel per centre, G(eta_p) = eta_p log eta_p (Lindstrom-Rocek / Ivanov-Rocek, typed [C] as a dictionary); its derivative kernel chi = sum_p log eta_p = log P4 is THE LOG OF THE v515 TWISTOR-FAMILY POLYNOMIAL. [E] KERNEL BRIDGE: the O(2) section is a NULL coordinate -- Laplacian(3d) G(eta_p) = 0 identically for an ARBITRARY kernel (the uplift is well-typed); RESIDUE IDENTITY d_x [transform(log eta_p)] = 1/r_p EXACTLY (residue 1/(2r)): the GH potential V = sum_p 1/r_p IS the x-derivative of the contour transform of chi; flux -4 pi per centre, total source charge 4 = |mu4| (the V-ledger matched). [E] FOUR COUPLED CENTRE-COUNT SCALES: (1) asymptotic kernel log chi = 4 log eta + a2/eta^2 + (a0 - a2^2/2)/eta^4 + ... -- LOG COEFFICIENT 4 = |mu4| = centre count (the CPS 'N log' analogue at kernel level); at the clock-fixed seam fibre zeta = 0 the a2 slot closes and the first correction is EXACTLY a0/eta^4 = the (4, +-4) multipole in kernel clothes; exact m-grading (zeta^k/eta^4 carries field charge m = 4 - k: m = +-4 pieces PURE a0, m = 0 piece -6 t0^4 phi0-free -- the U(1) -> Z4 breaking sits exclusively in the a0 pieces); NO log x power terms in the Laurent tower (the v514 honest zero -- no asymptotic METRIC log -- preserved); (2) the once-integrated GLT kernel sum_p eta_p log eta_p = 4 eta log eta + p_4/(12 eta^3) + p_8/(56 eta^7) + ... with p_n = 0 unless n = 0 mod 4 (selection rule at potential level) and p_{4k} = 4 (-a0)^k: EVERY correction carries the prefactor 4 = centre count; (3) the exceptional-locus log chi(eta = 0) = log a0 = 4 log t0 + i(4 phi0 + pi) -- t0-coefficient 4 = |mu4|, clock phase 4i; modulus response a0 d/da0 chi = a0/(eta^4 + a0) equals 1 AT the exceptional locus and falls off as a0/eta^4 at infinity (no constant, no log): the a0-log lives at the exceptional locus, exactly as v514 S6.3 demanded; (4) periods (v515 consistency): Pi_j(la) LINEAR in t0 (uniform first-order shift, lockstep + phase ratios (1, i, i^2) preserved exactly); d log Pi_j / d log a0 = 1/4 = 1/|mu4| uniformly; integrated over the a0 circle: monodromy e^{2 pi i/4} = i = ONE Coxeter clock step (the v493 monodromy REPRODUCED from the perturbative period shift; the finite monodromy is the cyclic centre permutation q_p -> q_{p+1}); node la-support t0-FREE (the Z8 seam bridge is a0-rigid); all 6 pair discriminants ~ t0^2 != 0 (K4 connectivity: the forced uniform flux vector persists); the BM kernel is modulus-free with int_{S^3} K = (2 pi i)^2 (the quantised fluxes cannot move). [E] NEGATIVE CONTROLS: the (4, 0) multipole is phi0-independent in V AND in the kernel (-6 t0^4): clock-invariant, breaks nothing; Z2/Eguchi-Hanson analogue: EVERY dial reads 2 = |Z2| ((2, +-2) amplitude (3/2) sin^2 th, kernel log 2, exceptional log 2 log t0, period response 1/2, monodromy -1 = the Z2 clock); wrong centre counts k = 3, 5: kernel log coefficient k, modulus slot O(6)/O(10) != O(8) -- the observable MOVES with the centre count; forbidden family Z^4 - Z: p3 = 3 != 0 (selection rule broken), e4 = 0 (no a0-log at the exceptional locus) -- fails BOTH uplift dials. [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 (no closed form exists); the quantised BCOV coefficient stays with the delta-1 strand; SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). VERDICT: SUCCESS -- the log-type correction delivered on four coupled dials (kernel log 4, exceptional-locus log 4 log t0, tower prefactor 4, period response 1/4 -> monodromy i), every one tied to the centre count (EH: 2, k-orbit: k); the KILL branch does NOT fire; NO marker moves anywhere.
Recorded status: Formal/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)
Dependencies, status detail and source
Canonical status: M3 [E]/[C]/[O] (SUCCESS on the preregistered criterion): M3 executed via v517 -- the (4, +-4) multipole uplifted to the GLT kernel chi = log P4 (the log of the v515 family polynomial; harmonic for any kernel, residue identity matches the V-ledger with flux -4 pi per centre); the log-type correction is delivered on FOUR COUPLED CENTRE-COUNT SCALES: asymptotic kernel log 4 = |mu4| (first seam-fibre correction exactly a0/eta^4, exact m-grading, no log x power terms -- the v514 honest zero preserved), GLT tower p_{4k} = 4(-a0)^k with the n = 0 mod 4 selection rule, exceptional-locus log chi(0) = log a0 = 4 log t0 + i(4 phi0 + pi), and period response d log Pi/d log a0 = 1/4 = 1/|mu4| integrating to the clock monodromy i (v493 reproduced) with a0-rigid node support and topological (2 pi i)^2 fluxes; controls: (4,0) clock-invariant, EH/Z2 reads 2 = |Z2| on every dial, k = 3/5 orbits move the coefficient with the centre count, the forbidden family fails both dials (p3 = 3, e4 = 0) -- the KILL (decoupling from the centre count) does NOT fire. FENCE [C]: the GLT dictionary; [O]: the full nonlinear Kahler potential of the resolved A3 ALE; no marker moves
Dependencies: CELEST.SEAM.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; SEAM.EQUIV.TWISTOR.01; v493; v505; v509; v514; v515; v516
Scripts: v517_celestial_wp5e_m3_a0_uplift.py
Location: tfpt_3;tfpt_research_contracts
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