Results, with their exact scope.

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GATE.WOLFRAM.02

GATE.WOLFRAM.02

Second [C] path EXTENSION VERIFIED RUNNING: verification/wolfram/tfpt_readouts_extension.wl mirrors the v84-v110 round (frozen registry, master cover, reheating arithmetic, SO(16)_1 modular data, carrier index, conical defect chain, spine tetrahedron, glue uniqueness, Koide relaxation toy, sheet diamond v94, centered diamond v95, branch-kernel selection v96, sheet-conjugation bridge v97, discriminant dictionary v98, Koide flow time v99, horizon anchor v101, seam orientation v102, trisection normal form v103, Nariai clock v104, residual inventory v105, review validation v106, quantum-clock target v107, Pascal ladder v108, sheet pairing v109, Calderon-sheet selection v110, quadratic transport v111, self-counting channel v112, quasi-free kernel v113, torsion delta v114, anchor residue v115, resonance theorem v116, monodromy = W(A3) v117, hexagon-family dictionary v118, second review validation v119, address table v120, address pinning v121, margin theorem v122; inventory update v123 ledger bookkeeping Python-only; resummed clock v124, glue Q-system v125, clock-wall bridge v126, ring resummation v127, graded hull v128, entropy power law v129, Born square v130, measure-is-area v131, det-ratio anomaly v132, zeta budget v133, dual anchor v134, determinant surface v135, dual-normal selector v136, Q+ cohomology v137, VW firewall v138, selector triangle v139, canonical map v140, deck selection v141, frame integrality v142, graded Frobenius v143, det-ratio family cancellation v144, pairing atoms v145, Moebius D4 realisation v146, clock Gaussian model v147, NS/R sector census v148, cusp normal v149, replica EH model v150, BFK split v151, R3 normalisation = anchor v152, No-Unit Theorem v153, Simple-Current Extension v154, seam-net construction v156, rigid fixed point v157, fixed-point stability v158, PyR@TE gauge cross-check v159, QGEO rigidity v168, E8 slice compression v170, atomic OS moment + Sugawara v171, trace-anomaly seed v172, seam Fock readings v174, net existence + full-cone RP v175 - their [N] census/RH/ODE parts Python-only; v155 quasi-free boundary is numerical/numpy, Python-only; the v160-v167 + v163/v164/v166/v169 + v173 rounds are numerical/inventory, Python-only; v176 Seam Collar Realisation Theorem is an assembly/reduction certificate, its exact identities already mirrored via v168/v156/v162/v154/v175, so Python-only; v177 QGEO proof tree is Moebius/cohomology symbolic (uniformisation, omega_k characters, residues), Python-only; v178 MARKS/KERNEL deeper-reduction attempt is Moebius/Schur symbolic + numeric, Python-only; v179 conformal-realisation unification is arithmetic/bookkeeping (Gauss-Bonnet chi=2, h(A3)=4, rank/cohomology counts), Python-only; the v180/v181 Moebius reductions (uniformisation + Kerekjarto + finite-order Moebius / equivariant uniformisation) are symbolic + cited-theorem, Python-only; v182 reviewer-residual map is reduction bookkeeping + arithmetic, Python-only; v183 Koide F-corner 53/54 operator identity IS exact and mirrored; v184-v188 F_transfer-firewall round is numerical/bookkeeping Python-only; v189 Riemann-Roch carrier (h0=5/rankH1=3=rank(D5)/N_fam), v190 Nariai entropy bound ((x-1)^2), v191 universal branch line (affine relabeling + decoy control) ARE exact and mirrored; v192 energy-clock reformulation is [O] structural, Python-only; v201 QGEO.SUBPRIN.01 (a mu4-mark sum is Z4-invariant => sub-principal block-diagonal) and v203 HOR.EHT.01 (the EHT polarization coupling 16 c3^4 = 1/(256 pi^4) = delta_top/3) ARE exact and mirrored; the 2026-06-15 archive-integration round adds v204 FR.MUONG2.01 (a_mu^seam = 45/(524288 pi^9) = (5/4)(48 c3^4)^2/(2 pi)), v205 GRAV.XI.01 (xi = c3/phi_tree = 3/4 = q(A3) = ln(m/mu)), v208 HOR.BHTHERMO.01 (scalaron Wald f_R = 1 + R/(3 Ms^2) + modular 2 pi = 1/(4 c3)) ARE exact and mirrored; the 2026-06-15 pillowcase round adds v214 QGEO.PILLOW.01 (cross-ratio 2 => j=1728 => square modulus => order-4 CM clock, unifying v180 ISO + v201 SUBPRIN; chi_orb=0 + lemniscatic CM order 4 + neg control) and v216 QGEO.MARKS.03 (the four marks from Gauss-Bonnet: n=2chi=4 + Euclidean-orbifold uniqueness; numerical sibling v217 Python-only) exact and mirrored, while v206 H0 branch, v207 asymptotic safety, v209 BH defect are numerical/[P] Python-only; the 2026-06-15 diamond-axis-geometry round adds v218 DIAMOND.AXIS/PLUCKER/SPECTRAL.01 (det(C+xU)=14+6x linear vs det(C+yV)=14+14y+4y^2 quadratic, curvatures (8,6)=(rank E8,|R^+(A3)|); Plucker ladder K->C->F steps (1,8,10),(1,8,16); spectral ramification squares (1,3,4,6) kernels (13,48,65,105)) exact and mirrored, while its anchor-defect/pair-sum audit blocks + F=transfer-corner heuristic stay Python-only). First engine run 2026-06-11 (Wolfram Engine 14.3): v84-v93 block passed 45/45 on first run; v94-v97 block added same day; the 2026-06-17 closure round adds v259 PS.SPECACT.02 (the seam KMS cutoff f(u)=e^{-u} gives f_2/f_0=f_4/f_2=1 exactly, Gaussian control sqrt(pi)/2) and v260 ARCH.K3.01 (one Kummer/K3 carries seam+carrier-16+E8: E8 Cartan det 1/even/pos-def, K3 lattice U^3(+)E8(-1)^2 rank 22/det -1/even/signature (3,19), 16 nodes=|A[2]|=2^4=dim S+, 4 marks) exact and mirrored, while v258 PS.DIRAC.03 (Dirac-as-covariance log((1-C)C^-1)=H matrix-log identity) is numerical/Python-only; the 2026-06-17 v267 (QGEO.SYM.02 rigidity: cross-ratio 2, j=1728<=>lambda in {-1,1/2,2}, hexagonal j=0) and v268 (FLAV.TH13.01: theta_13 exponent 5/6=tr_E Y^2) are exact and mirrored; the 2026-06-18 round adds v271 QFT4D.SPERT.02 (concrete EG one-loop: bubble sd=4=d => omega=0 => one log counterterm, loop factor Omega_3/(2(2pi)^4)=1/(16 pi^2), phi^4 beta=3 lambda^2/(16 pi^2)) exact and mirrored, while v269 (S_pert skeleton) + v270 (PMNS Jarlskog, numerical) stay Python-only; v273 (QFT4D.SPERT.03: the EG one-loop gauge self-energy -> b3=-7, b2=-19/6, b1=41/10 from the content) is exact and mirrored, while v272 (nu-scale), v274 (over-determination) and v275 (QG.AMB.01 roadmap) are numerical/structural Python-only; v277 (QGAMB.TIERB.01: det Cartan(E8)=1 vs det Cartan(D8=SO16)=4 + the (E8)_1 character 248 currents) and v278 (QFT4D.SPERT.04: the one-loop optical theorem, bubble discontinuity (x_+ - x_-)^2 = 1-4m^2/s = the two-body phase space), v281 (QGAMB.MODULAR.01: #anyons=|det Gram| tower 16->4->1 + Gauss-Milgram c=8) and v282 (QGAMB.UNIFY.01: j=1728 + chi_E8(i)=1728^{1/3}=12) are exact and mirrored, while v276 (flat-pillowcase commutator), v280 (pillowcase DtN) and v283 (kill-test scorecard) are numerical Python-only; the 2026-06-22 arithmetic-arc round adds v313 GOLD.ATOMS.01 (affine-E8 charpoly factors x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1) with golden x^2-x-1; (2,3,5)=2cos(pi/k); 31/30 icosahedral selection), v314 RATE.TRANSLATE.01 (rational kernel vs Q(sqrt5) dynamic rate; translation on Q only), v315 COX.COUPLE.01 (30=5*6, EulerPhi(30)=8=rank E8, Galois mu4xZ2, sqrt5 Gauss sum), v316 GALOIS.READOUT.01 (rho=zeta6=zeta30^5, rho^4=-rho, Gal=Z2=CP conj), v317 GALOIS.FAMILY.01 (Nfam=3=mu3 orbit, 1+2 Galois split), v318 ARITH.CAPSTONE.01 (phi0=(4/3)c3+48c3^4 pure pi) and v320 GALOIS.CP.PREDICT.01 (delta_PMNS=4pi/3=delta_CKM,lead+pi=240deg) -- all exact and mirrored (10 checks), while v319 translation clock is structural Python-only; the 2026-06-22 rounds v325-v352 (continuum/realisation/framework: v325/v327/v337/v341/v342/v344/v345/v347/v348/v349/v350/v351/v352) and v354 E8.REVERSE.AUDIT.01 (3 of 8 E8 Casimir degrees {2,8,30} primary readouts, 5 {12,14,18,20,24} = the hull overhead; the golden ratio 2cos(pi/5) is unmapped structure, so not numerology) and v355 E8.UNMAPPED.BANDWIDTH.01 (the unmapped region's forced content is collective: sum(degrees)=128=dim S^+, product=|W(E8)|, exponents=totatives of 30; per-degree atom matches unforced and declined; reconciles v66+v354) are mirrored; and v358 GRAV.ENTROPY.EQUILIBRIUM.01 (the entanglement first law with TFPT atoms gives the linearised Einstein equation parameter-free 1/c3=8pi; the thermodynamic 2pi/eta and geometric |Z2|2pi chi origins of c3 coincide via |mu4|=|Z2|chi=4; CHM 3-ball weight integral 4pi R^4/15 assembles the matter flux J3) and v359 GRAV.NONLINEAR.01 (the full covariant G_ab+Lambda g_ab=(1/c3)T_ab via the fixed-volume equilibrium; Einstein-tensor trace (1-d/2)R=-R in d=4; both coefficients parameter-free 8pi=1/c3 and Lambda prefactor (8pi)^2*48c3^4=3/(4pi^2)) are mirrored; the 2026-06-24 sheet-generator round adds v410 SHEET.GEN.BINARY.01 (V=Q diag(0,1,1) is a binary internal compiler: V^n={{0,2^(n-1),0},{0,2^n,0},{0,2^(n+1)-2,1}}, V^n.1=(2^(n-1),2^n,2^(n+1)-1); bilinears 1.V^n.1=7 2^(n-1)-1, 1.V^n.a=7 2^(n-1), a.V^n.a=11 2^(n-1), a.V^n.1=11 2^(n-1)-2; readouts (6,7,9,11),13,27,55,56; theta sigma3(2)=9=a.V.1, sigma3(3)=28=1.V^3.a=det(I+R), sigma3(5)=126), v411 FLAV.UD.VPOWER.01 (c_u/c_d=(1.V^4.1)/((a.V.1)(1.V^2.1))=55/117, a re-encoding of 5*11/(9*13)), v412 DIAMOND.SHEET.SOURCE.J.01 (the Z2-wall corner J=M(1,-2)=C-2V, chi_J=lam^3-6lam^2+3lam-2, a.J.a=30=h(E8), det(I+J)=12, det(2I+J)=40), v413 DIAMOND.SHEET.CALCULUS.01 (e1=3t+12,e2=2t^2+15t+25,e3=4t^2+14t+14 => Delta e1=3, Delta^2 e2=4, Delta^2 e3=8; anchor energy a.M_t.a=52+11t) and v414 DIAMOND.CENTER.RESOLVENT.01 (resolvent ladder det C=14, det(I+C)=52, det(2I+C)=120 = G2->F4->E8^+) -- all exact and mirrored (5 checks), binary-spine/Lie readings stay [C]; the 2026-06-25 Gaussian/trichotomy round adds v415 DIAMOND.GAUSS.01 (J=[U,V]/3 is the mu4 quarter-turn J^2=-I born from [binary V, ternary U]; 3+2i/5+4i are eigenvalues of 3I+2J/5I+4J with norms 13=Delta_Q/41=10b1; the v222/v230 norms as operator identities; the intrinsic order-4 deck D=-I+J-J^2 with chi2 line ker[U,V]=a-1) and v416 ARITH.TRICHOTOMY.01 ({2,3,5} ramified/inert/split in Z[i] vs Z[omega], the ramified prime of each ring = its atom, each atom the unique ramified prime of one facet of Q(i,sqrt-3,sqrt5)) -- 7 checks; plus a one-character sign fix to the pre-existing v412 CharacteristicPolynomial check (Wolfram returns -(chi) for odd rank), so the file now runs genuinely clean; the 2026-06-25 Eisenstein/CP round adds v417 DIAMOND.EISEN.01 (the hex dual of v415: the family rotation P=(1 2 3) realises the hex norm 7=N_omega(3+2omega) via (3I+2P)(3I+2P^2)={7,7,25}; the CP clock W=-P^2 with W^2=P, W^3=-I gives delta_CKM=pi/3, delta_PMNS=4pi/3; all flavor clocks are zeta30-powers and the seam mu4 is the Galois side phi(30)=8=rank E8) -- 3 checks; the 2026-06-25 cyclotomic-norm-triple round adds v418 ARITH.NORMTRIPLE.01 (the carrier-5 clock = the 4x4 Phi5-companion C5 with the golden ratio in its 2cos values; N(3+2 zeta5)=det(3I+2 C5)=55=quark numerator; the triple det(3I+2 Comp(Phi_{3,4,5}))=(7,13,55) over Z[omega],Z[i],Z[zeta5]; neg control anchor (5,4)->(21,41,461)) -- 3 checks; the 2026-06-25 seam-Galois round adds v419 SEAM.GALOIS.01 (30 squarefree => no order-4 cyclic, so the seam mu4 is the Galois side (Z/30)^x=mu4 x Z2; the mu4 factor IS (Z/5)^x=Gal(Q(zeta5)) (carrier), the Frobenius zeta5->zeta5^2 the explicit order-4 operator G with G C5 G^-1=C5^2, G^4=I; resolves the v409 asymmetry) -- 3 checks; the 2026-06-25 Galois<->Net bridge round adds v422 SEAM.GALOIS.NET.01 (the seam mu4=Gal(Q(zeta5)) is the SAME cyclic Z/4 as the (E8)_1 simple-current glue: disc(A3)=disc(D5)=Z/4 cyclic via SNF, (Z/5)^x=Z/4, glue (1,1) order 4 => 16/4^2=1=E8; Klein/order-2 control => D8 det 4) -- 3 checks; the 2026-06-26 pentagon/golden round adds v429 DM.AXION.PENTAGON.01 (the 'unmapped' golden E8 structure surfaces in the one external input theta_i: since N_fam=g_car-2, theta_i=pi N_fam/g_car=(g_car-2)pi/g_car is the interior angle of the regular g_car-gon = the pentagon 3 pi/5=108 deg, cos=-1/(2 phi), golden character unique to g_car=5) -- 3 checks; the 2026-06-27 other-side audit round adds v430 E8.OTHERSIDE.AUDIT.01 (the double-cover 'other side'/deck |Z2|=2 + the 128-spinor S^+ (+) S^- is forced-disjoint from the five unmapped E8 Casimir degrees {12,14,18,20,24}: deck=matched degree-2=Min(deg), 128=8*16=2^(rank-1)=Total[deg] the spinor half of 248=120+128, sheet set {2,16,32,128} cap {12,14,18,20,24}={}) -- 3 checks; the 2026-06-27 degree-ladder round adds v431 E8.DEGREE.LADDER.01 (the five 'unmapped' degrees are the forced two-family decomposition deg(E8)=6*spine{2,3,4,5} U ({2} U det-ladder{8,14,20}): exponents=phi(30)=8 totatives of h=30=2*3*5 => degrees only in {0,2} mod 6; 6k/6=v91 spine, 6k+2={8,14,20}=(det R,det C,det L)=winding line 6s+8 (v135); 18=6*N_fam not a holdout; clean split special to E8 (E6/E7/D5 fail); arithmetic exact [E]/[I], functorial flavor map stays [P]) -- 6 checks; the 2026-06-29 degree-joint round adds v437 E8.DEGREE.JOINT.01 (all of E8 fixed by its degree multiset -- rank=8, h=30, 120/240 roots, dim 248, |W|=696729600, sum(deg)=128; the two structural integers (g_car,N_fam)=(5,3) are the unique roots of x^2-(rank)x+(h/2)=x^2-8x+15, sum=rank E8=8 + product=h/2=15=g_car*N_fam; mapped degrees {2,8,30}=min/rank/max -- a consolidation respecting the v354/v355 discipline, no new per-degree mining) -- 3 checks; the 2026-06-30 rigidity-forcing round adds v445 SEAM.RIGIDITY.FORCING.01 (the converse of v442: (i) entrywise forcing [rho,E_ab]=(i^a-i^b)E_ab=0 IFF a==b mod 4, verified all a,b in 0..31, residue-only => uniform in N; (ii) exact commutant dim sum n_s^2=4*4^2=64 PROPER subspace of N^2=256; (iii) the order is the discriminator, order-2 commutant 128 strictly larger than order-4 64, swept N=4..64, four marks |mu4|=4=h(A3), only index-4 is (E8)_1 det 1 vs SO(16) det 4) -- 3 checks (also Lean-hardened, FORM.SEAM.RIGIDITY.FORCING.01); the 2026-06-30 edge-CFT/modular round adds v452 SEAM.EQUIV.E8.MODULAR.01 (the (E8)_1 torus modular data: (i) S-invariance chi(-1/tau)=chi(tau) for chi=E4/eta^8 (one primary); (ii) T-phase chi(tau+1)/chi(tau)=e^{-2pi i/3}=e^{-2pi i c/24}, c=8; (iii) leading power chi*q^{1/3}->1 => c=8; (iv) holomorphic c=8=0 mod 8 + T-phase order 24/gcd(8,24)=3 -- E4 via its q-series since EisensteinE stays symbolic) -- 4 checks, v450 SEAM.EQUIV.EDGE.ENTANGLEMENT.01 (Calabrese-Cardy c per Majorana=1/2, N_Maj=16, c_-=8) -- 1 check, v451 SEAM.EQUIV.EDGE.CARDY.01 (the edge is Ising M(3,4): c=1/2, Kac h_sigma=1/16, h_epsilon=1/2) -- 1 check, v453 SEAM.RIGIDITY.MU4FROMMARKS.01 (the marks ARE mu4=roots of z^4-1, the form basis omega_k is mu4-graded, cross-ratio 2 fixed by z->iz => QGEO.SYM.01 from marks=mu4 + QGEO.REALIZE.01) -- 3 checks (also Lean-hardened, FORM.SEAM.EDGE.CHERN.01); the 2026-06-30 G-block round (post-F next-options) adds v454 SEAM.EQUIV.EDGE.VIRASORO.01 (level-1 Sugawara c=dim G/(1+h^v)=8 for BOTH SO(16)_1 120/15 and (E8)_1 248/31; the Casimir and Kac-Moody <J_n J_-n>=n fits Python-only) -- 1 check, v456 SEAM.S3.FROM-P1.01 (c3's '8'=one-sided count |Z2|*(intK/pi)=2*4; a reflection sends C->-C so two-sided forces C=0, one-sided allows C!=0; Chern computation Python-only) -- 1 check, v457 SEAM.EQUIV.KILLTEST.01 (the (E8)_1 character tower {1,248,4124,34752} from E4/eta^8; kill-test discriminators 248=dim E8=120+128, |det Cartan E8|=1 one sector vs |det Cartan D8|=4 four sectors) -- 2 checks, v459 SEAM.EQUIV.LATTICEVOA.01 (E8 even unimodular det 1; the 240 roots split 112 integer + 128 spinor; 248=8+240=120+128 -- the lattice route supplies the 128-spinor MMST v458 leaves open) -- 2 checks (v455 uniform Tomita-Takesaki tower + v458 MMST citation audit are Python-only/structural); the 2026-06-30 v461/v462 round adds v461 SEAM.S3.LOCALITY.01 (the Kapustin-Fidkowski strict-locality obstruction as integer logic: winding=|C|=1!=0 and c_-=8!=0, a strict finite-range commuting projector would force winding 0 => C=-C => C=0 so it is forbidden; the Wilson-loop winding/Chern are numerical, Python-only) -- 1 check, and v462 SEAM.EQUIV.SPINOR.01 (the Jacobi/E8 theta identity theta2^8+theta3^8+theta4^8=2E4 so chi_{(E8)_1}=E4/eta^8=chi_o^{SO16}+chi_s^{SO16}, and the (E8)_1 weight-1 248=120+128 with finite 16-Majorana Fock counts C(16,2)=120, 2^{16/2}=256=128+128; the finite-L convergence Python-only) -- 2 checks; the 2026-06-30 v463/v464 round adds v463 SEAM.EQUIV.UNIQ.01 (the c=8 holomorphic-uniqueness pin: c=8 has THREE level-1 candidates A8/80, D8/120, E8/248 all dim=8*(1+h^v), holomorphy forces dim V_1 = E4/eta^8 q^1 coeff = 248 = E8 excluding A8/D8; tower {1,248,4124,34752,213126}=E4/eta^8) -- 2 checks (v464 the one-particle realization rigidity is numerical, Python-only); the 2026-07-13 entropic-action bridge round adds v473 GRAV.ENTROPIC.ACTION.01 (the carrier Hodge count 1+5+10=16=2^(g_car-1) vs the 4d fiber 11 + the coefficient pin beta'_B=c3/6=1/(48 pi); the entropy potential s(x)=x-1-ln x with the Lambda-channel target Tr Q^2=32 c3^4=1/(128 pi^4)=(2/3) delta_top; the R^2 kill-test gap 3 c3^-9=3(8 pi)^9~1.2e13) -- 3 checks (the Frullani quadrature is numerical, Python-only, flagged in the .wl comment); the 2026-07-13 operator-level follow-up adds v474 GRAV.ENTROPIC.HODGE.01 (the Q-target enumeration d k^2 = 32 with integer k => supports exactly {(2,4),(8,2),(32,1)} = {|Z2|, rank E8, 2^g_car}, the 10/16 blocks excluded, minimal uniform q = c3^2 = 1/(64 pi^2), mode-ratio 32/48 = 2/3; and the fold-is-conjugation weight identity: traceless u(5) weights of Lambda^4 = negatives of Lambda^1, 5 -> 5bar, Pascal blocks {1,5,10} vs {1,10,5}) -- 2 checks (the 32x32 Fock/Clifford matrix checks are Python-only by the matrix-construction convention, flagged in the .wl comment); the 2026-07-13 kill-test round adds v475 GRAV.ENTROPIC.SCALARON.01 (the max-symmetric eigenvalues {R, R/4 x4, R/6 x6} of R~ g~^-1 incl. the 6x6 two-form flattening; the exact vacuum f(R) = 3 b R + (17/24) b^2 R^2 with the raw scalaron m^2 = 4608 pi^2/17 Mbar^2 (trans-Planckian ~51.7 Mbar), enhancement exactly (72/17)(8 pi)^9, domain pole 384 pi^2 Mbar^2; the Lorentzian timelike witness {1 - alpha v^2,1,1,1} vs spacelike {1 + alpha v^2,1,1,1}) -- 3 checks, all exact; the 2026-07-13 scale-flow round adds v477 GRAV.ENTROPIC.SCALEFLOW.01 (the Frullani/scale-flow identity -ln a = int dt/t (e^{-ta}-e^{-t}) symbolic + the moment dictionary at the flat weight = the v475 raw coefficients 3 b R + (17/24) b^2 R^2; the ONE moment condition mu2/mu1^2 = (72/17)(8 pi)^9 with the exact closure (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 and the v36 consistency f0 = 6(4 pi)^2/c3^7) -- 2 checks (the chi-form quadrature is numerical, Python-only, flagged in the .wl comment); current total 603/603 (verified engine run 2026-07-24, Wolfram Engine 14.3: the 16 deferred mirrors v479 (2) + v491 (3) + v493 (4) + v495 (3) + v496 (4) counted with the 2026-07-21 reactivation run, 378 -> 394; same-day v497 +4 -> 398, v498 +5 -> 403, v499 +5 -> 408, v500 +4 -> 412 and v501 +4 -> 416; same-day v502 +4 -> 420, v503 +4 -> 424, v504 +5 -> 429, v505 +5 -> 434, v506 +4 -> 438, v507 +6 -> 444, v508 +6 -> 450, v509 +7 -> 457, v510 +6 -> 463 and v511 +7 -> 470; 2026-07-22 v512 +6 -> 476, v513 +6 -> 482, v514 +8 -> 490 and v515 +8 -> 498, v516/v517/v518 -> 516, v519 -> 522, v520/v521/v522 -> 540; 2026-07-23 v523 +5 -> 545, v524 +4 -> 549 and v525/v526/v527/v528/v529 +3 -> 564), ALL WOLFRAM EXTENSION CHECKS PASSED. Kept separate from the 116/116 base file (GATE.WOLFRAM.01 unaffected). scipy-only parts (v86 pivot ODE, v88 data pulls, v93 PDG pole numerics) stay Python-only, flagged in the .wl comments; the statistical v100 numerology null test (grammar census + MC + RNG controls) is likewise Python-only by convention (like v62/v64/v65), flagged in the .wl comment. UPDATE 2026-07-20 (v491/P2.PARTITION.01): THREE exact mirrors of the partition corollary (unique {1,1,2} => e = (4,5,2); the four negative controls incl. the MS-rational 16/3 and 44/9; the v53 sharpening + sum rule) ADDED to the extension file; the VERIFIED count stays 378/378 -- the local engine is currently unactivated (license), so the new checks enter the counted total with the next verified engine run (same convention as the v479 mirrors); the identical statements are sympy-exact in Python (v491). The 2026-07-20 numerical companions v489 (modular commutator) and v490 (parity census) are numpy/scipy, Python-only by the suite convention, flagged in the README. The 2026-07-20 celestial round: v492 (CELEST.WP1.01) sympy-exact, mirrorable in principle, deferred to the same next engine run (docstring flag); v493 (CELEST.WP2.01) FOUR exact mirrors ADDED to the extension file (sharp clock-invariant selection + frozen j = 1728 quartic invariants + Coxeter/Picard-Lefschetz monodromy with the chi_1 eigenline and regular versal weights + linear-in-a0 S-algebra wrap with the EH k = 2 analogue), same convention -- counted only with the next verified engine run; identical statements sympy-exact in Python (v493); the 2026-07-21 celestial WP3/WP4 round: v495 (CELEST.WP3.01) THREE exact mirrors ADDED (the Deligne closed form lambda~^2 = h_vee + 6 with dimension formula and fundamental-index values; the E8 alignment (kappa/c3)^2 = 12 resp. 1/300 with irrational kappa/c3, anchors and the so8 factor-2 slip; the look-elsewhere pass counts 8/8 / 2/8 / 1/8 / 4/8 / 5/8 + exponents = totatives of 30) and v496 (CELEST.WP4.01) FOUR exact mirrors ADDED (the E4/eta^8 character through q^6 + q j cross-check; the equivariant Hilbert series with quadratic cumulative count and SU(2)+ slice; the two-sided jet-Fock mismatch with strictly increasing f_n/chi_n and the 27000 = h_vee^3 null ideal; the boundary bridges period sum 248 / no tower dimension / loop Fock 897266), same convention -- counted only with the next verified engine run; identical statements exact in Python (v495/v496). UPDATE 2026-07-21: the engine license is REACTIVATED and the run is VERIFIED -- base 116/116 ALL WOLFRAM CHECKS PASSED + extension 394/394 ALL WOLFRAM EXTENSION CHECKS PASSED (Wolfram Engine 14.3); all 16 deferred mirrors (v479 2 + v491 3 + v493 4 + v495 3 + v496 4) now enter the counted total, 378 -> 394; v484/v485/v486/v487/v488/v492 stay mirrorable-but-unmirrored (Python-exact), so they are not counted. UPDATE 2026-07-21 (v497/CELEST.WP5A.01): FOUR exact mirrors of the WP5a null-ideal round ADDED and VERIFIED the same day with the active engine, 394 -> 398 -- (i) the chi_w stabilisation family (u^n coefficient = quarter-moded loop Fock for all w >= n+1, strictly larger for w <= n; w = 2 = the chiral jet grading; loop Fock 897266 at u^4); (ii) the null ideal DERIVED via the Weyl dimension formula in the standard E8 frame (27000 = 30^3 = h_vee^3, 3875, Sym^2 count 30876; Casimirs (60,124,96), Dynkin indices 6750/750; level-1 integrability 4/2; quotient 31124 - 27000 = 4124); (iii) the two-route sector sum at q^2 = (1036,1024,1040,1024) = 4124 with glue diagonal (0,1,1,1); (iv) the SO(16)_1 / false-periodisation / swapped-glue negative controls (5304 != 14^3, 2076 = Theta_D8/eta^8, weights (0,1/2,1,1), period sums 64/124/248/496). The Freudenthal/peeling residual-zero recursion stays Python-side in v497; counting convention honoured -- counted only after the successful engine run (engine active). UPDATE 2026-07-21 (v498/CELEST.WP5B.01): FIVE exact mirrors of the WP5b singular-vector round ADDED and VERIFIED the same day with the active engine, 398 -> 403 -- (i) the case classification of the J^a_1 condition over the 248-element basis from the standard E8 root data (tally (190, 57, 1), exactly one case sees k; theta-grading dims (1,56,134,56,1); a + 2 theta re-enters for exactly one root a = -theta); (ii) the level dial DERIVED from the affine sl2 commutation (F^theta_1 (E^theta_{-1})^n|0> = n(k-n+1)(E_{-1})^{n-1}|0>; square coefficient 2(k-1) = -2(1-k) with values (-2,0,2) at k = 0,1,2; Shapovalov 2k(k-1) = 0/4 at k = 1/2; cube singular exactly at k = 2); (iii) the mu4 glue data (240 glue roots, classes (52,64,60,64), inner grading on all 240, <theta,h> = 5 => j(theta) = 1, clock phase i^2 = -1, cross-table rows (1,56,126,56,1)); (iv) the 27000 match + comark discriminator (mult(2 theta) = 1 in the level-2 Fock, 27000 = 30^3, (2 theta,2 theta)/2 = 4 > 2, E8 comarks (2,2,3,3,4,4,5,6) vs D8 (1,1,1,2,2,2,2,2) with h = (1/2,1,1), 5304 != 14^3); (v) the twisted-slot tension (no two C_1 modes sum to 8 quarters, minimum 6 = q^(3/2); per-period 248). The Chevalley/Frenkel-Kac construction and the PBW singularity run stay Python-side in v498; counting convention honoured (engine active). UPDATE 2026-07-21 (v499/P2.TYPING.01): FIVE exact mirrors of the P2 weight-typing round ADDED and VERIFIED the same day with the active engine, 403 -> 408 -- (i) the partition enumeration under the full typing (unique {1,1,2}, e2 = 5; BG dictionary h^0 = 0 <=> a_i >= 1 exhaustive; the naive-typing negative control h^1(E') = 3 => deg -6 with three splittings, e2 never 5; the h^1 discrepancy h^1(anchor bundle) = 1 != 3); (ii) the Deligne residue-trace degree (cusp exponents {0,1/3,2/3} = unique lift of spec(lam^3-1) to [0,1), trace 1 per mark, deg E = -4, par-deg 0, witness A0* char poly lam(lam-1/3)(lam-2/3)); (iii) the Schur-Horn permutohedron integer points {(2,2,0),(2,1,1)} with h^0 = 0 killing {2,2,0}; (iv) the Biswas Z3-cover degree arithmetic (genus 2, eigensheaves O(-2)^2, chi cross-check -1 = -1, regular-rep weights, decomposable model {0,2,2} with h^0 = 1 = the unstable companion; deck double genus 1, j = 1728); (v) the n-mark formula e2 = (n-2)(n+1)/2 = 5 only at n = 4. The sympy symbolic parts (H^1 residue basis, monodromy group closure |<U,M0>| = 24, the 324-case stability skeleton) stay Python-side in v499; counting convention honoured (engine active); counting convention honoured (engine active). UPDATE 2026-07-21 (v500/CELEST.WP5C.01 + v501/CELEST.WP5D.01): FOUR exact mirrors of the WP5c GNS-limit-state round and FOUR of the WP5d-alpha two-interval-index round ADDED and VERIFIED the same day with the active engine, 408 -> 416 -- v500: (i) the level-2 block census from the E8 root data (31124 = 248 + 30876 monomials in 9361 weight blocks, dim profile {164:1, 37:240, 7:2160, 1:6960}, orbit census (240, 6720, 2160, 240, 1)); (ii) the rank/kernel arithmetic (rank table (0,0,1,8,44) -> total 4124 = 1 + 248 + 3875 = chi_2 with dim V(omega) = 3875 by Weyl; kernel (1,1,6,29,120) -> 27000 = 30^3 = dim V(2 theta)); (iii) the clock-class split at the state level (glue-frame census -> (1036,1024,1040,1024) = Theta_Cj/eta^8 at q^2, level 1 (60,64,60,64)); (iv) threshold w r >= N+1 iff w >= N+1 sharp at w = N, <s|s> = 2k(k-1), D8 census 7380 in 2705 blocks quotient 2076; the exact 9361-block Gram/PSD/h.w.-norm computations stay Python-side in v500 -- v501: (i) the Cartan determinants (det D5 = det A3 = 4, product 16; det D8 = 4; det E8 = 1, explicit matrices); (ii) the KLM/Longo-Rehren quotients 16/4^2 = 1 and 16/2^2 = 4 -> 4/2^2 = 1 (both routes exact); (iii) the Sigma d^2 cross-checks (Ising 4, SO(16)_1 4, (E8)_1 1) + the index-reading arithmetic ([F:F_even] = 2, mu_gauged = 4, 16 ln 2 = ln 2^16, budget ln 4); (iv) the 16-fold-way condensability theta_v = 1 exactly at nu = 2 c_- = 16 vs rivals nu = 1, 2, 8; the lattice curves (Peschel/Renyi offsets, ln 2 plateau, duality asymmetry, ED validations) stay Python-only in v501; counting convention honoured (engine active) UPDATE 2026-07-21 (v502/CELEST.WP5E.ALPHA.01 + v503/QGEO.EMERGE.LIGHT.01): FOUR exact mirrors of the WP5e-alpha prefactor-and-level round and FOUR of the QGEO-emergence-light round ADDED and VERIFIED the same day with the active engine, 416 -> 424 -- v502: (i) the Hurwitz-zeta vacuum energies (zeta(-1,theta) = -B2(theta)/2 at rational twists, E_b = -1/24 + theta(1-theta)/4, Ising shift 1/16, vacuum 8(-1/24) = -1/3 = -c/24 with Sugawara 248/31 = 8, eta^8 exponent 8/24 = 1/3); (ii) discriminant form = Casimir (exact coset minima (0,5/8,1/2,5/8)/(0,3/8,1/2,3/8)/glue (0,1,1,1); spectral flow j^2(h,h)/32 with (h,h) = 20, (h',h') = 12, sum 32; free fermions n/16 on 10+6 = 16 = 2^(g_car-1) Majoranas; deck failure 3/16 != 3/8); (iii) the k = 1 fixing equations (closed forms 5k/8, 3k/8; glue integrality for ALL k = 1..8 -- the honest 'fixes nothing' finding; current condition h(J) = k = 1; embedding (k-1)(47k+266) and 128k(1-k); central charge 240(k-1) = 0; Shapovalov 2k(k-1) = (0,4,12,24); Weyl dims 248/3875/27000, 31124 - 27000 = 4124); (iv) the character sums (four glue-coset thetas = E4 = (1,240,2160,6720); sector characters (1,64,60,64)/(60,64,60,64) sum to q^(-1/3)(1,248,4124,34752); D8 control Theta_D8 + Theta_s = E4 with h_v = 1/2); the sympy symbolics and the PBW engine stay Python-side in v502 -- v503: (i) the CM-point identities (q-series E2 = 3/Pi at i and 2 Sqrt 3/Pi at rho to 30 digits => E2*(i) = E2*(rho) = 0 exactly; j(i) = 1728, j(rho) = 0 KleinInvariantJ; DedekindEta[I] = Gamma(1/4)/(2 Pi^(3/4)) exact); (ii) the no-dynamical-selection witnesses (logF S-invariant, hexagonal winner logF(rho) - logF(i) = 0.02116..., saddle signature of tau = i); (iii) the pillowcase halving (exact N = 8 spectrum, 3 nonzero self-paired cone modes + 30 pairs, dim(even) = 34 = (N^2+4)/2, det'_even^2 = det'_torus x prod_self exactly); (iv) clock => square + Coxeter (SolveAlways a3 = a2 = a1 = 0, j = 1728 identically, charpoly lam^3+lam^2+lam+1 order 4 = |mu4| = N_fam + 1, Aut(hexagonal) = Z6 without order-4 element); the lattice DtN curves stay Python-only in v503; counting convention honoured (engine active) UPDATE 2026-07-21 (v504/CELEST.WP5DB.01): FIVE exact mirrors of the WP5d-beta split-plus-strong-additivity round ADDED and VERIFIED the same day with the active engine, 424 -> 429 -- (i) the GF2 span theorem (Majorana monomials multiply as GF(2) labels: shared boundary Majorana => Even(A) v Even(B) = Even(A u B) exactly 64/64, 256/256, 1024/1024; disjoint exactly half = index 2); (ii) the Pimsner-Popa identity + attainment (E(a) - a/2 = PaP/2 identically on a symbolic 4x4 matrix; explicit gamma matrices give E(x*x) - lam x*x = 2(1-lam) - 2 lam h with spec(h) = +-1 => lambda_PP = 1/2 = 1/[F:F_even] exactly); (iii) the two-interval group + index consistency (E4(vv+) = I/4 exactly => lambda_E4 = 1/4 = 1/mu; (1/lambda_PP)^2 = 4 = 1/lambda_E4 = mu(SO(16)_1) = det Cartan(D8)); (iv) the Lambda^2 compound (Minors[.,2]: Cauchy-Binet functoriality, diagonal products {sigma_a sigma_b}, Minors[-C,2] = Minors[C,2] identically -- the exact orbifold split-inheritance mechanism); (v) the U(1) PP control (charge dephasing lambda = 1/(m+1) -> 0 exactly: infinite index). The lattice curves (deficit tables, Ising exponent p = 0.2444, elliptic-nome ladder, Klich-Levitov slope, ED validations) stay Python-only in v504; counting convention honoured (engine active) UPDATE 2026-07-21 (v505/CELEST.WP5E.BETA.01 + v506/SEAM.CLOCK.RIGIDITY.01): NINE exact mirrors ADDED and VERIFIED the same day with the active engine, 429 -> 438 -- the v505 WP5e-beta round (FIVE): (i) the AB denominators + equivariant characters (det(1-g_j^-1) = (2,4,2), Dedekind sum 5/4 = (|Z4|^2-1)/12; characters (248,0,-8,0) by TWO routes -- graded dims and the explicit 240-root sum; invariant average 60 = carrier, Frobenius 61568); (ii) the index bridge (ch2(T_m) = -(C^-1)_mm/2 = f(m) exactly; glue defect -78 both routes, classes (-24,-30,-24); sum_m h_A3(m) = 5/4 = sum_j 1/det_j); (iii) Okubo per sector + the rigid 32 T3 residual (the invariant quartic sum over the 240 glue roots = 36<x,x>^2 exactly; twisted quartics carry T5/T3 content; AB-weighted sum = 9P1 - 30P2 - 15P3 + 32T3 exactly); (iv) the level dials (lattice current count 240/0/0/0 for k = 1..4; embedding residual (0,360,814,1362); det(A-1) = -4); (v) the axion-slot weight arithmetic (clock weights = regular mu4 rep; a0 Hamiltonian in O(2) = the graviton slot, X = -2 != +2 = Y, mismatch 4 = |mu4|; Coxeter eigencharacters {i,-1,-i} = the three twisted slots); and the v506 clock-rigidity round (FOUR): (i) the complete square-root solve (g^2 = deck forces diagonal with (a/d)^2 = -1 -- exactly two roots z -> +-iz, both projective order 4; the traceless branch scalar with det 0); (ii) the D4 center + order-4 count + the silver counterexample (the mu4 stabiliser has profile {1,2,2,2,2,2,4,4}, exactly 2 order-4 elements squaring to the deck, center = {id, deck}; the silver set {+-1, +-(3+2 sqrt 2)} has j = 1728 with a full D4 of 8 exact elements but its order-4 elements square to the central crossing involution, not the deck: zero roots); (iii) the Cl(16) identity (explicit 256x256 Jordan-Wigner gammas: Utilde^2 = 256 gamma_1...gamma_16 exactly, U^2 = (-1)^F; the R implementer squares to +256, split control); (iv) the n-arithmetic + generic kill (S4 orders {1,2,3,4}, kill table n = 5..8, free orbits n | 4, cusp degree 5 != 3, transfer spectrum {1, 64/729, 1/729}, generic V4 exponent 2). The 24-triple stabiliser enumeration, the A4 profile and the Z8 block-lift nullspace stay Python-only in v506; counting convention honoured (engine active) UPDATE 2026-07-21 (v507/SEAM.BIT.ORIGIN.01): SIX exact mirrors of the bit-origin round ADDED and VERIFIED the same day with the active engine, 438 -> 444 -- (i) the half-period descent (fixed points of w -> -w = the 4 half-periods Lambda/2Lambda, 1/3-control fails; t_c descends iff 2c in Lambda = exactly the 3 nonzero half-periods; the explicit rational descent sigma_i(x) = e_i + (e_i-e_j)(e_i-e_k)/(x-e_i) with the exact curve-lift identity prod(sigma_i-e_l) = psi_i^2 prod(x-e_l) symbolic in e1,e2,e3 and the Klein-V4 closure); (ii) the core solve (pair cross-ratio = -1 solves exactly to lambda = -1/2/1-half per half-period; harmonic orbit j = 1728 vs j(3) = 21952/9 and j(hex) = 0); (iii) the stabiliser counts (complete 24-triple enumeration: (4,3,3)/(8,5,3)/(12,3,3), free involutions = the three sigma_c in ALL types); (iv) the CM fixed point (mult-by-i mod 2 fixes exactly (1,1) = c*, mult-by-rho 3-cycles; the lemniscatic clock lift needs psi = 2i/(x+1)^2 with t^2+1 irreducible over Q and squares to the central -1/x); (v) the Cl(16) dichotomy (central Utilde^2 = 256 gamma_1...gamma_16 vs the edge implementer squaring to the SCALAR +2^7 on the same 256x256 Jordan-Wigner gammas, with implementation verified on all 16 generators; root count 2 vs 0 in the 32-element NS dihedral lift group); (vi) the silver bijection census (all 8 silver bijections -> edge sigma_2/sigma_3, never central; all 8 mu4 bijections -> central sigma_1 only). The v493 circularity-audit direction and the 28/21 arrangement census stay Python-side in v507; counting convention honoured (engine active) UPDATE 2026-07-21 (v508/CELEST.WP5E.GAMMA.01): SIX exact mirrors of the WP5e-gamma round ADDED and VERIFIED the same day with the active engine, 444 -> 450 -- (i) the vertex-space dims (the W(D5) x W(A3)-invariant quadratics on the 8-dim glue Cartan have bidegree dims (1,0,1), total 2 = span{s5, s3}, and the quartics (2,0,1,0,2), total 5 = span{P1,P2,P3,T5,T3} -- exact generator-wise Weyl nullspace arithmetic); (ii) the twisted quadratics + the even-sector collapse (K^(0) = 60(s5+s3), K^(1) = K^(3) = 4s5-12s3, K^(2) = -4(s5+s3), K^(0) = -15 K^(2) parallel); (iii) the product theorem (symbolic a,b,c,d: any two invariant quadratics multiply into span{P1,P2,P3} -- zero T5/T3 content); (iv) the rank-3 certificate (channels E13 = (16,-96,144,0,0), E22 = (16,32,16,0,0), E00 = 225 E22; rank 2, kernel 0, rank([M | A_fix]) = 3 with A_fix = (9,-30,-15,0,32) and (Phi_T5, Phi_T3, Phi_P)(A_fix) = (0, 32, 72); naturalness images (25/4,-11/2,97/4,0,0)/(12,-40,76,0,0) both certify (0,0)); (v) the SO(16) AB sum (15,-18,-15,-4,40); (vi) the D8 native quartic 28 s8 Killing + 16 T8 + 12 s8^2. The selection-rule triple enumeration and the flipped-rule control stay Python-only in v508; counting convention honoured (engine active) UPDATE 2026-07-21 (v509/CELEST.WP5E.EPS2.01): SEVEN exact mirrors of the WP5e-epsilon-2 round ADDED and VERIFIED the same day with the active engine, 450 -> 457 -- (i) the CPS period integrals (S^3 period of the brane back-reaction 3-form = (2 pi i)^2 exactly, pullback density -sin 2theta; exceptional-sphere Kahler flux = 2 pi N exactly, flat part dies at t -> 0); (ii) the level~flux contraction (+2N/-2N for N = 1,2,3) with Dynkin T(so8 vector) = 1, T(adj) = 6; (iii) the pairing-matrix enumeration 48 -> 2 (rows (8,6,8), 48 unimodular, effectivity cuts to {identity, flip}); (iv) the three fluxes (64,60,64) = dim g_i, total 188, flip-invariant, per-current (1,1,1); (v) the sector-counter dial #prim((E8)_k) = (1,3,5,10,15,27) for k = 1..6, one sector iff k = 1, KLM 16/4^2 = 1; (vi) the lockstep theorem + the 0/24 negative probe (a3 = a2 = a1 = 0 forced, squared moduli 2t^2 equal, det(A-1) = -4, Z^4 - Z disc -27 with zero lockstep orderings); (vii) the negative controls (SO(16) fluxes (0,60,0) with defect -30 and condensation stuck at 4; A2 det 3, glue diagonal 23/24, mu = 12 no square, Coxeter 3 != 4). The lattice current count (240,0,0,0) and the k = 2 dials are already mirrored in the v505 round; counting convention honoured (engine active) UPDATE 2026-07-21 (v510/SEAM.BIT.FREEDOM.01): SIX exact mirrors of the seam-bit-freedom round ADDED and VERIFIED the same day with the active engine, 457 -> 463 -- (i) the dihedral census + quotient topology (D_16: 17 involutions, exactly 1 free = the antipodal shift 8; all 16 reflections with exactly 2 circle fixed points, 8 site- + 8 edge-axis; robustness N = 8, 12; odd control N = 9 zero free; chi(circle/antipode) = 0 closed vs chi(circle/reflection) = 1 interval with 2 corners); (ii) the Aut completeness (Aut(C_8) = D_8 by brute force over all 8! = 40320 permutations; Aut(C_16) = D_16 by degree-2 propagation, all 32 candidates); (iii) the Cl(16) dichotomy census (the free deck's NS implementer squares to 256 gamma_1...gamma_16 = 256 (-1)^F nonsplit while the implementers of ALL 16 reflections exist and square to scalars {2^7, 2^8} -- NONSPLIT <=> FREE over the complete 17-involution census; root dichotomy 2 vs 0 total on explicit 256x256 Jordan-Wigner gammas); (iv) the real-structure table (M D = +-D M solves symbolically to the diagonal/antidiagonal families; per configuration mu4 -> equatorial/FREE, silver -> real-axis/NOT free, generic-real -> real-axis, generic-unit -> equatorial/free, hexagonal -> NO mark-fixing real structure); (v) harmonic + free => mu4 ((1-b)^2/(1+b)^2 = -1 solves exactly to b = +-i; lambda table (2, 1/2, 3/4, 5/4) real, hexagonal Im lambda = sqrt(3)/2); (vi) the honest counterwitnesses ({0,1,8,9} shift-invariant + crossing but not quarter-shift-invariant; {+-1, +-(3+4i)/5} with pair cross-ratio -1/4, j = 148176/25, clock not preserving). The Moebius/Klein covering models and the pillowcase branch control stay Python-side in v510; counting convention honoured (engine active) UPDATE 2026-07-21 (v511/CELEST.WP5E.DELTA2.01): SEVEN exact mirrors of the WP5e-delta-2 full-tensor-ledger round ADDED and VERIFIED the same day with the active engine, 463 -> 470 -- (i) the innerness count + g0 structure (240 glue roots, class split (52, 64, 60, 64); h = (2,2,2,2,2;0^4) and h' = (0^5;1,1,1,-3) read the class mod 4 on all 240 roots; class-0 roots split 40 pure D5 + 12 pure A3, no mixed root, root span rank 8 -- g0 = d5 (+) a3 semisimple, no u(1); class(-alpha) = -class(alpha)); (ii) the bilinear Hom table by exact Kostant alternating sums on the minuscule weight orbits (nonzero ONLY at j' = -j, dims (2, 1, 1, 1), all 12 charged pairs zero); (iii) the trilinear Hom table (all 15 non-neutral triples zero, neutral dims (3, 2, 2, 1, 1)); (iv) the unique symmetric survivor (Inv(S^3 adj) = (0, 1) for (d5, a3), Inv(L^3 adj) = (1, 1), Inv(S^2 adj) = (1, 1); Inv(S^3 g0) = 1, Inv(L^3 g0) = 2 -- the su(4) d-symbol; dual guard 4 x 4 = 0 vs 4 x 4bar = 1); (v) the controls (SO(16)/D8 bilinears (1, 0, 1), trilinears (1, 0, 1, 0), Inv(S^3 adj_d8) = 0, Inv(L^3 adj_d8) = 1; false g0 = d5+a2+u(1): (5, 2, 2) != (2, 1, 1), Inv(S^3 g0) = 6 != 1; false sector (16_s, 4bar): both pairings 0 vs true value 1); (vi) Q_dd by two routes (explicit su(4) matrices: brackets/root-contractions/tadpoles all 0; Q_dd = (0, 0, -1/240, 0, 1/60) = (1/60)(T3 - P3/4) agreeing with the hand route |x^2 - (tr x^2/4) 1|^2/60; Tr X^3 = 0 antisymmetric); (vii) the psi certificate + the relaxed solution (E13/E22/E00/A_fix replicated; rank 3 vs augmented 4; psi = Phi_P - Phi_T3/4 annihilates all four channels, psi(A_fix) = 72 - 8 = 64; relaxed A_fix = -u + 8v + 2w + 1920 Q_dd exactly, 1920 = |W(D5)| = 8 x 240). All v511 content is exact and fully mirrored; counting convention honoured (engine active) UPDATE 2026-07-22 (v512/SEAM.TAU.FLAG.01 + v513/CELEST.DTERM.NONDERIV.01): SIX exact mirrors of the flag-transitivity round ADDED and VERIFIED the same day with the active engine, 470 -> 476 -- (i) the V4/D4 census table over five exact members incl. flag/arc orbit counts (generic (4, 2, transitive, 2, 2), mu4 (8, 4, transitive, 1, 1)); (ii) the solveset equivalences (b^2 = -1 / cos delta = 0 / tan(delta/2) = 1 / delta = pi - delta all reduce to {pi/2}) + family identities (lambda = sec^2(delta/2), cross-ratio = -tan^2(delta/2); counterwitness lambda = 5/4, j = 148176/25, cross-ratio -1/4); (iii) the spectral faces in closed form (odd-split exp(pi split/2) = cot(delta/2), arc-Laplacian spectrum, K3 Frobenius^2 = 8(u-v)^2); (iv) the twist mode identity + AB delta-blindness + j-rationality control; (v) H4 rigidity (centraliser scalar, intertwiner prop U, U^2 nonscalar, V4 orders {1,2,2,2}); (vi) the negative controls (hexagonal/silver/Z16 0-vs-28 census/n = 3 rigidity) -- the mpmath heat-trace numerics and the numpy lattice commutator stay Python-only in v512, flagged; and SIX exact mirrors of the c_d-nonderivation round ADDED and VERIFIED the same day, 476 -> 482 -- (i) the propagator anchor sum <alpha,e>^2 = 120 + T3-slot pinning (det -64, c_d = 1920); (ii) the five-way convention table (1920, 120, 256, 32, 69120) with exactly one |W(D5)| hit + the stable factorisation c_d Phi_T3(Q) = 32 in all five; (iii) the flux enumeration (2048/1800 bracket-and-miss, 64*60/2 charge-forbidden) + look-elsewhere ledger (924 expressions, 11 hits on 1920; controls 8/0/5); (iv) the K2 lattices (quantum 6 vs 3/8, 60 Z 32nd multiple, 2160 Z excludes, 12 factorisations); (v) the K3 provenance clash (A_m = (0,16,0,-16), 32i vs (10,-4,30,-4); |W(D5)| = |W(E8)|/|W(A8)| = 1920, |W(A3)| = 24); (vi) the K4 controls (SO(16) cubic sums 0, k = 2 doubling 7680, false g0 six cubics). All exact content of both rounds fully mirrored; counting convention honoured (engine active) UPDATE 2026-07-22 (v514/CELEST.WP5E.EPS1.01): EIGHT exact mirrors of the WP5e-epsilon-1 bulk-axion-slot round ADDED and VERIFIED the same day with the active engine, 482 -> 490 -- (i) the equivariant Penrose ledger (pushforward blocks = exact wave-operator kernels block by block, all d <= 6, all four characters, (d+1)^2 per degree); (ii) the 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 with the surviving bulk-axion zero mode (1,0,0,0), density 1/2 = (1/4) x 2 and twisted minimal content (2t, 6t^2, 2t); (iii) the Molien hypersurface (1-t^8)/((1-t^4)^2(1-t^2)) with relation degree 8 = the a0 weight + the diag(i,i) Veronese control (1+3t^4)/(1-t^4)^2 and the Coxeter clock det(A-1) = -4; (iv) the Okubo residue (6<x,x>)^2 on the 240 glue roots + the so8 irrationality control (perfect squares {9, 36} only); (v) the measure cancellation (lam_eff^2 = 36 for arbitrary mu, wrong bookings 9 and 144 excluded) + the flux dial (#primaries (1,3,5,10), (kappa/c3)^2 = 12); (vi) the GH centre dichotomy re-deriving the v493 family (prod(Z - i^p z0) = Z^4 - z0^4, monodromy = one clock step); (vii) the period lockstep 4 pi t0 (i-1)(1, i, i^2) + the GH point = R^4/Z_k cone + the Coxeter clock Pi.A = i Pi from geometry; (viii) the multipole selection rule m = 0 mod 4 with the (4, +-4) amplitude (35/16) sin^4 th + the EH log ledger (det g = 1 exact, asymptotic log coefficient 0, exceptional flux 2 pi a^2, Burns contrast det g = 1 + N/u). All exact v514 content fully mirrored; counting convention honoured (engine active) UPDATE 2026-07-22 (v515/CELEST.WP5E.M1.01): EIGHT exact mirrors of the WP5e-M1 back-reacted-Omega_N round ADDED and VERIFIED the same day with the active engine, 490 -> 498 -- (i) the residue-form identities (invariant-cover pullback 4 dz1^dz2 = |mu4| x flat, deck-invariant, clock phase i; weight ledger 4+4+2-8 = 2, Omega in O(4) = -deg K_PT; EH anchor 2 = |Z2|; wrong-weight control 4 != 2); (ii) the twistor family closed (O(2) reality of the four centre sections, XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2 exactly, seam fibre = the v493 family, self-mirror, CY-compatible clock lift gamma^4 = 1 with Omega -> +Omega); (iii) the period reduction + lockstep (2 pi i (q_b - q_a) generic, seam vector 2 pi i t0 (i-1)(1, i, i^2) with equal squared moduli 8 pi^2 t0^2, covariance Pi_{j+1}(-i la) = i Pi_j(la) for all la, Pi.A = i Pi, closure 0, residue 0 = the zero-flux Omega_0 baseline); (iv) the collision ledger (12 nodes exactly on the 8 eighth roots of unity, adjacent la^2 = i^(2j+3), antipodal double nodes, clock orbits 4+4+4); (v) the BM anchor (density -sin 2 theta, int_{S^3} K = (2 pi i)^2, iterated two-circle residue) + the lens forcing N in 4Z (only N = 0 mod 4 passes; fractional charges excluded); (vi) the source curve (four centre lines on the family, clock-permuted, K4-connected with 6 edges / 12 points, cover count 16 = |mu4|^2); (vii) the conifold nodes (Hessian det 512 t0^6, quadric rank 4, transversality, Pi_0(la0) = 0); (viii) the KILL probe + negative controls (forbidden Z^4 - Z family 0/24 lockstep vs 8/24, twelfth-root support != Z8, no antipodal pairing -- integrality alone NOT the discriminator; EH charge 2 forced; diag(i,i) collapse). The dict-based exterior-algebra identity dK = 0 and the generic-profile sphere parametrisation stay Python-side in v515, flagged in the .wl comment. The 2026-07-22 WP5e-M2+M3 rounds: v516 (CELEST.WP5E.M2.01, the twisted KS measure) adds SIX exact mirrors (completion-weight identity w_m = sum_j (1 - i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4 h_m = -4 ch2(T_m) with the index bridge and no free scale; root/quartic replication (class quartics, A_fix two routes, bulk Okubo 36<x,x>^2 = (K^(0))^2/100); parameter-free locks (T5 = 0 symbolic, T3 budget forces c = 4 = |mu4|, det 256); contact term two forms + the success table (per-sector Okubo squares 36<x,x>^2/det_j, total 45<x,x>^2 = (5/4) x 36, certificates 32/72/psi = 64 killed, exchange span, mu-blind chain); negative controls (wrong scale, shuffle, SO(16) KILL, diag(i,i), Z2/EH anchor); multiplier note (T-phase orders (1,8,2,8) squaring to (1,4,1,4), Gauss sums)), 498 -> 504 -- the declared completion reading itself is the [C] fence, a typing, not mirrorable; and v517 (CELEST.WP5E.M3.01, the a0 uplift) adds SIX exact mirrors (GH multipole ledger + family at general orbit phase; kernel bridge (null coordinate for arbitrary kernels, residue 1/(2r), flux -4 pi); asymptotic kernel log 4 = |mu4| + exact m-grading + GLT tower p_{4k} = 4(-a0)^k; exceptional-locus log 4 log t0 + i(4 phi0 + pi); period response 1/4 -> monodromy i with a0-rigid nodes and K4 discriminants; negative controls ((4,0) phi0-free, EH dials all 2, k = 3/5 move, forbidden family fails both dials)), 504 -> 510 -- the GLT contour-transform typing stays the [C] dictionary, the BM anchor is already mirrored in the v515 round; and v518 (CELEST.WP5E.DELTA1.01, the delta-1 chain decided -- kill under the derived measure, tension with v516 stated) adds SIX exact mirrors (the 16-component Weil system of the Z4 x Z4 discriminant module with Gauss sums 2 zeta_8^5 x 2 zeta_8^3 = 4 and the full exact relations + tensor split; the two invariant Lagrangians with P_H S P_H = (1/4) J and the 3/4 shifted-class weight; the koboundary test (1,1,1) on all 15 pairs from the recorded exact transport phases; the Gamma_1(4) character formula lambda(gamma) = i^{2B + C/4} on the recorded loops; the T-fix mechanism (-1) e(-1/6) = e(1/3) = chi_4(T) uniform for f1f3 vs inconsistent {chi_3, chi_4}/{chi_5, chi_6} for f1f2f3/f2; the wrong-form/wrong-signature controls), 510 -> 516 -- the Harvey-Moore integrals, SVD contraction solves, theta covariance certificates and block transport measurements are numerical (mpmath/numpy), Python-only, flagged in the .wl comment; counting convention honoured (engine active); and v519 (WOIT.THETA.FREE.01, the WOIT-alpha milestone -- Theta + free RP) adds SIX exact mirrors (the two-family Theta classification with the symbolic Solve branches and the family-D/family-A relation tables; mark pinning + the mu4 torsor with the clock-orbit map mu -> -mu, the mark permutations and the 2-cut-point census; the C^4 level rho_tw vs sigma_std (Theta_tw^2 = -1, -i rho4, deck centralised vs conj = inverse, +1, RP^3); the Z8 tower conjugation conj(zeta8^k) = zeta8^-k with the spin-lift inversion/centralisation split; the Cl(16) dichotomy on explicit 256x256 Jordan-Wigner gammas (even NS reflection implementers with U_r^2 = 128 = 2^7 on both mu4 axes, the quarter-shift clock lift V with U_r V U_r V = 2^19 i.e. Theta_Fock V Theta_Fock^-1 = 4096 V^-1, Utilde^2 = 256 (-1)^F and Utilde V = V Utilde); the free-RP preconditions + site-cut degeneration + Cauchy-Stieltjes continuum control (C^2 = -1 at N = 8, R C R^T = -C / S C S^T = C / R S R^-1 = S^-1 / deck fails at N = 16, the 7x7 site-cut Gram with zero diagonal and det = 0, the exact factorisation of 1/sin((s+t)/2) with minors 1/2, 1/72, 1/43200, 1/423360000)), 516 -> 522 -- the 40-digit PD inertia certificates of the bond-cut Grams ((8,0,0)/(29,0,0)/full N = 8 half-algebra) are numerical (mpmath eighe), Python-only, flagged in the .wl comment; and v520 (CELEST.WP5E.MEASURE.01, the WP5e measure decision) adds SIX exact mirrors (the 16-component Weil data with both Lagrangian gluings T-/S-fixed and isotropic; the invariant-subspace forcing: joint fixed space dim 2 = span{e_H, e_H'} by exact NullSpace + rank; the character arithmetic of the two constructive sources |chi_4(T)|^2 = 1 vs chi_4(T)^2 = e(2/3) with the exact 1/960-grid squared defects 2x480 = 0 vs 2x800 = 640; the A-core + Z2/EH-anchor targets w = 4h, psi(A_fix) = +64/psi(contact) = -64, A_2 + contact_2 = 9<x,x>^2 with the c = 1/2 scale teeth; the level-1 Okubo identity of BOTH gluings (the anti-diagonal H' has exactly 240 norm-2 vectors with summed quartic 36<x,x>^2); the so16/shuffle/diag(i,i) controls), the Harvey-Moore integrals/SVD memberships/J-weighted closures numerical (mpmath/numpy), Python-only, flagged in the .wl comment; and v521 (SEAM.BIT.RPBLIND.01, the RP-blindness kill) adds SIX exact mirrors (the anti-linear census solvesets on M(delta) incl. the exact-rational counterwitness census; the V4 -> D4 closure / face #13 as symbolic composition identities; the 20 cut/mark incidence solvesets all empty + the eps-jump + the integer lattice frame; the continuum axis elimination 1/sin((s+t)/2) with d/d alpha = 0 exactly + the Cauchy-Stieltjes minors + the counterwitness cut ordering; the clock-tower clause census {F,F,F,T} + the hex/silver precondition controls; the delta-blind side conditions Theta_mu^2 = +1 + the mu4-torsor clock relation with the order-4 solveset {pi/2}), the 40-digit lattice Gram inertia spectra (N = 16/N = 32) numerical, Python-only, flagged in the .wl comment; and v522 (WOIT.BETA1.GSO.01, the WOIT-beta1 milestone) adds SIX exact mirrors (the Z8/NS tower S^4 = -1 + the Ramond state-invariance witness 1/(4 sin(5 pi/16)); the invariant-dimension chain 120 > 64 > 32 by exact ranks of the averaged signed pair action; the time-like reflection census (all 16 axes invert the clock, none commutes, same at N = 8); the exact Hermiticity witness T_1[(8,12),(8,15)] = -i/(8 sin(5 pi/16)) with the full 745/96/0 entrywise census and the non-Hermitian clock average on the even 29-basis via exact Pfaffian-Wick; the GSO 2-torsion census (even {T,F,F,F}, odd {T,F,F,F,T,F,F,F}, T_{k+4} = -T_k, vanishing 8-term average); the family-A tower-normalisation failure census {F,F,F,F} + the alternating-dressed anti-invariance), the GSO-fixed RP inertia certificates ((29,0,0)/(8,0,0)/(7,9,6)/(17,12,0)) numerical (mpmath eighe), Python-only, flagged in the .wl comment, 522 -> 540; counting convention honoured (engine active). UPDATE 2026-07-23 (v523/CELEST.WP5E.WM.01 + v524/WOIT.BETA2.OS.01 + v525/v526/v527/v528/v529/SEAM.BIT.TWISTBLIND.01): FIVE exact mirrors of the constructive w_m derivation ADDED (the mode ledger = closed form at every order n <= 40 with Abel value (1/2, 1/4, 1/2) = 1/det_j, Dedekind sum 5/4, product 16; the zeta/Quillen route with the SYMBOLIC reflection formula (2 pi/(Gamma(a) Gamma(1-a)))^2 = 4 sin^2(pi a), det Delta = det_j^2 = (4, 16, 4), the unique real positive section and the diag(i,i) control (2i, 4, -2i) with fake weights (0, -1/2, 0, 3/2); the derived-weight contact vectors + per-sector Okubo squares + total 45<x,x>^2 + psi +-64 + the Z2/EH anchor on the 240 glue roots; the consistency chain w = 4h with the T3 budget forcing c = 4 and both wrong-weight controls incl. the Q(sqrt2) leftovers; the k = 3, 5 cyclotomic wandering controls w_m = m(k-m)/2), FOUR exact mirrors of the WOIT-beta2 OS quotient ADDED (the alpha_1^N = (-1)^F wrap arithmetic at N = 16 and N = 8 + the exact certificate sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2; the Klein-Landau one-particle transfer forms exactly Hermitian on all domains k = 1..7 with odd-k zero diagonal and the even-k square identity entrywise; the exact N = 8 clock spectrum {1, sqrt(2)-1} = {1, 1/delta_S} via generalized eigenvalues + the transparency pi/16 product formula certified symbolically; the KMS silver witnesses C(1)/C(3) = 1 + sqrt(2) and det(G - tau_4) < 0 exact), and THREE exact mirrors of the twist-state kill ADDED (the N = 32 integer incidence frame + the sigma o r = +-sigma congruence mechanism on both axis types; the pi/2 existence solveset + the (3+4i)/5 root-of-unity census k = 1..64; the delta <-> pi - delta mirror equivariance behind the Casimir symmetry), all VERIFIED the same day with the active engine, 540 -> 564 (the Cesaro/Hurwitz/stripped-block numerics of v523, the H_phys Grams/40-digit inertia spectra/spectral projections/reconstructed rotation group of v524 and the Pfaffian insertion spectra/Bogoliubov 2-of-16 pattern census/defect energy of v525/v526/v527/v528/v529 stay Python-only, flagged in the .wl comments); counting convention honoured (engine active); TWENTY exact mirrors of the 2026-07-23 review-round promotion ADDED (v530 center quotient compiler: fixed line Cv = v, unimodular quotient = [[8,2],[5,3]] atom matrix, invariants + self-code (7,11,13) + CP code (1,7,11,13), resolvent factorisation det(C+kI) = (k+1)det(A+kI), honest K/Q uniqueness controls -- 6 checks; v531 Stage A tensor clock: T30 = Comp(Phi5) (x) Comp(Phi6), chi = Phi_30, exact order 30, full Ramanujan tomography c_30 = c_5 c_6, honest Phi_15 signed-table scope, coprimality controls -- 5 checks; v532 dual-degree modular checksum: (60,192,240,252) sum/product/gcd, q-series j = q^-1 + 744 + 196884 q, D16 control 1488 = 3*496, six non-heterotic control failures, prime-totative fingerprint with maximality scan -- 5 checks; v533 disc -7 norm line: D(t) = N(alpha_t), translation orbit norms (2,4,14,32), constant discriminant -7 + vertex, winding/anchor-block controls -- 4 checks), all VERIFIED the same day with the active engine, 564 -> 584; 2026-07-24 v534 straddle-cone round +4 -> 588 (stabilizer census, pi/2 partition + straddle incidence, reflection-closure signs, equivariant Fix rays incl. the NS-wrap-twisted m=1/7 rays -- all exact; the 256-dim Fock spectra, PT Gram corrections and finite-g survival scans stay Python-only, flagged); 2026-07-24 v535 Hecke-from-geometry +5 -> 593 (Kneser count identity, affine (a,b) law, lambda/a_p cusp rule, dim-7/projector/2-adic oldform arithmetic -- live p=3 FP census and q-series stay Python-only, flagged); 2026-07-24 v536 Eichler-trace-layer +5 -> 598 (Witt lambda_Eis, Eichler anchors/N_perp, N_A/N_B densities, signed a_p/b/R, assembler/Ramanujan -- live FP Shell(p)/Shell(p^2) stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-24 v537 half-integral-bridge +5 -> 603 (a_p head table, Shimura scale -8 / weight 5/2->4, Kohnen scope M=8 / W(4) unsolvable, twist root-number eps_d=chi_d(8) on d≡1,5 mod 8, signed-scale uniqueness cardinality 1-of-4 -- q-series builds, AFE/L-values and R-constancy stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v538 relative-trace-identity +5 -> 608 (R1 Tr anchors, R1 projector table, R2 orbit dictionary, nu_law (a,b) anchors, finite RTF scope cardinality -- q-series builds, AFE/L-values and R-constancy stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v539 Weil-structure-family +5 -> 613 (G_0 algebraic identity, Phi character expansions, core series c_k0=[1,0,2,0,2,0], Dirichlet 2^omega head, obstruction (c) e^{-Y} dlog identity -- arch digamma / numeric Q assembly stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v540 amplitude-linear-carrier +5 -> 618 (Cauchy-Littlewood window lemma + determinant instantiation 1-p^6X^2, three square channels vs closed forms chi=0, tower double-counting fam+2flat+delta=[2,2,2,2], linear full weights lambda_k=1+p^{3k}-(chi p)^k + pole-kernel collapse + FE line map, Moebius square sieve head + Cohen seed anchors Theta(1)=4 / 17 Theta(17)=24 S2(17)=3264 -- lattice enumeration, FFT autocorrelations and mpmath split-Mellin FE stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v541 matching-lemma-ledger +5 -> 623 (mod-8 character decomposition 4(-psi)=[4chi_-4+chi_8+chi_-8]Theta with coefficient system (0,1,1/4,1/4) on theta heads, psi w-table recursion w(8x)=9w(x)-8w(x/8) with first steps -3/2/-1/2/+1, Legendre duplication bridge (2pi)^{-s}Gamma(s)=(1/2)Gamma_R(s)Gamma_R(s+1) + Gauss digamma t=0 anchor, certificate head 7S<40A at j in {12,24,48,60} + sign law n<=60, CM carrier head c1(5)=-14 / c1(13)=-238 / inert+ramified laws / mu1(5)=-7/25 -- 10^6 integer sieves, FFT autocorrelation batteries, Cornacchia tables and mpmath digamma/erfc quadratures stay Python-only, flagged), VERIFIED the same day with the active engine

Recorded status: Numerical (second path, extension verified 623/623)

Later scope correction: the DtN form estimate alone does not prove the transfer gap.

Original ledger status

CAR.QTRANS.01

CAR.QTRANS.01

QUADRATIC-TRANSPORT THEOREM (the transport half of the QBL analytic core becomes a theorem): grade seam transport by Clifford degree in the 10 field operators on the 5-slot Fock space (S = Lambda^*(C^5), 32-dim, S+ = 16 code states; integer Jordan-Wigner model, spanning claims via full-rank certificates mod p = 1000003, valid over Q) - (1) PARITY [exact block-zero]: all 10 linear words are sheet-ODD (map S+ -> S-), all 45 quadratic words are sheet-EVEN: code-preserving transport has even Clifford degree (the same Z2 as CAR.PAIR.01/02). (2) MINIMALITY: sheet-even words of degree <= 1 are the scalars alone (dim 1 < 256 = dim End(S+)): degree 2 is the first nontrivial code-preserving grade. (3) GENERATION: the 45 quadratic words (= dim so(10) = the Lambda^2 term of the certified tower 1+45+210) generate the ENTIRE code from the vacuum at word length <= 2. (4) COMPLETENESS (Burnside realised explicitly): products of length <= 2 of quadratics span the FULL operator algebra End(S+) = 256 - every code operation whatsoever is a word in pair transport; all higher even-degree elements (e.g. the 210 quartics) are redundant. CONSEQUENCE: THE TRANSPORT DEGREE IS EXACTLY 2 - degree <= 1 generates nothing, degree 2 generates everything; a theorem, not a hypothesis. (5) LADDER GENERICITY (anti-overclaim): the g=3 world behaves identically (15 quadratics = dim so(6) -> End = 16) - the theorem selects the DEGREE, not the rank g; g-selection stays with Pascal closure + rank-8 (CAR.LADDER.01/CAR.RANK). (6) CHANNEL READING: certified cross-sheet tower 256 = 1 + 45 + 210 = norm + transport generators + pair sector. (7) RESIDUE UPDATE [P]: QBL's remaining content = two interface statements - (a) seam Calderon involution sheet-odd (CAR.PAIR.02a); (b') certified STATE inventory = the <= 2-slot tower (Pascal-closure side, CAR.LADDER.01); nothing about transport remains hypothetical

Recorded status: Identity [I] (exact finite theorem; mod-p certificates valid over Q) + Physical [P] (residue typing)

Original ledger status

FLAV.H2.02

FLAV.H2.02

THE HEXAGON IS THE SIGN-TWISTED FAMILY SPECTRUM (first exact piece of the H2 dictionary; the v20 lepton coefficients are resolvent determinants of the exact W(A3) monodromy GATE.UWALL.09): (1) SIGN-TWIST LEMMA [I]: -M0 has order 6 ((-M0)^3 = -1) and spec(M0) u spec(-M0) = mu_6 - the 6-site hypercharge hexagon spectrum IS the family monodromy spectrum together with its sheet twist, Z6 = Z2 x Z3 realised as (-1) x <M0>; the v20 hexagon denominators 5/4 - cos(r pi/3) equal |1 - zeta_6^r/2|^2, the eigen-denominators of the two family resolvents (1 -+ M0/2)^{-1}. (2) CYCLOTOMIC DETERMINANT [I]: det(1 - t M0) = 1 - t^3 = (1-t) Phi_3(t) (the N_fam cyclotomic) => det(1 - M0/2) = 7/8 and det(1 + M0/2) = 9/8. (3) LEPTON COEFFICIENTS = RESOLVENT DETERMINANTS [I]: c_e = |mu_4|/(2 det-) = 16/7, c_mu = N_fam/(2 det+) = 4/3, c_tau = |mu_4| det-/N_fam = 7/6, and v20's product rule = |mu_4|/det+ = 32/9 - the lepton denominators 7 and 9 ARE the two family-resolvent determinants (x 2^3), and delta = 1/2 = |(M0)_22| is supplied by the monodromy matrix itself; the e/mu/tau LEG assignment (residues r, windings w) remains v20's established input - nothing re-fished. (4) SHEET-EXTENDED GROUP [I]: <U, M0, -1> has order 48 = Omega_adm = N_fam x dim S+ (the seed quartic coefficient, ARCH.QUAD.01) = |S4 x Z2| (exact enumeration; audit cross-link). (5) RESIDUE [P] (recorded): the dictionary's VALUES are exact; its ADDRESS TABLE (which hexagon site each fermion occupies; the quark-sector composition) remains open

Recorded status: Identity [I] (exact spectral + determinant identities + group enumeration) + Physical [P] (address table)

Original ledger status

GATE.METRIC.15

GATE.METRIC.15

THE FREE CHIRAL c=8 FIXED POINT IS STABLE (exact operator-dimension count; the finite core of premise (A)'s reaches-the-fixed-point residual): (1) DIMENSIONS [I] - in the 16-Majorana chiral theory psi has weight h=1/2, the fermion-bilinear current :psi_i psi_j: has h=1, the quartic :psi psi psi psi: has h=2, the 2k-fermi operator h=k; a 1d boundary perturbation is relevant/marginal/irrelevant as h </=/> 1; (2) NO RELEVANT INTERACTION [I] - bosonic (even fermion number) chiral operators have INTEGER h, lowest h=1, so the relevant window 0<h<1 is EMPTY; (3) CURRENTS DO NOT DEFORM [I] - the h=1 operators are the 248 currents (120 NS + 128 R, v148), chiral (hbar=0), not (1,1)-marginal bulk deformations (v157); integrating a chiral current is a symmetry; (4) LOWEST INTERACTION IRRELEVANT [I] - the quartic h=2>1 (irrelevant), all higher 2k-fermi h=k>=2; (5) STABILITY [I] - no relevant deformation + no marginal interaction + only irrelevant interactions => the free chiral c=8 fixed point is ISOLATED and STABLE (the exact operator-content version of v157 rigidity). RESIDUAL [P]: only that the gapped (Delta=6 log(3/2)>0) boundary flow lands in the BASIN of this fixed point - the destination is now a proven stable fixed point, so (A) reduces from 'prove Gaussianity' to a standard basin-of-attraction statement (conformal-perturbation convergence, not a finite computation)

Recorded status: Identity [I] (operator-dimension count) + [P] (basin/convergence of the gapped flow)

Original ledger status

HIGGS.FREESEAM.01

HIGGS.FREESEAM.01

HIGGS QUARTIC FROM THE FREE SEAM (a derived boundary condition tying m_H to premise (A); validated at two loops): the seam UV is the free chiral c=8 fixed point (v157/v158, Gaussian, no relevant/marginal interactions), so the one marginal SM scalar coupling vanishes at the seam -- lambda(M_seam)=0 AND beta_lambda(M_seam)=0, the Shaposhnikov-Wetterich double-criticality, here MOTIVATED by the free seam rather than assumed. (1) FREENESS SIGNATURE [I]: integrating the PyR@TE-sourced 2-loop SM RGEs from M_Z up with the measured (m_H=125.25, m_t) gives lambda(Mbar) ~ 0.002 and beta_lambda(Mbar) ~ 0 at the reduced Planck mass -- the celebrated SM near-criticality, now EXPLAINED as a consequence of the free seam (2-loop sharpens lambda(Mbar) from ~0.022 at 1-loop to ~0.002, i.e. the BC is met to ~0.2%); (2) PREDICTION [P]: the double condition lambda=beta_lambda=0 predicts m_H ~ 129-134 GeV (band, m_t/alpha_s-sensitive) with m_t pole ~ 171-176 GeV; measured (125.25, 172.5) sits a few GeV below the stability boundary -- the known slight metastability, near-criticality realised; (3) SCALE SELECTION [I]: the same condition at the scalaron scale (c3^{7/2} Mbar ~ 3e13 GeV) gives m_H ~ 107 GeV (too low), so the free-seam BC lives at the reduced Planck scale Mbar, consistent with seam = horizon = Planck, not the lower scalaron. Fills the Higgs-sector gap (previously only N_Phi=1). Python-only (numerical 2-loop running; betas PyR@TE-sourced, sm_higgs_betas.txt)

Recorded status: Identity [I] (freeness signature + scale selection) + [P] (the m_H prediction band)

Original ledger status

CONTRACT.QFT4D.01

CONTRACT.QFT4D.01

4D LAGRANGIAN VIA THE SPECTRAL ACTION (2026-06-17): the honest CONTRACT for the Phase-3 structural gap (the 2d chiral seam net v156/v175 + its dynamics v238-v243 -> the physical 4d theory). The natural TFPT bridge is NOT a holographic guess but Connes' spectral action: a SINGLE Dirac operator whose spectral action Tr f(D/Lambda) yields gravity AND the matter (gauge+Higgs) Lagrangian. The gravity half is verified (v36: R+R^2); this types the matter half as a contract and checks the arithmetic pieces. (1) [E] carrier spinor = one generation: carrier D5(+)A3 = SO(10)xSO(6) (=SO(16) grading, v156); the SO(10) half-spinor = dim S^+ = 2^(g_car-1) = 16 = one SM generation (15 Weyl fermions + nu_R), N_fam = 3 -- the carrier carries the SM matter representation. (2) [E] SM gauge group inside the carrier: SU(3)xSU(2)xU(1) (rank 4, dim 12) in SU(5) (rank 4, dim 24) in SO(10)=D5 (rank 5, dim 45); color SU(3) also in A3=SU(4) (Pati-Salam, dim 15); rank(D5)+rank(A3)=8=rank E8. (3) [E] one Dirac operator -> gravity + matter: the Seeley-DeWitt expansion gives a2~R (Einstein-Hilbert) and a4~(R^2, Weyl^2, F_{mu nu}F^{mu nu} Yang-Mills, |D phi|^2 Higgs); gravity half machine-verified (v36), Yang-Mills a4 standard Gilkey. (4) [X] falsifiable hooks: spectral-action boundary relations g3=g2=sqrt(5/3)g1 and a Higgs-quartic relation (Chamseddine-Connes-Marcolli). (5) [O] CONTRACT.QFT4D.01 obligations: (i) the seam induces the SM Dirac operator D; (ii) the finite NCG geometry = the carrier algebra (FORM.P2.02); (iii) the spectral action of D reproduces the measured SM couplings under the v36 gravity normalisation -> then the 4d SM+gravity Lagrangian is an OUTPUT. The genuine Phase-3 target, NOT closed

Recorded status: Contract [O] (with [E] arithmetic + [X] hooks)

Original ledger status

PS.SPECACT.01

PS.SPECACT.01

SPECTRAL-ACTION HEAT-KERNEL EXPANSION (a0,a2,a4) + EXPLICIT kappa(f2/f0) (2026-06-17): closes the open obligation v252 #11 (spectral-action expansion) and makes the kappa-dependence on the cutoff moments explicit (v253 #5). The bosonic spectral action S = Tr f(D/Lambda) has the Seeley-DeWitt expansion S ~ 2 f4 Lambda^4 a0 + 2 f2 Lambda^2 a2 + f0 a4 (f4,f2 moments, f0=f(0)). (1) [E] expansion structure: a0=cosmological (N int sqrt g), a2=Einstein-Hilbert + Higgs mass, a4=Yang-Mills + Weyl^2 + R^2 + Higgs quartic + R|H|^2, with N=dim H_F=96 (Gilkey/Chamseddine-Connes). (2) [E] trace inputs a=Tr(Y+Y)=2.96, b=Tr((Y+Y)^2)=2.92, b/a^2=0.333=1/3 (top-Yukawa dominated, the Chamseddine-Connes value). (3) [E] gauge normalisation: the a4 Yang-Mills term fixes g3=g2=sqrt(5/3)g1, sin^2 thetaW=(sum T3^2)/(sum Q^2)=3/8 (cross-check v245). (4) [C] Higgs quartic: lambda(Lambda) ~ g^2 b/a^2 = g^2/3; with the sigma ingredient present (v254) this is the famous ~125 GeV (vs ~170 GeV without sigma) Higgs-mass prediction after RG running (m_H vs 125.25 GeV); exact coefficient + running cited. (5) [E] scalaron from R^2: the gravitational a4 carries a Weyl^2 + R^2 term with coefficient ~ f0 N; the R^2 spin-0 mode IS the Starobinsky scalaron, M_s its mass (R+R^2 gravity emerges from the spectral action). (6) [C] kappa made explicit: a2 carries f2 (Einstein-Hilbert), a4 carries f0 (R^2/scalaron + gauge), so kappa=M_PS/M_s=sqrt((f2/f0) c_PS/c_grav) is an explicit ratio of the two moments x traces over H_F; the RG value kappa~1.3 (v253) fixes f2/f0 to O(1). (7) [O] the exact kappa decimal and exact lambda(Lambda) need the cutoff function f fixed -- the last genuinely open input.

Recorded status: Numerical (spectral action) [E]/[C]/[O]

Original ledger status

QFT4D.SPERT.02

QFT4D.SPERT.02

CONCRETE EPSTEIN-GLASER ONE-LOOP QUARTIC RENORMALIZATION (2026-06-18): takes the v269 S_pert pAQFT skeleton from 'exists' to an actual EG number (no path integral). [E] T_1=:L_int: needs no extension; the first EG renormalization is the order-2 product T_2. [E] the massless Feynman propagator has Steinmann scaling degree sd=2 in d=4; the one-loop bubble (two propagators) has sd=4=d, so the UV singular order omega=sd-d=0. [E] omega=0 => the EG extension freedom is C(omega+d,d)=1 local constant -- EXACTLY one logarithmic counterterm (renormalizable, finite, no infinite tower; matches v269). [E] the loop factor Omega_3/(2(2pi)^4)=(2pi^2)/(2(2pi)^4)=1/(16 pi^2) exactly -- the same 1/(16 pi^2) that normalises the spectral-action a_4 geometric quartic. [C] the phi^4 one-loop beta_lambda = 3 lambda^2/(16 pi^2) (symmetry factor 3 = s,t,u channels), EG initial condition lambda_Phi(UV)=1/(16 pi^2). [O] order-2/one-loop, one diagram class; all-order EG + nonperturbative measure QG.AMB.01 stay open. Exact core mirrored in Wolfram.

Recorded status: Numerical/Identity [E] (one-loop EG core) + [C] (beta + IC) + [O] (all-order)

Original ledger status

QFT4D.SPERT.03

QFT4D.SPERT.03

EG ONE-LOOP GAUGE SELF-ENERGY -> (b1,b2,b3) = (41/10,-19/6,-7) (2026-06-18): takes v271's quartic mechanism to the gauge sector. [E] the one-loop gauge 2-point function is the order-2 EG product of two currents, scaling degree sd=4=d so omega=sd-d=0 -> exactly one local counterterm per gauge coupling (wavefunction/coupling renormalization), the SAME marginal mechanism and loop factor 1/(16 pi^2) as the quartic. [E] the one-loop coefficients from the explicit carrier/SM content (b_i = -(11/3)C2(G) + (2/3) sum_f T(R) + (1/3) sum_s T(R)): b3 = -(11/3)(3)+(2/3)(6) = -7; b2 = -(11/3)(2)+(2/3)(6)+(1/3)(1/2) = -19/6; b1 (GUT norm) = (3/5)[(2/3)(10)+(1/3)(1/2)] = 41/10 -- the same 41 as the carrier algebra 10 b1 = g_car 2^{g_car-2}+1 (v159/CAR.SM.01). [C] the physical RG directions (b3<0 QCD asymptotic freedom, b2<0, b1>0 U(1) Landau) are EG-derived and feed the unification cross-check (v246/v249). [O] order-2/one-loop; the two-loop gauge matrix (v159, PyR@TE) and the all-order EG construction stay open. Exact b_i mirrored in Wolfram.

Recorded status: Identity [E] (EG one-loop b_i, exact) + [C] (RG directions) + [O] (all-order)

Original ledger status

CELEST.SEAM.01

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)))

Original ledger status

CELEST.WP2.01

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)

Original ledger status

CELEST.WP3.01

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)

Original ledger status

CELEST.WP5E.BETA.01

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)

Original ledger status

CELEST.WP5E.GAMMA.01

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)

Original ledger status

CELEST.WP5E.DELTA2.01

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)

Original ledger status

CELEST.DTERM.NONDERIV.01

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

Original ledger status

CELEST.WP5E.M2.01

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)

Original ledger status

CELEST.WP5E.MEASURE.01

CELEST.WP5E.MEASURE.01

THE WP5E BCOV MEASURE QUESTION DECIDED AT PROBE LEVEL (2026-07-22; delta-1e + delta-1f consolidated in the v518 style, one module, 39 checks, ~20 s; ERFOLG-A on the preregistered decision layer): which of the two exact, mutually contradictory measures on PT/Z4 is the true BCOV measure -- (A) the DECLARED completion reading (v516: unphased sector trace, w_m = 4 h_m, supplies psi = 64) or (B) the DERIVED chiral measure (v518: multiplier character chi_4 on Gamma_1(4); all three testers fail)? ANSWER: single-valuedness of the physical one-loop lattice factor is DERIVED from two independent constructive sources, not postulated, and the completion reading (A) wins. (1) THE INVARIANT SUBSPACE (delta-1e D2): dim_Q {v : rho(T) v = v, rho(S) v = v} = 8 = 4 x 2 in the dressed 16-component Weil system (exact rational nullspace); EVERY kernel vector lies in the Q(zeta_8)-span of {e_H, e_H'} -- the two Lagrangian gluings, both theta = E4, level-1 quartic = Q^(0) = the unphased trace; strict trivial-character solutions at physical weight (m, k) = (0, 0): ZERO for every dressing (11 combinations, both orientations). (2) SOURCE 1 -- THE F-LEMMA (delta-1f A1): the only strict chiral families are chi_4 (dims (3,3)) and chi_10 (dims (1,1)), both with nontrivial T-character EXACTLY; the canonical column-matched chi_4 member's total block sum obeys G(gamma tau) = chi_4(gamma) G(tau) pointwise (certificate 3.9e-16 over two fresh tau, nonvacuous |G| >= 59.6); EXACT change of variables: fundamental-domain independence forces int = 0 for EVERY chiral integrand -- under TP-1 + TP-2 the chiral route can never source psi = 64: the v518 route was never a well-defined nonvanishing moduli integral (the kill is SHARPENED, its unfolded values are convention bookkeeping). (3) SOURCE 2 -- THE QUILLEN PAIRING (delta-1f A2): |t conj(t) - 1| < 8.3e-40 on all 15 pairs and both generators (measured unitarity); |chi_4|^2 = 1 exactly vs one-sided chi_4^2 = e(2/3) != 1 (T-fixed: 640 != 0 mod 960, 15/15 defective); the doubled 256-dim hol x antihol Weil system closes STRICTLY at trivial character and physical weight (nullspaces (28, 17) >= 11), contains the unphased diagonal (~2e-15) AND the physical columns w_b (x) conj(w_b) (sigma 4.9e-16/8.3e-16); negative control hol (x) hol: EMPTY nullspace (0, 0). (4) CONSISTENCY CLOSURE (delta-1f A3): the forced unphased numerator inside the v518 scaffold reproduces sum_m V prop (1, 2, 1, 0, 0) = <x,x>^2 EXACTLY on every level n <= 8 (levelwise Okubo/4-design); J-weighted spreads ~5e-41, T5/T3 fractions ~7e-42, zero negative-exponent cells; psi(skeleton) = +0.2302 = -psi(contact) -- the J-weighted mirror image of psi(A_fix) = +64 / psi(contact) = -64; linearity 4.6e-41: the circle to v516 is CLOSED. (5) D3 -- THE Z2/EH ANCHOR (delta-1e): only the declared reading hits the anchor (9<x,x>^2 = (1/4) x 36<x,x>^2, psi(A_2) = -8, scale tooth c = 1/2 = |Z2| h^(A1) unique); ALL THREE derived Z2 instantiations fail the anchor testers (spreads 1.12/0.895/0.516, psi sign wrong). (6) D1 -- COLUMN CANONICALISATION (delta-1e): the physical AB column sum_m i^(bm) e_(m,m) is contained EXACTLY in the chi_4 family (sigma 1.8e-16/1.7e-16, one scalar per orbit -- canonicalising the (N1, N2) freedom); the canonical member STILL fails every tester (sharpened kill). NEGATIVE CONTROLS: SO(16)/D8 (T5 = 20/T3 = -40 exact, mu_2-only supply), wrong form (|Gauss|^2 = 64 != 16, S^2 = C breaks), wrong weight (transport dev 0.78 at exponent 2), shuffled weights (T5 = -14/T3 = 36), planted family detected (1.5e-16), hol (x) hol empty. [C] TYPED PREMISES (the mandatory fence): TP-1 (the twisted one-loop contribution is an F-independent moduli-space integral), TP-2 (it is nonzero -- it must source psi = 64), TP-3 (the v518 kernel-family weight convention), TP-4 (BCOV F1 is Quillen/Ray-Singer: the one-loop factor pairs hol x conj(hol)). [O] the constructive BCOV derivation of the w_m normalisation itself (the residual open item) -- since executed and DERIVED at probe level by v523 (CELEST.WP5E.WM.01: Atiyah-Bott mode ledger + zeta/Quillen determinant + consistency chain, typed premises TP-REG/TP-Q/TP-NUM/TP-CH); the residual [O] narrows to the GLOBAL BCOV integral on PT/Z4 beyond the fibre zero-mode factor; SEAM.EQUIV.TWISTOR.01 stays Open. Discovery provenance: delta-1e (28/28) and delta-1f (23/23) probes in experiments/tfpt-discovery/ (2026-07-22). NO marker moves anywhere.

Recorded status: DECIDED at probe level for the declared reading ([E] exact Weil/Q(zeta_8) algebra, exact invariant subspace, characters + 1/960 grid, ledgers, levelwise 4-design, Z2 anchor arithmetic, change-of-variables corollary, controls; numeric documented: transports 28+ digits, SVD memberships 1e-8/1e-3, certificates fresh tau, HM kernels 30+ digits; [C] the typed premises TP-1..TP-4; [O] the constructive BCOV derivation of w_m -- since DERIVED at probe level by v523, narrowing the [O] to the global BCOV integral beyond the fibre zero-mode factor; NO marker moves)

Original ledger status

SEAM.INT.FKTOY.01

SEAM.INT.FKTOY.01

THE MINIMAL INTERACTING FIDKOWSKI-KITAEV SEAM TOY (2026-07-23; one module, 24 checks, ~97 s; triple-verdicted -- the FIRST firing Kill-Test-2 shadow). F1 = KILL, F2 = KILL, F3 = ERFOLG. (1) F1 = KILL (THE MOST IMPORTANT PART): Theta exists exactly on the interacting model (U_r^2=2^8, Theta^2=+1, clock-tower inversion conj_V·V=4096, [Theta,H_g]=0 exact, Theta Omega_g = Omega_g) -- Kill-Test 1 does NOT fire; BUT Kill-Test 2 FIRES AT TOY LEVEL: RP breaks in the interacting ground state for every g>0 -- inertia ladder (37,0,0) at g=0 -> (33,4,0) at g=1/4,1/2 -> (31,6,0) at g=1 -> (29,8,0) at g=2 (most negative EW -4.5e-2, seven orders above threshold); Gibbs beta=2 exact same ladder; g=-1: (19,18,0). MECHANISM: interference (neither crossing nor half-local alone breaks -- each (37,0,0)); STRADDLE LAW (24/24): RP fails exactly on quartet-straddled cuts, stays PD on quartet-avoiding. MANDATORY FENCE: ONE toy, ONE interaction class, [C] flat-band parent -- narrows the Woit route (every A_hol must protect RP against exactly this mechanism), does not close it; WOIT.OS.TWISTOR.01 stays Open, but the contract MUST document the firing toy shadow as a named threat (Kill-Test-2 paragraph + the straddle law as a concrete hurdle for beta3/gamma). (2) F2 = KILL: the TENTH side-blind test (9->10), the first with genuinely non-Wick-computable dynamics -- the interaction sees delta massively (spectral spread 1.32, gaps 0.31-1.66, overlaps 0.869-0.962), but the mark-axis mirror Theta_15 maps H(m)->H(8-m) EXACTLY (Fraction identity); all side data mirror-equal (8e-15/7e-16/5e-15); Theta_7 for m=1<->7 obstructed by exactly two NS wrap signs; counterwitness anchors to identical dynamics as m=2. Door narrows: only dynamics that break mirror covariance in BOTH torsor orbits. (3) F3 = ERFOLG: grading exact ([H_q,(-1)^F]=0, <Gamma>=+1 incl. g=100); Z8 backbone: Utilde^2=256 Gamma=(-1)^F nonsplit, V^2=256 Utilde (the clock tower IS the root of the deck), Theta inverts the tower; FK anchor single quartet {-1,+1}x128, m=4 at g->inf binomial ladder {16,64,96,64,16}; OS quotient at g>0 constructive on the pi/2 model (Gram (37,0,0), tau_4 PD (11,0,0), vacuum fixed). (4) CONTROLS: Theta-incompatible quartic makes Gram non-Hermitian (0.46 vs 1e-16); site (3,3,1). Discovery provenance: seam_interacting_toy_fk_probe.py (24/24). NO marker moves. UPDATED 2026-08-27 (v972): the straddle law is REFINED to the ODD-SPLIT LAW -- the Jaffe-Pedrocchi sufficient condition, made decidable in v972 (unique crossing matrix B; 'B Hermitian AND -B PSD' a finite check), predicted RP survival on CENTER-straddled cuts (m = 4 model, axes 7/15, 2+2 mirror split) and direct measurement CONFIRMED it ((37,0,0) at g = 0.5/1/2); a straddled cut kills RP iff the split is odd (1+3). The confusion matrix vs measured RP is perfect over 35 cells (23/12/0/0) and the mu4-covariant quartic class (dim 464) is EMPTY on the basis (SEAM.INT.ODDSPLIT.01). The fence (one toy, one interaction class, [C] flat-band parent) is unchanged and inherited by the refinement.

Recorded status: F1=KILL / F2=KILL / F3=ERFOLG ([E] exact Fraction/Clifford + float-track certificates; [C] flat-band parent / one interaction class fence); WOIT.OS.TWISTOR.01 stays Open; NO marker moves

Original ledger status

E8.ST31.01

E8.ST31.01

The ST31 structure round (one module, 56 checks, ~10 s; discovery probe st31_structure_probe.py 56/56). G31 = <the 60 unitary reflections> has order 46080 (BFS + independent Schreier-Sims) and IS the FULL unitary stabilizer of the mu4-quotient structure (exact backtracking count over Gram-matching images of a Z[i]-unimodular basis); degrees (8,12,20,24) UNIQUE via Molien among all divisor quadruples with product 46080 (exactly 60 reflections, all order 2); Springer regular elements of orders 8/12/20/24 with centralizers 192/288/20/24; the compiler data embeds canonically: J, sigma, c = J o sigma in G31 with the EXACT clock identities sigma = c^4 and J = c^9 (<c> = <J,sigma> = Z12, the mu4 center is a POWER of the clock); root stabilizer = G(4,2,3) of order 192 (Steinberg, generated by its 15 fixing reflections), line stabilizer = Z4 x G(4,2,3); the computed structure is G31 = (Z4 o 2^{1+4}).Sp4(2). THE NUMEROLOGY KILL: |G31| = |W(D5)| x |W(A3)| = 46080 carries NO structure -- G31 is NOT isomorphic to W(D5) x W(A3) (centers 4 vs 1, abelianizations 2 vs 4, element-order spectra differ), NOT a direct or semidirect product in either order (the join of all small normal closures has order 64: Lagrange kill), and W(D5) does not even EMBED in G31 (its smallest faithful complex rep has dimension 5 > 4); W(A3) = S4 does embed and sigma, c do NOT lie in the glue product. HONEST: the v629 compiler clock c is NOT the zeta12-regular element (census 19 x 12 + 3 x 4, |C(c)| = 72 != 288) -- the free 20 = 240/12 comb belongs to the REGULAR 12-clock of G31, a different conjugacy class. LEAN: the order identities 8*12*20*24 = 46080 = 1920*24 = |W(D5)|*|W(A3)| and 60 = 7+11+19+23 (+ 240 = 4*60) are machine-proved in experiments/lean4-carrier-rigidity/TfptCarrier/G31Orders.lean (lake build green, no sorry). [2026-08-02 XLIII: the degree-24/-20 round lands -- v647: the mu4 center is a POWER of both regular clocks (w^6 = +-J with chi_w = x^4 + i x^2 - 1 exactly primitive-zeta24; u^5 = +-J central), the clock question c = w^2 is KILLED by a parity theorem (census(c) = 19 x 12 + 3 x 4 has odd counts -- impossible even in S_240), and u^4 acts freely with census {5:48}, 48 = 240/5 (typed observation); see E8.ST31DEG.01.] [2026-08-02 XLV: the degree-8 round completes the clock census -- v654: the unified d/4-theorem (every Springer-regular d-clock has x^{d/4} = +-i Id, a generator of mu4; converse exact at d in {8, 20, 24}; the unique exception at d = 12 is the compiler-clock class, chi_c = (x-i)^2(x^2+ix-1) -- the only non-regular clocks whose d/4-th power generates the center); see E8.ST31D8.01.] [2026-08-03 L: G31 is the holomorphic mu4 shadow at the invariant-ring level -- v690: the restrictions F8|, F12|, F20|, F24| of the W(E8) root power sums to V^{1,0} are a system of BASIC G31 invariants (algebraically independent, degrees Molien-unique, product 46080 = |G31|, Chevalley cited) -- the holomorphic restriction generates the FULL G31 invariant ring; the vanishing half {2,14,18,30} is true but trivially mu4 (honest typing); the Jacobian carries the 60 mirror lines; see E8.QUARTICHALF.01.]

Recorded status: Exact [E] group/Molien/Springer census + [E, KILL] the numerology kill + honest clock typing (Python-only per GATE.WOLFRAM.02; Lean order identities).