Original claim records.

Original ledger status

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

FTRANSFER.KMSCOSET.01

FTRANSFER.KMSCOSET.01

the coset assignment of the fibered clock functor is a THEOREM from KMS/modular typing (18/18 checks incl. the three must-fire controls, ~3 s; discovery probe ftransfer_kms_coset_probe.py, 2026-08-05, verdict KMS-COSET-THEOREM; frozen before the first run). THE THEOREM: S_transported = -(ln lambda)^2/2 with lambda the PGL2 multiplier of the flow's time-1 holonomy per clock e-fold -- the coset is COMPUTED from the typing datum lambda, never case-inspected. THE CLASSIFIER uses NO Schwarzian and NO derivative of any order (non-circularity gate on its own source): (i) the holonomy grounded as an exact functional identity y(x+1) = T(y(x)), (ii) the PGL2 class of T (scalar -> degenerate; parabolic -> autonomous, lambda = 1; else hyperbolic candidate), (iii) the detailed-balance multiplier cross-ratio m(x) -- the discrete KMS signature -- exactly constant -> modular with lambda = m. DERIVATION CHAIN sympy-exact: Lemma A {M(e^{ax}), x} = -a^2/2 identically for GENERIC Moebius M; Lemma B {M(x), x} = 0; Lemma C -a^2/2 = -Delta^2/2 iff a = +-Delta, affine dictionaries change nothing. BLIND PROTOCOL: the four deployed clock charts classify F_pole/F_Boltzmann MODULAR with |ln lambda| = Delta = 6 log(3/2) exactly, F_QCD AUTONOMOUS (parabolic), F_relic DEGENERATE; unblinding: predicted cosets == transported Schwarzians EXACTLY (sympy zero) -- the fiber classification of the executed contract (v777, FTRANSFER-FIBERED-CARRIES) is now DERIVED. CONTROLS: C1 fake-KMS witness caught (functional identity fails AND the ratio is non-constant); C2 affine reparametrization x -> -x + c leaves typing and |ln lambda| unchanged; C3 the v578 Moebius clock t = e^{Delta sigma} flips the QCD chart to modular and the theorem PREDICTS -Delta^2/2, confirmed by the transported Schwarzian -- the K1-duty transplant as the positive case. Completes the three-probe chain executor -> fibered groupoid -> coset theorem: the CONTRACT.F.01 fibered refinement's clock-fiber classification is a derived statement. Freeze guards: prereg YAML + both data tables byte-identical. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Theorem-grade [E] sympy-exact derivation chain + blind classification with the non-circularity of the classifier gated on its own source; completes the fiber classification of the executed clock contract; no marker move (Python-only per GATE.WOLFRAM.02).

Original ledger status

CURVE.CODE.OUTERSPIN.01

CURVE.CODE.OUTERSPIN.01

the canonical OUTER bridge between the code's duad side and the curve's syntheme side of the S6 duality (24/24 checks, ~11 s; discovery probe duad_syntheme_bridge_probe.py, 2026-08-05, verdict BRIDGE-CANONICAL). THE CONSTRUCTION (Sylvester duality made explicit): a bridge is a bijection beta: Omega_code -> T_curve (code odd Arf forms -> curve isotropic spreads/totals), canonically inducing the incidence-preserving duality Delta_beta: V\{0} -> IsoLines(A[2]) through the K6 duad model and the shared-syntheme rule (every pair of totals shares exactly one line -- verified). THE CENSUS: 18 + 18 bridges intertwine sigma with tau|T resp. tau^2|T (each family a torsor under C_S6(3-cycle), order 18); after pinning beta(q*) = S* -- the UNIQUE curve spread fixed by the FULL <tau, rho> automorphism image (rho = the mu4 rotation = the complex structure J = mult by zeta^9; existence + uniqueness COMPUTED, not assumed) -- exactly 6 + 6 remain = ONE ORBIT (torsor) under the order-6 code joint stabilizer, with the tau/tau^2 families exchanged by an inner code symmetry (one statement). B2 ARF RESOLUTION: the pullback of the curve's even q_Theta through the two frozen even<->3|3-split dictionaries is CONSTANT over all 36 B1 bridges and equals q_even* = the unique sigma-invariant even code refinement, with the NAMED operation q* = q_even* + hbar(., A) EXACT -- the anchor shift resolves the v784 Arf mismatch canonically. THE OUTER WITNESS exact: sigma is [3,1,1,1] on odds but [3,3] on spreads, tau the reverse, the anchor transvection t_A is [2,1,1,1,1] on odds but [2,2,2] on spreads, and the class swap is UNIFORM (0 violations over all 720 group elements) -- the twist IS the outer automorphism of S6, not an inner accident. TRANSPORTS (structure labels only): {q* = 0}\{0} maps EXACTLY onto the 5 lines of S* (5bar) and the 10 onto the complement; the 3+2 split maps onto the [3,1,1] tau-orbit structure of S*; t_A maps to rho = J on every census bridge; chi_NSR transports as an honest exact TWO-VARIANT torsor ('incidence with one of the two tau-fixed lines of S*') -- it is pinned code-side by the v752 NS/R lattice parity, a datum the bare quadruple (V, hbar, sigma, q*) does not carry, and rho swaps the two candidate lines (the two-fold ambiguity intrinsic). CONTROLS: inner cross-side census 0 (class mismatch), same-side census 0, no bridge pulls q_Theta to an ODD form (never to q*), scrambled sigma census 0. The v784 INNER identification stays dead (CURVE-CODE-PARTIAL untouched); the bridge is a NEW statement about the outer twist. NO matter semantics (5bar/10, 3+2, chi_NSR are structure labels; ROOTCLASS-MIXED v775 stands). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT DECIDED (registered here as the decided outer-bridge statement): the outer identification is CANONICAL; the residual is the intrinsic two-variant chi_NSR ambiguity (pinned only by the v752 NS/R lattice parity, which the bare 2-torsion quadruple does not carry) -- typed, not open work; cross-referenced with CURVE.CODE.2TORSION.01, whose inner-identification kill stands unchanged.

Recorded status: Exact [E] finite census theorem (torsor uniqueness incl. the tau/tau^2 swap; constant Arf resolution; exact transports) behind the structure-label fence; the inner identification stays dead (v784); no marker move (Python-only per GATE.WOLFRAM.02).

Original ledger status

SEAM.CLIFFORD.MODULAR_S.01

SEAM.CLIFFORD.MODULAR_S.01

the missing Clifford index-2 bit IS the metaplectic lift of the modular S transformation at its fixed point tau = i -- coset decision positive, deck placed, anchor named; the frozen uniqueness decider fails sharply (27/27 checks, ~50 s; discovery probe seam_clifford_modular_s_probe.py, 2026-08-06, verdict MODULAR-S-PARTIAL). COSET DECISION POSITIVE (exact cyclotomic arithmetic): the per-binary-factor Weil/metaplectic lift K1 = H (x) I (and K2 = I (x) H) lands EXACTLY in the missing coset zeta8 G31 of v783 (BFS membership certificates for zeta8^{-1} K_i in G31), with the exact tau = i fixed-point relations S^2 = +I, T^2 = the Pauli parity element, and (T K)^3 = zeta8 I -- the Weil constant gamma = zeta8, c = 1 per binary factor. R2 SOLVED (deck placement): the TOTAL symplectic Fourier K3 = H (x) H (the toric-code modular S) lies INSIDE G31 (chi = 0) and its metaplectic anomaly is (T3 K3)^3 = i I = J -- THE DECK IS THE METAPLECTIC ANOMALY OF THE TOTAL MODULAR LIFT (c = 2); zeta8 = (1 + J)/sqrt2 is the real orthogonal HALF-DECK ((1+J)^2 = 2J exact) swapping the seam pair {L, zeta8 L}: lift = half-deck turn x lattice automorphism. The Z4 deck-clock Fourier F4 is NOT a carrier symmetry (fails the Clifford gate; typed out). OS reflection gate exact: Theta K1 Theta = K1 and Theta T1 Theta = T1^{-1} -- the antiunitary reality involution acts as the GL(2,Z) orientation flip tau -> -tau-bar, which FIXES tau = i. THE FROZEN UNIQUENESS DECIDER FAILS SHARPLY (the PARTIAL): the strict census over all 46080 coset elements (order gate 480 -> Theta-reality 96 -> q*-fixed + metaplectic pairing) is EMPTY (0/46080) -- NO Theta-real metaplectic involution in the missing coset fixes the v774 Arf refinement q*: the Arf defect q* -> q* + hbar(., c) is UNAVOIDABLE; the retyped census (defect recorded instead of forbidden) has 32 survivors = 16 (+-1)-pairs = EXACTLY ONE G31-conjugacy class (w1 ~ w2; tier-A 4 = the Pauli/phase gauge fiber of H (x) I). R3 NAMED, NOT FIXED: the sigma-orbit of lifts {w1, sigma w1 sigma^{-1}, w2} has Arf defects EXACTLY {A + F_Sigma + F_i : i = 1, 2, 3} (one per family) whose XOR is THE ANCHOR A itself; the total Fourier's defect is A + F_Sigma; every retyped survivor's defect carries the anchor flag chi_NSR(c) = 1 -- the lift does not fix the anchor, it NAMES it through the Arf defects. CONTROLS: CZ (real, order 2, Clifford, non-metaplectic) lies INSIDE G31 (chi = 0); the wrong phase convention zeta8 K1 breaks S^2 in {+-I} ((zeta8 K1)^2 = iI); census selectivity 480 -> 96 -> 32 of 46080. ROOTCLASS-MIXED (v775) unaffected -- finite stabilizer/Weil mathematics about the carrier, no physics claim. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT (registered, partial state): coset decision POSITIVE and R2 solved (the deck = the metaplectic anomaly of the total Fourier); R3 NAMED through the sigma-orbit Arf defects (XOR = the anchor A); remaining: derive the physical selection of the defect class (the anchor naming from seam data) or type the two-variant/defect residue as intrinsic gauge; kill: a Theta-real metaplectic coset element fixing q* (excluded here by the exhaustive 0/46080 census) would falsify the defect law.

Recorded status: Exact [E] cyclotomic coset/metaplectic identities (BFS certificates; exact tau = i relations) with the strict uniqueness census EMPTY by its own frozen gate -- the Arf defect unavoidable, a sharp typed fact; R2 placed, R3 named; no marker move (Python-only per GATE.WOLFRAM.02).

Original ledger status

SEAM.CODE.TYPEII.01

SEAM.CODE.TYPEII.01

Type II (doubly-even self-dual) is FORCED from three physical seam axioms -- the direct attack on residual R1 of ARF.BOUNDARY.CODE.01, decided exhaustively (19/19 checks, ~5 s; discovery probe seam_code_typeii_probe.py, 2026-08-06, verdict TYPEII-FORCED). THE THREE EQUIVALENCE CENSUSES, exhaustive over ALL 308,993 subspaces of F2^8 of dim 0..4 (totals = the Gaussian binomials 1, 255, 10795, 97155, 200787; dim > 4 excluded by rank-nullity): (1) MUTUAL LOCALITY of the Construction-A boundary vertex words <=> C self-orthogonal, ZERO exceptions (braiding exponent <(c+2m)/sqrt2, (c'+2m')/sqrt2> = c.c'/2 + Z -- the lift-shift identity verified exhaustively; word-pair census all 65536 pairs; survivors per dim 1/28/165/280/135); (2) INTEGER CONFORMAL SPIN (all T sector phases i^{wt(c)} trivial, |c+2m|^2 == wt(c) mod 4 exhaustive; the eta^8 multiplier typed universal/code-independent) <=> C doubly even, ZERO exceptions (survivors 1/8/38/52/30); (3) HOLOMORPHY / INDEX ONE (|disc| = det(rescaled Gram) = 2^{8-2k} exact in Fractions; |disc| = 1 = single modular-closed sector) <=> C self-dual <=> k = 4 -- and EVERY non-self-dual doubly-even code has a word in C-perp \ C of weight != 0 mod 4, i.e. a discriminant sector of FRACTIONAL conformal weight on which the S-image cannot close (proved exhaustively; detailed witness: the [8,3] subcode of the boundary-selected C* with exact coset theta series to h <= 2, theta_{L*} != 2 theta_L). THE KILL EXCLUDED: the survivor set of (locality, integer spin, |disc| = 1) over ALL subspaces IS the Type II set = the 30 e8-hat copies (weight enumerator 1 + 14x^4 + x^8); no boundary-legal non-Type-II code exists at length 8. R1 -> R1': the universe premise 'the boundary realizes a doubly-even self-dual code' is REPLACED by the three physical axioms A1 mutual locality (trivial monodromy / bose statistics), A2 integer conformal spin (bosonic T-alignment), A3 holomorphy / index one (single modular-closed sector) -- each a standard physical demand, not a coding-theory premise; given A1-A3 the chain boundary axioms + deck + orientation bit -> unique code -> E8 -> F2^4 -> g_car = 5 stands with this single sharpened [C] residual. IMPORTED INGREDIENTS typed and machine-verified in every instance used (I1 Frenkel-Kac-Segal braiding factor, FLM 1988/Kac, Adamo-Moriwaki-Tanimoto arXiv:2407.18222; I2 h = |lambda|^2/2 lattice-VOA grading; I3 Conway-Sloane/Ebeling theta S-transform with L* = A(C-perp) verified); the OPE-linearity scope note (admissible word sets are linear codes) typed as a scope assumption. Controls: braiding monodromy -1 witness; T-phase witnesses i and -1; the [8,3] S-closure violation. Uniqueness (30 copies; the boundary data select ONE) CITED from v776, count echo verified. NO P2 marker move. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT (registered; the sharpened [C] residual replacing R1 of ARF.BOUNDARY.CODE.01): the physical seam satisfies A1 mutual locality (trivial monodromy of admissible boundary vertex words), A2 integer conformal spin (bosonic T-alignment), A3 holomorphy/index one (a single modular-closed sector); given A1-A3, Type II is FORCED at length 8 (this module), e8-hat is unique (30 copies) and the v776 boundary data select one copy -> E8 -> F2^4 -> g_car = 5; kill: a seam construction violating A1-A3 while staying boundary-legal, or a boundary-legal non-Type-II code at length 8 (excluded exhaustively here).

Recorded status: Exact [E] exhaustive finite equivalence chain at length 8 (all 308,993 subspaces) modulo the cited classical lattice-VOA/theta ingredients, each machine-verified in every instance used; the OPE-linearity scope note typed; R1 -> R1' [C] residual registered; NO P2 marker move (Python-only per GATE.WOLFRAM.02).

Original ledger status

E8.TORSOR.FOURIER.01

E8.TORSOR.FOURIER.01

the projective Fourier modes of the 15 Gaussian root torsors -- carrier-block purity at CHARACTER level with the mutually-unbiased honest cap (30/30 checks, ~60 s; discovery probe e8_torsor_fourier_probe.py, 2026-08-06, verdict TORSOR-FOURIER-PARTIAL; FROZEN_SPEC SHA-hashed before any root data). THE CONSTRUCTION: because the v782 class fibers are torsors WITHOUT origin, their character data is canonical where their point data is not -- 120 = 15 x 8 projective Fourier rays m(v, u) with coefficients (-1)^{hbar(u, x(r))} on the 16-root fiber; PHASE LAW EXACT on all 960 origin shifts (a shift by t multiplies by the GLOBAL sign (-1)^{hbar(u,t)}: the rays are canonical); orthogonal ray basis of the (-1)-symmetric 120-dim pair space (integer Gram = 16 I; the antisymmetric half receives NO canonical mode -- the v782 sign theorem). EQUIVARIANCE: deck J diagonal with eigenvalue (-1)^{hbar(u,v)} (60 even + 60 odd, 4 + 4 per fiber); ALL 16 Pauli translations diagonal by characters (15 x 15 x 8 exact); G31 MONOMIAL (7200 generator matches; label action closes to Sp(4,2), order 720, kernel 64 = mu4 x Pauli16; label orbits {15 uniform, 45 deck-even, 60 deck-odd}); Burnside exact (transitive on pairs and on lines); sigma (Z3) multiplicities integral. THE CARRIER BLOCK CLEAN AT CHARACTER LEVEL (P1a/P1b/P2): the spinor sector = EXACTLY the 8 R-fibers (split-independent), so M_S = the span of their 64 whole Fourier rays -- a Fourier-ALIGNED invariant block (impossible at point level per v775); M_S factorizes as a W(D5) x W(A3)-set as 16 x 4 (W(D5)/W(A3) rebuilt by BFS from the certified v775 split, 1920/24), W(D5) transitive of permutation rank 3 on the 16 half-spinor weight pairs with the folded-5-cube/CLEBSCH charpoly (x-5)(x-1)^10(x+3)^5 EXACT => isotypic dimensions 1 + 5 + 10 -- the carrier Pascal AT CHARACTER LEVEL -- and the tetrad is S4-NATURAL (faithful, transitive, stabilizer S3, permutation rep 1 + standard 3) with sigma = the family 3-CYCLE + fixed anchor; W(A3)-tables exact (the 20 D5-adjoint pairs family-invariant; M_S = (1+5+10) x (1+3)); W(D5) Burnside on the spinor pairs: 4 orbits (the tetrad), sum m_i^2 = 48. P3 FAILS AS PREDECLARED (the honest cap): every mode has full 16-root support and |<pair|mode>|^2 = 1/8 EXACTLY on every cell -- the D5-weight basis and the character basis are MUTUALLY UNBIASED on each carrier fiber; no weight axis is a Fourier ray; the pointwise code -> matter reading stays dead (ROOTCLASS-MIXED v775 intact, re-confirmed). OVERLAP CENSUS: every certified stabilizer ray expands exactly in the 4 deck-even modes of its fiber (squared overlap 1/4 each; deck-odd and foreign fibers orthogonal -- the deck-odd half is a pure phase register invisible to stabilizer rays); the 45 nontrivial deck-even modes ARE, under the certified chart U, the 45 (context, Pauli) incidences (each nonzero Pauli covered exactly 3x; uniform modes -> identity, Sum z z^dag = 4I) -- a canonical G31-equivariant operator dictionary, noted (report only) as exactly the SHAPE the v791 carrier-intertwiner demand asks for on the finite register. CONTROLS: fiber scramble destroys deck equivariance; the wrong character group F2^4 = V collides 2:1 (rank 8 < 16); seeded random sign systems fail the certificates. Everything exact integer/Fraction/Z[i]/F2 (sympy gated to two charpolys). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact [E] integer/cyclotomic character theory (canonical rays, exact Clebsch spectrum, exact overlap census) with the predeclared P3 failure typed -- the weight and character gradings mutually unbiased; ROOTCLASS-MIXED (v775) intact and re-confirmed; no marker move (Python-only per GATE.WOLFRAM.02).

Original ledger status

GNET.MARTINGALE.LIMIT.01

GNET.MARTINGALE.LIMIT.01

the noncommutative-martingale route to the G_net GNS limit -- its finite hypotheses MEASURED, all met (15/15 checks, controls 4/4, ~7 s; discovery probe gnet_martingale_probe.py, 2026-08-06, verdict MARTINGALE-HYPOTHESES-MET). THE PROGRAMME: replace continuum heuristics by martingale convergence -- if the local-state defects along the certified Haar inductive system are SUMMABLE in trace norm, the limit state exists uniquely by elementary Banach telescoping; the operator-algebra content becomes a typed list of analytic steps with finite evidence in hand. S1 FILTRATION EXACT: Haar isometry V*V = 1 with support coherence iota(win(N,0,l)) inside win(2N,0,2l) EXACT (leak 0); the Watatani quasi-basis explicit at the predeclared depths (185 elements at l = 2, 3041 at l = 3, sum v v* = 4*1); martingale compatibility omega_k o E = omega_k at every level; window data literally level-independent (H_W identical at N = 48 and N = 3072 -- one index-4 expectation for the whole tower). HONEST TYPING: the omega_k are an ALMOST-martingale (approximate, with measured defects); the convergence mechanism is norm summability + dual-space completeness (Umegaki expectations; Cuculescu cited as context, not used). S2 SUMMABLE DEFECTS (load-bearing): delta_k = ||rho_{k+1} - rho_k||_1 geometric with fitted rates 0.250 / 0.263 / 0.265 per doubling for l = 1/2/3 -- ONE POWER BETTER than the 2^{-l} dim-law envelope (reported against 0.5, not forced; bar 0.55); envelope delta_k <= 8 delta_1 2^{-k} quantifies summability (l = 2 tail sum 2.9e-2). S3 TELESCOPING / UNIQUE LIMIT: the Cauchy tail bound holds on every rung against the Richardson extrapolant; the frozen 6-observable battery gaps fall at <= 0.6; extrapolation uniqueness within 2.25 delta_5. S4 SPLIT/NORMALITY PRECURSORS uniform along the tower: sector-resolved cross-window profiles anchored with shrinking tails; determinant witness D_k = det(1 - X X*) >= 0.985 and stabilizing at 0.992756 -- the uniform estimate a normality/split argument consumes. STEPS 5-7 TYPED, NOT EXECUTED (the analytic remainder): GNS reconstruction of omega_inf, local normality (S4 witnesses as the finite input), split property + the mu4 simple-current extension (GATE.METRIC.11) on the limit net. THE QGEO ARCHITECTURAL NOTE (typed, no gate): the open majorant residues (b) 'L2 uniform FH/RH' and (c) 'L3 uniformity' of QGEO.GRIDSUP.01/v745 are NOT needed for limit EXISTENCE (summable state-norm defects suffice) -- the route relocates them into the typed normality step: an architectural reallocation, not a proof of (b)/(c). CONTROLS: the shifted-bond isometry leaks out of the doubled window (0.707 > 0.3 -- the filtration breaks); the Z2-only average has lambda = 1/2 != 1/4; the scrambled sea gives O(1) defects (summability breaks); DECIMATION CONVERGES TO THE WRONG DEGENERATE LIMIT (||rho_dec - 1/16||_1 < 1e-3 vs Haar 0.1+) -- a summable filtration with the wrong bond data forgets the sea: the coherent Haar bond is load-bearing. Every measurement is finite-level evidence FOR the cited continuum steps, never the theorems. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT (registered OPEN [O], the analytic remainder = steps 5-7): GNS reconstruction of omega_inf on the quasi-local algebra (existence/uniqueness finite-level delivered here); local normality of the GNS representation on each A(W) (nuclearity-type estimate; the uniform S4 witnesses are the finite input); split property and the mu4 simple-current extension (GATE.METRIC.11, [B:A] = 4 = |mu4|) on the limit net; kill: non-summable defects at a deeper reachable rung or a determinant-witness collapse.

Recorded status: Measured [MEASURED] martingale hypotheses under frozen gates (all met; rate one power better than the dim-law envelope, reported not forced) + typed analytic remainder; the QGEO majorant residues relocated into the normality step, not proven; no marker move (Python-only per GATE.WOLFRAM.02).

Original ledger status

E8.RAMIFIED.JETCODE.01

E8.RAMIFIED.JETCODE.01

E8.RAMIFIED.JETCODE.01 (+ E8.AFFINE.NSR.01): the jet-code normal form of the Gaussian E8 compiler + the NS/R bit as an affine 8-point geometry, ONE module from two probes (18/18 + 15/15 checks, ~3 s; discovery probes e8_ramified_jetcode_probe.py JETCODE-EXACT and e8_affine_nsr_probe.py AFFINE-NSR-EXACT, both 2026-08-06, FROZEN_SPEC SHA-hashed before any root data). PART 1, THE JET CODE (all [E neu], machine-certified): Z[i]/(2) = F2[eps]/(eps^2) with eps = 1 + i and 2 = -i(1+i)^2 (2 RAMIFIES; the ring iso is UNIQUE -- the only nonzero nilpotent is 1+i -- and the wrong split ring F2 x F2 is NOT isomorphic: all 24 bijections fail, control C1); W = L/2L is FREE of rank 4 over the dual numbers (explicit Z[i]-basis by Hermite reduction over the Euclidean domain Z[i], covolume-certified index 1; the 4^4 = 256 dual-number combinations enumerate W bijectively); 0 -> eps W -> W -> V -> 0 is an exact SELF-extension with V = eps W canonically and ker(eps) = im(eps) -- the self-dual differential code; the deck IS the jet unit J = 1 + eps, whence the certified v782 transition-bus torsor translation rho(Jr) = rho(r) + iota(v) on ALL 240 roots (the full 15-row fiber-translation table re-derived from ring algebra); the extension admits ZERO mu4-equivariant splittings among all 65536 = 16^4 linear sections (-1 = +1 mod 2 makes mu4-equivariance = J-equivariance) -- the no-canonical-origin obstruction IS non-splitness; the 120 root pairs are EXACTLY the q = 1 unit shell of the plus-type quadratic form (census 120/136, Arf 0, polarization b = <x,y>/2 mod 2 with trivial radical), so every root is canonically the triple (address v, admissible error jet in the affine v_perp coset, orientation +-): 240 = 15 x 8 x 2. PART 2, THE AFFINE NS/R GEOMETRY (A1-A6 [E neu], chain reading [H neu]): V \ {0} = (7 NS classes = ker chi_NSR \ 0) u (8 R classes = the affine F2^3-torsor A = chi^{-1}(1)); the 7 NS classes carry EXACTLY the 112 D8 roots and the 8 R classes the 128 Ramond spinor roots -- 248 = (112 + 8) + 128 anchored in v148/v227/v301; Stab_GL(4,2)(chi_NSR) has order 1344 = 20160/15 and EQUALS -- as a permutation group with its natural 8-point action -- AGL(3,2) = Aut(H8) (brute force over all 40320 coordinate permutations; the chart composition is the explicit iso of actions); the evaluation code of affine functions on the 8 points of A is the [8,4,4] code RM(1,3) = H8 with EXACTLY 1344 equivalence permutations; the deployed Sp(4,2) stabilizes chi in an order-48 subgroup acting INSIDE AGL(3,2) (the kill does not fire). HONEST TYPING: the reconstruction chain NS/R bit -> affine 8-point space -> H8 SHARPENS the ARF.BOUNDARY.CODE.01 attack surface but the route to V still passes through Gaussian E8 -- an exact internal reconstruction, NOT a P2 removal. Controls fire in both parts (wrong ring, seeded scrambles, unique parity functional 14/14, flipped code bit). No matter semantics (v775 ROOTCLASS-MIXED cited). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact [E] ring/module/extension structure theorems (integer/F2/Z[i] arithmetic, exhaustive censuses) behind the interpretation fence; the reconstruction-chain reading typed [H neu]; carries E8.AFFINE.NSR.01

Original ledger status

E8.SYNDROME.ALGEBRA.01

E8.SYNDROME.ALGEBRA.01

E8.SYNDROME.ALGEBRA.01: the honest kill -- E8 is NOT an error-correction hull in the sharpest sense, one probe (13/13 checks incl. the 3 must-fire controls, ~6 s; discovery probe e8_syndrome_algebra_probe.py, 2026-08-06, verdict SYNDROME-DEAD, the kill fired exactly as preregistered). THE QUESTION: do the 240 root operators U_r |x> = i^{c(r).x} |x XOR v> plus the 16 neutral kernels form a 256-dimensional mu4-twisted V-graded basis of End(S+)? THE KILL FIRES: the frozen mu4-linear-lift realization family reaches max rank 222/256 over ALL FOUR frozen phase conventions (Im/Re 222, Re+-Im 160; sympy-exact over Q(i)) -- with a STRUCTURAL obstruction: any phase rule Z-linear in the root gives c(-r) = -c(r), so +-r pairs with c in {0,2}^4 produce PROPORTIONAL operators (collision witness printed; 240 - 228 = 12 c-vector collisions), unreachable for the ENTIRE diagonal-phase x translation family; TWO post-hoc diagnostics score WORSE (nonlinear BFS transversal 215, zeta8-linear float 160) -- the death is robust, not a convention artifact; a successor would need root-individual (non-linear-in-r) operator data. WHAT SURVIVES: the syndrome-table counting 16 + 240 = 256 = 1x16 + 15x16, the neutral sector (the 16 v112 kernels ARE the diagonal maximal abelian subalgebra; same-class products land in A_0 with mu4 coefficients EXACTLY), the fine (class, d, k) groupoid composition law (associative on all 3375 section triples), and the W(3,2) commutation skeleton (scalar commutators 15.1%, matching (-1)^hbar on only 51.5% -- the class-level symplectic reduction fails in this realization). NO scalar V-cocycle exists (section closure 0/210). Controls fire (scrambled classes, wrong cocycle i^hbar fails the 2-cocycle identity on 960/4096 triples, random c-assignments). The sharpest E8-is-a-hull form (twisted V-graded basis) is DEAD as frozen; successor criterion named. No matter semantics (v775 ROOTCLASS-MIXED stands). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: DEAD as frozen [E kill, preregistered adjudication fired]: the twisted V-graded basis reading of E8 killed at rank level; the surviving structures typed exactly; successor criterion named

Original ledger status

CARRIER.PETERSEN.RADIAL.01

CARRIER.PETERSEN.RADIAL.01

CARRIER.PETERSEN.RADIAL.01 (+ TRANSPORT.SIXTHROOT.01): the K5 edge machine, the Petersen carrier geometry with the radial dynamics whose sixth power IS the deployed transport spectrum, + the bit-exact decider on the PSD sixth root, ONE module from two probes (19/19 + 16/16 checks, ~1 s; discovery probes carrier_petersen_radial_probe.py PETERSEN-RADIAL-EXACT and transport_sixthroot_probe.py SIXTHROOT-SPECTRAL-ONLY, both 2026-08-06). PART 1, THE K5 EDGE MACHINE (all exact integer/Fraction): X = (-2,-2,-2,3,3) is the UNIQUE primitive traceless block-constant charge for the F|W = 3+2 split (sign pinned by the certified v774 table); E(K5) = 10 pairs carry the charge census {-4: 3, +1: 6, +6: 1} = the SU(5) decuple 3_{-4} + 6_{+1} + 1_{+6}, every pair cross-checked against the certified v774 hypercharge table (classes u_c/Q_up/Q_dn/e_c); under disjointness the pairs form the PETERSEN graph SRG(10,3,0,1) (charpoly (x-3)(x-1)^5(x+2)^4 exact, girth 5, |Aut| = 120 by full backtracking census = the faithful S5); the distance shells around the W vertex are (1, 3, 6) and CHARGE-PURE (+6/-4/+1 = e^c/u^c/Q), and the distance-2 shell induces a C6 -- THE FLAVOR HEXAGON AS GRAPH GEOMETRY [E neu] (explicit cycle verified edge by edge; 6 = |R+(A3)| = the transfer exponent, v124); the shell partition is EQUITABLE with quotient [[0,3,0],[1,0,2],[0,1,2]] whose eigenvector (6,-4,1) -> -2 IS the hypercharge vector, and P = Q_Pet/3 has spec(P^6) = {1, 64/729, 1/729} == the deployed TFPT transport spectrum EXACTLY (v54/v56/v171/v317; v124 lambda_n = (1-n/3)^6; GATE.METRIC.18). THE PREDICTION [H neu]: the full 10-dim Petersen walk has spec(P10^6) = {1, (1/3)^6 x5, (2/3)^6 x4} -- fast mode multiplicity 5 = g_car, slow hypercharge mode 4 = |mu4| (trace crosscheck 990 = 729 + 5 + 256); NO corpus 10-dim operator exists (T is 3-dim on the cusp space) -- a prediction, not a verification. PART 2, THE SIXTH ROOT (exact Fractions): B = (1/18)[[13,1,4],[1,13,4],[4,4,10]] -- every entry a compiler number (18 = p4(anchor), 13 = N_fam^2 + |Z2|^2, 4 = |mu4|, 10 = A_Lambda, 1 = N_Phi) -- is the UNIQUE symmetric doubly stochastic PSD sixth root with the frozen corpus-basis eigen-assignment ((1,1,1) -> 1, (1,-1,0) -> 2/3, (1,1,-2) -> 1/3; spectral reconstruction entry-exact; exactly 2 assignments in the corpus basis, a 1-parameter circle without the pin); B^6 carries the deployed spectrum exactly BUT the decider lands SPECTRAL-ONLY: B^6 is GL(3,Q)-conjugate to v486's lazy-walk M^6 (exact rational W and orthogonal O conjugations verified) and to the cusp-diagonal canonical form, yet bit-equal to NEITHER -- the corpus freezes the SPECTRUM, not the basis; the stochastic types differ (doubly vs sub-stochastic with absorbing channel). Controls fire in both parts (wrong edge 4+1 kills the (1,3,6) structure; Johnson complement kills the hexagon; both matrix perturbations break the spectrum). Typed carrier representation geometry -- ROOTCLASS-MIXED (v775) cited, no particle semantics. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. [2026-08-06 LXVI dated correction + addendum, v814: the sixth-root census missed the deployed v221 evaluation -- T_v221 (the exact spectral sum of v221's transfer (1-w)^6) == B^6 BIT-EXACTLY, so the TRANSPORT.SIXTHROOT.01 verdict upgrades from SPECTRAL-ONLY to BITEXACT-vs-v221 (the frozen SPECTRAL-ONLY stands as the probe's adjudication; ABEL-DEAD correction precedent); and the [H neu] 10-dim prediction now has a promotion-ready canonical PROPOSAL: T10 = P10^6 in the Lambda^2 basis passes all gates with the equitable v774 quotient == (Q_Pet/3)^6 bit-exact (TRANSPORT10-PROPOSED, no deploy claim), while the six-step composition is typed CLOCK-ONLY -- the deployed 6 is the clock exponent, every spatial route carries a typed obstruction.]

Recorded status: Exact [E neu] finite graph/matrix structure theorems (integer/Fraction; charpolys sympy-gated) behind the carrier-geometry fence; the 10-dim multiplicity prediction typed [H neu]; carries TRANSPORT.SIXTHROOT.01 (spectral-only decider)

Original ledger status

CURVE.CODE.DOILY.01

CURVE.CODE.DOILY.01

CURVE.CODE.DOILY.01: the certified outer bridge sharpened into an incidence-operator statement with a spectral payoff, one probe (21/21 checks incl. the 3 must-fire controls, ~1 s; discovery probe curve_code_doily_probe.py, 2026-08-06, verdict DOILY-EXACT). THE EXACT IDENTITY: the K6 doily incidence N (15 duads x 15 synthemes on the six odd Arf refinements) satisfies NN^T = 3I + A_KG(6,2) = B + 2I entrywise, with B the certified v752/v774 symplectic incidence (synthemes <-> isotropic lines bijectively; row and column sums 3; two duads share a syntheme iff disjoint, then exactly one). THE SPECTRAL PAYOFF: spec(NN^T) = spec(N^T N) = {9^1, 4^9, 0^5} exact (annihilating polynomial + traces + rank over Q), so the normalized channel N/3 has singular values {1, 2/3, 0} -- THE RECOVERY BASE RATE 2/3 (v221) as the unique nontrivial singular value of the canonical duad-to-syntheme incidence; the identification with the corpus rate (2/3)^6 = 64/729 is TYPED [H neu -> decidable, not decided here]: it requires the six-step composition semantics (the hexagon). PETERSEN LOCALIZATION: at the q*-distinguished point the 15 duads split 5 + 10, the 10 internal duads form the Petersen graph, and syntheme -> internal pair is a BIJECTION onto E(Petersen) 15/15 -- Petersen vertices = code duads, Petersen edges = curve synthemes under the certified bridge. THE REFINED BRIDGE: every census bridge (rebuilt with the frozen BRIDGE-CANONICAL conditions, census 6) factors EXACTLY through N -- Delta(d) = {Gamma(s) : d in s} and the matrix identity N_code = P_Delta^T N_curve^T P_Gamma: the outer bridge conjugates the code incidence to the TRANSPOSE of the curve incidence, a DUALITY, not an isomorphism (control C3: 720 symplectic isomorphisms exist, 0 intertwine sigma with tau -- the inner route stays dead). THE ARF LAYER in doily coordinates: the switch set T6 = the six within-block duads of the q_Theta point-split, bijective onto the 6 odd theta characteristics -- the census-6 anchor. N/3 is doubly stochastic (typed CLASSICAL incidence channel, not CPTP; 1 + 9 + 5 spectral decomposition, rate 2/3 on the 9-dim window) -- measured, no descent claim. No matter semantics (ROOTCLASS-MIXED, v775); the inner identification stays CURVE-CODE-PARTIAL/dead. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact [E] incidence-operator identities and spectra (integer/Fraction, annihilator-certified) with the (2/3)^6 identification typed [H neu -> decidable] pending the six-step composition; the inner identification stays dead

Original ledger status

CARRIER.CUTCODE.01

CARRIER.CUTCODE.01

CARRIER.CUTCODE.01 (+ CARRIER.MOMENT.INCIDENCE.01): the Arf split as the cut classification of K5 + the one-line master moment with the frozen 210 decision, ONE module from two probes (17/17 + 16/16 checks, ~1 s; discovery probes carrier_cutcode_probe.py CUTCODE-EXACT and carrier_moment_incidence_probe.py MOMENT-INCIDENCE-PARTIAL, both 2026-08-06). PART 1, THE CUT CODE [E neu]: the cut map delta: F2^5 -> F2^10 (edges of K5) is INJECTIVE on C_even(5) and ISOMETRIC (delta(x).delta(y) = x.y mod 2 on all 256 pairs; kernel = the repetition code; control C2: the isometry fails on exactly the 256 odd x odd pairs), with the quadratic identity q*(v) = wt(delta iota v)/2 mod 2 on all 16 words -- so the Arf split 16 = 1 + 5bar + 10 IS the cut classification (0|5) + (1|4) + (2|3) (the five wt-4 words = the elementary vertex cuts, the ten wt-2 words = the duad cuts, both bijections certified); the Clebsch graph SRG(16,5,0,2) (40 edges = |R(D5)|, tfpt_1 Theorem 2) grades over the Arf shells as 40 = 5 + 20 + 15 with forbidden blocks EMPTY, and the within-10 edge graph IS the Petersen graph (full isomorphism census 120 = |S5|); the 240 Gaussian roots / 60 mu4-lines grade 80 + 160 / 20 + 40 over the shells -- typed as Arf/cut grading, explicitly NOT a matter split (ROOTCLASS-MIXED, v775; the 5/20/15 readings are audit fingerprints [C]). PART 2, THE MASTER MOMENT: the symbolic identity sum_{i<j}(x_i + x_j)^2 - 3 sum x_i^2 = (sum x_i)^2 for ALL 5-vectors gives the one-line derivation Tr_{S+} X^2 = 4|X|^2 = 4 h(E8) = 120 (|X|^2 = 30 = h(E8), Tr_10 = 90, Tr_5bar = 30; the chain 120/3 + 1 = 41 = 10 b1 re-anchored); chi_X(t) = (t+2)^3 (t-3)^2 exact with elementary symmetric values (0, -15, 10, 60, -72) typed audit fingerprints [C] (72 AUDIT-ONLY, no morphism). THE 210 DECISION (frozen rule, either outcome the deliverable): p4(X) = 210 = C(10,4) stays DECORATIVE -- the equivariant-bijection candidates are KILLED by the exhaustive fixed-point obstruction (both groups S3xS2 and Z2^5:(S3xS2), both indexings: the fixed 4-subset exists, no G-fixed slot does, total equivariant maps = 0), none of the six frozen slot-local weight families reproduces p4, and only the PARTIAL miss-slot map survives (80 type-(1,1,1,1) subsets -> missed slot with uniform fibers 16 = 2^4, exact on the colour slots, deficit 65/65 on the weak slots) plus the Newton tie p4 = 2 e2^2 - 4 e4 (generic symmetric-function algebra, [C] audit). Controls fire in both parts. No physics claim, no marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact [E neu] cut-map/shell-grading censuses (F2/integer, exhaustive) + the frozen 210 decision landing PARTIAL (decorative, per the preregistered rule); the 5/20/15 and charpoly readings audit fingerprints [C]; carries CARRIER.MOMENT.INCIDENCE.01

Original ledger status

FLAVOR.GRAPH.FILTRATION.01

FLAVOR.GRAPH.FILTRATION.01

FLAVOR.GRAPH.FILTRATION.01: the flavor-matrix filtration Q -> K -> L as an incidence-compiler fingerprint -- budgets exact, entries NOT reconstructible, one probe (16/16 checks incl. the 4 must-fire controls, ~1 s; discovery probe flavor_graph_filtration_probe.py, 2026-08-06, verdict FILTRATION-FINGERPRINT-ONLY). THE BUDGET IDENTIFICATIONS ARE EXACT AND ORDER-EXACT: Q row budgets (4, 5, 6) = (|E(K3) u E(K2)|, |V(K5)|, |E(K_{3,2})|), K row budgets (6, 9, 10) = (|E(K_{3,2})|, |E(L(K_{3,2}))| with the line graph certified isomorphic to the triangular prism AND the hexagon circulant C6(2,3), |E(K5)|), L row budgets (10, 14, 16) = (|E(K5)|, #wt-4 words of H8, |H8|) -- with the level-wise additivity (4, 5, 6) + (6, 9, 10) = (10, 14, 16) mirroring the deployed L = K + Q (v37/v4) exactly; the deployed matrices located (v11/v12/v37/v4; v11 uniqueness re-derived in pure integers); control C1 certifies the identifications hold in EXACTLY the identity order. THE DECISIVE BAR (frozen: row sums alone do NOT count): NO frozen structural candidate -- 25/20/47 flag- classifier and operator-quotient candidates per level, entry-exact comparison with all 6 column permutations -- reproduces ANY of the three deployed matrices entry-wise (per-row matches recorded as fingerprint data; K rows 2-3 and all L rows have ZERO candidates). THE CONTROLS MAKE THE MISS INFORMATIVE: the exhaustive false-positive census over ALL nonneg-integer matrices with the deployed row sums gives rates 0.748% / 0.035% / 0.005% (Q/K/L) -- the candidate set is not vacuously flexible, so the entry-level failure is a real structural boundary, not a weak test; the wrong split 4|1 and the non-extended Hamming [7,4] break the Q/K/L identifications as predeclared. TYPED: the identification is an AUDIT FINGERPRINT at the budget level, not an entry-level incidence construction. Matrix-structure archaeology only -- no mass/physics claim. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact [E] order-exact budget identifications (pure integers, exhaustive false-positive controls) with the entry-level reconstruction NEGATIVE; typed audit fingerprint at budget level

Original ledger status

P1.INDEX.KMS.01

P1.INDEX.KMS.01

P1.INDEX.KMS.01: the closed normalization identity c3 = 1/(I_Jones * beta_angle) = 1/(4 * 2pi) = 1/(8pi), its convention audit, and the sharpened P1 contract, one probe (13/13 checks incl. the 2 must-fire controls, ~0.1 s; discovery probe p1_index_kms_probe.py, 2026-08-06, verdict P1-IDENTITY-CLOSED). THE IDENTITY [E]: I * beta * c3 = 4 * 2pi * 1/(8pi) = 1 EXACT in formal Fraction x pi^k arithmetic (no floats; S2.1-S2.3 are one identity in three rearrangements, counted as one content; the v208 area-entropy factor 1/4 = 1/I; the AX.P1.01 anchor route 1/(2 e1(a) pi) with e1 = 4 aligns anchor-4 with index-4). THE DEPENDENCY AUDIT (the load-bearing honesty step): the index I = 4 = |mu4| is an INDEPENDENT c3-free theorem (ZERO 'c3' occurrences in all four sources v154/v726/v746/v779, gated by file scan); beta_angle = 2pi is an INDEPENDENT measurement (v526: beta = N clock steps by detailed balance, wrong-beta {N/2, N/4, 2N} excluded; the v519 radian pin is the ONE shared convention and CANCELS in the unit-invariant statement 'response per full modular circle = 1/I'); c3 STAYS the P1 axiom -- the identity EXPLAINS its value, it does not derive it. THE FINITE EVENNESS PRECURSOR (measured, real): the tracial weight per mu4 deck sector on the l-windows satisfies w_1 = w_3 = 1/4 EXACTLY at every level l = 1..6, with the w_0/w_2 offset the symmetric zero-sum pair +-2^-(l+1) halving per level -- equal quarters asymptotically, NO anomaly offset (= the no-anomaly shape the contract's bridge needs); the index-4 conditional expectation is trace-preserving (dev < 1e-12); and the SEA STATE does NOT equidistribute (deviation 0.188 from 1/4 at the N = 1536 Haar pullback) -- the bridge must normalize the MODULAR RESPONSE, not vacuum weights. THE CONTRACT (registered [O]): an index-I simple-current extension carries per modular period exactly the normalized response 1/(I * beta_angle) = c3; the ONE open bridge is the even distribution of the modular-response (Quillen/determinant-line) normalization over the I deck sectors; KILL: any asymmetric sector offset (w_1 != w_3 at any level, or a non-vanishing offset as l -> inf). Controls fire: the wrong index 2 gives 1/(4pi), which the corpus books elsewhere (un-halved Gauss-Bonnet; SdS 1/(4pi) = 2 c3); wrong beta = pi breaks the chain at 1/2. c3 remains the P1 axiom; no marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. NOTE (2026-08-28, v993): unique finite CE does NOT close the equidistribution bridge -- that identification stays the open [C] of this row.

Recorded status: Exact [E] identity chain (formal Fraction x pi^k) + gated independence audit + measured finite evenness precursor; OPEN [O] contract with the one open bridge (even modular-response normalization) and the kill criterion named; c3 stays the P1 axiom

Original ledger status

K5.SIXSTEP.TRANSPORT.01

K5.SIXSTEP.TRANSPORT.01

K5.SIXSTEP.TRANSPORT.01: the six-step composition and the 10-dim transport deploy -- the (2/3)^6 rate is canonical on three spatial routes, the deployed six is the clock exponent, and the canonical T10 proposal passes every gate, one probe (23/23 checks incl. the 3 must-fire controls, ~0.2 s; discovery probe k5_sixstep_transport_probe.py, 2026-08-06, verdicts SIXSTEP-CLOCK-ONLY / TRANSPORT10-PROPOSED). THE CORRECTION (dated note on the v808/TRANSPORT.SIXTHROOT.01 row): the deployed transport is LOCATED -- v221's transfer (1-w)^6 IS the exact spectral sum T_v221 = J/3 + (2/3)^6 u2 u2^T + (1/3)^6 u3 u3^T, and T_v221 == B^6 == (1/4374)[[1651,1267,1456],[1267,1651,1456],[1456,1456,1462]] BIT-EXACTLY -- the v808 sixth-root census missed v221, so the SIXTHROOT verdict upgrades from SPECTRAL-ONLY to BITEXACT-vs-v221 (the frozen SPECTRAL-ONLY stands as that probe's adjudication; ABEL-DEAD correction precedent). TASK A (all exact): the rate (2/3)^6 arises CANONICALLY on three independent spatial routes -- the doily alternation (three PSD double-steps duad -> syntheme -> duad: spec {1, (2/3)^6 x9, 0^5}), the hexagon compression (Pi_2 (A_Pet/3) Pi_2 == A_C6/3 with spec((A_C6/3)^6) = {(2/3)^6 x2, (1/3)^6 x4} -- BOTH deployed decay rates from pure hexagon geometry), and the hexagon circulation (((2/3) S6)^6 = (2/3)^6 I) -- but each route is blocked by a TYPED obstruction (doily: wrong space, decay multiplicity 9, no (1/3)^6 mode; hexagon: NO unit mode -- ||(A_C6/3)^6||_op = (2/3)^6 < 1 kills every norm-nonincreasing intertwiner [E neu]; radial: the bipartite sign -2/3, P NOT similar to the clock root B, and the exact conjugation S with S T_v221 S^{-1} = P^6 is deployed NOWHERE -- the missing shells -> cusp isomorphism); SIXTH-POWER BLINDNESS [E neu]: (A_C6/3)^6 and (A_{2K3}/3)^6 have IDENTICAL spectra -- the sixth power erases the cyclic geometry AND the sign, so only per-step data can distinguish walk from clock: the deployed six is the CLOCK EXPONENT (v124 lambda_n = (1-n/3)^6, v486 hand, fixed eigenbasis (1,1,1)/(1,-1,0)/(1,1,-2)). TASK B: corpus census -- NO deployed 10-dim transport operator exists (v37's Pluecker apparatus is 3x3, re-derived; v44/v45 carry Lambda^2(5) = 10 as dimension only); the CANONICAL PROPOSAL T10 = P10^6 in the Lambda^2 basis (v774 cut-code dictionary, a permutation conjugation) passes ALL gates: symmetric doubly stochastic PSD, spectrum {1^1, (1/3)^6 x5, (2/3)^6 x4} exact (multiplicities (5, 4) = (g_car, |mu4|)), the v774 class partition (e_c | u_c | Q) EQUITABLE, exact quotient == (Q_Pet/3)^6 bit-exact with the deployed spectrum {1, 64/729, 1/729} and reversibility -- a promotion-ready PROPOSAL, no deploy claim (Q6 != T_v221 bit-exact, not symmetric -- recorded: this is WHY PROPOSED, not REALIZED). Controls fire (wrong per-step rate 1/2, Johnson-complement scramble, non-equitable partition). ROOTCLASS-MIXED (v775) cited: carrier representation geometry, no physics claims beyond operator identities. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact [E neu] operator identities and censuses (three spatial routes, sixth-power blindness, corpus census, bit-exact v221 location); [H neu] the walk-vs-clock reading and the T10 deployment proposal; dated BITEXACT correction to TRANSPORT.SIXTHROOT.01 carried

Original ledger status

ANCHOR.AFFINE.01

ANCHOR.AFFINE.01

ANCHOR.AFFINE.01 + FLAVOR.BIDIR.01: the anchor affine normal form and the flavor bidirectional checksum -- two exact corpus compressions, ONE module from two probes (18/18 + 15/15 checks, verdicts ANCHOR-AFFINE-EXACT and FLAVOR-BIDIR-CHECKSUM (selective); discovery probes anchor_affine_probe.py and flavor_bidir_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM). PART A: the anchor a = (1,1,2) has power sums p_n = 2 + 2^n satisfying the affine recursion p_{n+1} = 2 p_n - 2 identically in n (sympy); T(x) = 2x - 2 has the UNIQUE fixed point 2 = |Z_2| (the sheet), and the T-orbit of 4 is exactly the compiler quintet (4, 6, 10, 18, 34) = (|mu4|, |R+(A3)|, A_L, N_fam |R+(A3)|, Z6-lift) with p_0 = 3 = N_fam prepended. THE BUDGET FROM ONE RECURSION: |R(E8)| = p1 p2 p3 = 240, dim E8 = 240 + q3 = 248 (ladder identity p4 - p3 = p3 - 2 = 8 = h(D5) = rank E8), h(E8) = p2 p3/2 = 30, |R(D5)| = p1 p3 = 40, Omega_adm = 2 p1 p2 = 48, chi_a(t) = (t-1)^2(t-2) with (e1,e2,e3) = (4,5,2) = (|mu4|, g_car, |Z_2|), 10 b1 = e1^2 + e2^2 = 41 -- every number a word in the T-orbit and the fixed point. PART B: R = [[1,3,0],[1,5,2],[2,5,3]] decodes the SAME anchor on both sides -- R a = (4, 10, 13) = (p_1, p_3, N(3+2i)) and R^T a = (6, 18, 8) = (p_2, p_4, p_3 - 2); tr R = 9 = p_0^2, det R = 8, principal 2-minors {2,3,5}, chi_R = t^3 - p_0^2 t^2 + p_3 t - (p_3 - 2), ||R||_F^2 = 78, SNF diag(1,1,8). THE SIBLING ABLATION: of the 12 down-row candidates (all orderings of {1,2,5} and the sibling {1,3,4}) EXACTLY ONE -- the accepted (1,5,2) -- satisfies the full identity set; the anchor contraction (R a)_2 = 10 = p_3 kills both siblings (11 and 12). Controls fire (wrong map, wrong anchor, sibling (1,4,3) reproducing tfpt_2's failure numbers verbatim, scrambled ordering). A normal form / compression, NOT a new claim: no marker moves. Exact sympy/integer; no floats, no RNG, no fit. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact identity-grade compression [Identity]: both parts exact sympy/integer arithmetic; the compression is a normal form of existing corpus surfaces (tfpt_1 anchor power compiler + budget lines, tfpt_2 flavor spectral selector + bilinear table); no marker move (Python-only per GATE.WOLFRAM.02)

Original ledger status

GAUSSIAN.RAMLADDER.01

GAUSSIAN.RAMLADDER.01

GAUSSIAN.RAMLADDER.01: the (1+i) ramification ladder -- the four documented roles of the ramified Gaussian prime pi2 = 1+i are ONE machine-checked object with four rungs, each re-certified from scratch, ONE module from one probe (33/33 checks, verdict RAMIFICATION-LADDER-EXACT; discovery probe gaussian_ramification_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM, ~5 s). RUNG 1, NORM DOUBLING: N(pi2) = 2 and (pi2^2) = (2); on the Gaussian E8 lattice (the unique mu4/sigma-equivariant Construction-A placement, 2 of 30, rebuilt) (1+J)^T (1+J) = 2 I_8 EXACTLY; min L = 4 by exhaustive census, the zero class of L/(1+i)L carries NO root (0/240 directly), 240 = 15 x 16 over the nonzero classes (each a union of 4 of the 60 Gaussian lines), elementary divisors (1,1,1,1,2,2,2,2), index 16 = N(pi2)^4. RUNG 2, THE FOUR-BIT ADDRESS: L/(1+i)L ~= F2^4 with the deck J acting TRIVIALLY on all 240 roots. RUNG 3, THE NON-SPLIT JET: Z[i]/(2) ~= F2[eps]/(eps^2) (ring iso UNIQUE among all 24 bijections); W = L/2L FREE of rank 4 over the dual numbers (256/256 combinations distinct); the self-extension 0 -> V -> W -> V -> 0 exact with deck = JET UNIT 1 + eps, and of ALL 65536 linear sections EXACTLY 0 are deck-equivariant -- the no-canonical-origin obstruction IS non-splitness (v803); plus-type form census 136/120, Arf 0, the 120 root pairs ARE the q = 1 shell (240 = 15 x 8 x 2). RUNG 4, THE METAPLECTIC LIFT: zeta8 = (1+i)/sqrt2; (1+J)^2 = 2J; (S H)^3 = zeta8 I per binary factor and (T3 K3)^3 = i I = THE DECK (v798); exact Z[zeta8] BFS census: |C2/mu8| = 11520 = 16 x 720 with ZERO Galois-mixed classes, |C2| = 92160 = 2 x 46080, C2 = K u zeta8 K (K = U G31 U^-1, v783 chart theorem cited [E]); witness parities chi(H(x)I) = 1 (the RM-CSS zeta8 phase bit, v819 S4 -> v798 [E]), chi(H(x)H) = chi(CZ) = 0. Controls fire (split ring F2 x F2 admits 0 ring isos -- 2 must ramify; split self-extension admits 65536/65536 sections; scrambled nilpotent breaks ker = im). A compression: the four corpus roles (note_e8_gaussian_code Thm 1/v689, v689, v803, v783/v798/v819) are ONE ladder -- the same ramified prime read at (pi2), (pi2^2) and pi2/sqrt2. Exact integer/Fraction/sympy; no floats, no RNG. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact identity/lattice-grade compression [Identity/Lattice]: all four rungs re-certified from scratch in exact arithmetic; unifies note_e8_gaussian_code/v689 (census, norm doubling), v803 (non-split jet), v783/v798/v819 (metaplectic zeta8, Clifford coset, RM-CSS phase bit); no marker move (Python-only per GATE.WOLFRAM.02)

Original ledger status

PG32.FLAGS.01

PG32.FLAGS.01

PG32.FLAGS.01: the PG(3,2) flag normal form for the 105 + 35 vacuum completion, with the incidence kill test typed as a split, ONE module from one probe (26/26 checks, verdict PG32-FLAGS-EXACT + INCIDENCE-SPLIT; discovery probe pg32_flag_completion_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM, < 1 s). COUNTING NORMAL FORM: [4 choose 2]_2 = 35; 140 affine 2-flats of AG(4,2) = 35 through the origin + 105 not; PG(3,2) has 15 points, 35 lines, 105 = 35 x 3 point-line flags, canonically bijective to the C(15,2) = 105 point pairs via {x,y} <-> (x+y, {x,y,x+y}). RM LAYER (v819 S4 / v821 / v822 rebuilt, all set equalities): [15,10,4] weight enumerator 1 + 105z^4 + 280z^6 + 435z^8 + 168z^10 + 35z^12 EXACT; the 105 weight-4 words ARE the flats avoiding 0; the 140 weight-4 words of RM(2,4) ARE all flats; Grams 22I + 6J, 6I + J, 28I + 7J entrywise. VACUUM READING: origin planes puncture to weight 3, NONE in the shortened code, ONE unshortens dim 10 -> 11 -- the vacuum coordinate is ORIGIN RESTORATION. THE INCIDENCE SPLIT (the kill, typed): with the corpus symplectic form (v774) the 105 = 45 + 60 split is the Sp(4,2)-ORBIT decomposition on BOTH sides (|Sp(4,2)| = 720 brute force); on the 60 NON-ISOTROPIC rungs the polarity map PHI: (U,t) -> ((t+U) n U^omega, U^omega) is well-defined, bijective, Sp(4,2)-EQUIVARIANT (all 60 x 720) and INCIDENCE-PRESERVING; on the 45 DOILY rungs the obstruction is typed: U^omega = U and (t+U) n U = EMPTY for 45/45 flats, the unique abstract equivariant bijection matches flags with support n line = 0, and the flag Gram is 18I + 3J != 22I + 6J. TYPED CONCLUSION: '105 checks = 105 flags' is exact as a deck-equivariant BIJECTION, NOT an incidence isomorphism -- the obstruction sits exactly on the doily (the typed reason v822's continuum dilation died). VACUUM CARRIER CONTENT: the zero class carries NO E8 root (0/240 rebuilt, v689 [E]) -- the vacuum point repairs the label Gram but adds no carrier content; v822's frozen continuum kill cited as the companion negative. Controls fire (scrambled plane row, degenerate form, the 45-sector obstruction as its own control). Exact integer; no floats, no RNG, no fit. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact lattice/identity-grade compression with the kill typed as a split [Identity/Lattice]: 60-sector exact via the symplectic polarity (equivariant incidence dictionary), 45-sector obstructed on the doily (typed negative); compresses v819 S4 / v821 / v822 RM surfaces into the PG(3,2) normal form; no marker move (Python-only per GATE.WOLFRAM.02)

Original ledger status

MESSAGE.LADDER.01

MESSAGE.LADDER.01

v844 -- MESSAGE.LADDER.01 + DOILY.PASCAL.RANK.01: the E8 addressing formula and the Doily-Pascal rank theorem -- two exact corpus compressions, ONE module from two probes (23/23 + 16/16 checks, verdicts MESSAGE-LADDER-EXACT and DOILY-PASCAL-RANK-EXACT; discovery probes binary_message_ladder_probe.py and doily_pascal_rank_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM with no downscoping, ~2 s). PART A, THE MESSAGE LADDER: M_n = 15 * Delta p_n = 15 * 2^n (anchor a = (1,1,2) with p_n = 2 + 2^n, v832 [E]; 15 = the nonzero classes of V = L/(1+i)L, v833 [E]) certified against the ACTUAL rebuilt objects with ONE NAMED STRUCTURE PER RUNG: M_0 = 15 = the address space (classes); M_1 = 30 = h(E8) derived HERE from the rebuilt root system, not cited (exactly 8 simple roots; connected simply-laced Cartan matrix, det 1, degree sequence [1,1,1,2,2,2,2,3] = the E8 diagram; every positive root with exact nonnegative INTEGER simple-root coordinates; the highest root UNIQUE at height 29 with marks {2,2,3,3,4,4,5,6}; h = 1 + 29 = 30 = |R|/rank); M_2 = 60 = the Gaussian root lines (mu4 orbits, all of size 4, exactly 4 per class); M_3 = 120 = |R+(E8)| (exact generic functional f(v) = Sum v_k 10^k) = the sign pairs; M_4 = 240 = |R(E8)| with the census 240 = 15 x 16 (zero class EMPTY); the CODA 248 = 15*16 + 8 = M_4 + rank checked against the actual census and the actual HNF rank; the structural doubling 60 -> 120 -> 240 (each line = exactly 2 sign pairs, each pair = 2 roots) and the halving 60 -> 30 = |mu4| vs rank typed. The tfpt_1 budget lines |R| = 240 / dim E8 = 248 / h = 30 and v832 S3.1-S3.3 are ALL instances of this one formula. Controls fire: the wrong anchor (1,2,2) gives (45,105,225,465) missing all four targets with recursion offset 1; the wrong multipliers 14 (the weight-4 codeword count) and 16 (the class size) miss all four targets and break the coda (232/264 != 248). PART B, THE DOILY-PASCAL RANK THEOREM: the duad-syntheme incidence N of the Cremona-Richmond doily (15 duads x 15 synthemes, 3-regular in rows AND columns) and the symplectic point incidence B on PG(3,2) (B B^T = 4I + 3J entrywise on all 225 cells, charpoly (x-7)(x-2)^9(x+2)^5 exact, v774 S9 rebuilt) are ONE object: via the K6 duad model D(v) = {q Arf-1 : q(v) = 0} (exactly 16 quadratic refinements by brute force over 2^16, Arf census 6 + 10) the bijection point <-> duad carries N N^T onto B + 2I ENTRYWISE -- duad collinearity = symplectic orthogonality, and the spectrum shift 7/2/-2 -> 9/4/0 makes v774's (-2)-eigenspace v814's kernel. THE RANK THEOREM (exact Fraction/sympy): charpoly(N N^T) = (x-9)(x-4)^9 x^5, so sing(N) = {3^1, 2^9, 0^5}: rank N = 10 = A_Lambda = C(g_car, 2) (the decuple sector), dim ker N = dim ker N^T = 5 = g_car (the five-slot carrier), multiplicity 9 = N_fam^2, top singular value 3 = N_fam, and the SIX ovoid indicators 3*1_{O_q} - 1 (q Arf-1) lie in ker N^T and SPAN it -- the P2 integers are the singular-value data of the Cremona-Richmond incidence. THE RECOVERY VALUE: sing(N/3) = {1, 2/3 x9, 0 x5} exact, 2/3 = (N_fam - 1)/N_fam (the corpus recovery survival, v327); THE EXPONENTIATION IDENTITY: charpoly(((N/3)(N^T/3))^3) = (x-1)(x-64/729)^9 x^5 exact with 64/729 = (2/3)^6 = the deployed lambda_2 of {1, (2/3)^6, (1/3)^6} (v330/v486; the sixth power is the compiler clock, v814); TYPED GAP re-affirmed with NO upgrade: 1/729 is NOT a root of the six-step charpoly -- the deployed (1/3)^6 stays on the clock side (v814 SIXSTEP-CLOCK-ONLY [E], cited, consistent). Controls fire: the circulant and the frozen-seed LCG 3-regular pairings (same row/col sums 3) give rank 13 != 10, kernel 2 != 5 and the wrong charpoly. Exact integer/Fraction/sympy arithmetic; no floats, no fit; the only RNG is the frozen-seed LCG of control C2 (v775 C2 precedent). No marker moves; the compression is a normal form, not a new claim. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact identity-grade compression [Identity]: both parts exact integer/Fraction/sympy arithmetic certified against the rebuilt lattice objects (h(E8) = 30 derived from the rebuilt root system, not cited); the (1/3)^6 typed gap re-affirmed with NO upgrade (v814 cited); no marker move (Python-only per GATE.WOLFRAM.02)

Original ledger status

NORMALFORM.CFIN.01

NORMALFORM.CFIN.01

v845 -- NORMALFORM.CFIN.01: the finite compiler normal form C_fin = (V, hbar, q*, sigma, iota) assembled as ONE machine-checked object from the Gaussian quotient, with the code-to-matter kill built in as a checked NEGATIVE and the remaining P2 bridge typed -- the round's central corpus compression, ONE module from one probe (28/28 checks, declared SPEC v2, verdict NORMAL-FORM-ASSEMBLED; discovery probe finite_compiler_normal_form_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM with no downscoping, ~4 s). THE OBJECT, read off the lattice and not postulated: V = L/(1+i)L rebuilt from the Gaussian Construction-A E8 (the unique mu4/sigma-invariant placement C*, 2 of 30; census 240 = 15 x 16, zero class EMPTY); hbar = the descended symplectic form with a FOUND family/anchor basis (F1, F2, F3, A) of Gram J - I, equal to the bit form on ALL 256 label pairs; sigma = the descended order-3 deck (sigma^3 = id, symplectic, EXACTLY 3 fixed nonzero classes {A, F_Sigma, F_Sigma + A}, the other 12 in exactly 4 free 3-orbits -- the family cycle); q* = the distinguished quadratic refinement (EXACTLY 16 refinements by brute force over 2^16, Arf census 6 + 10; the frozen selector sigma-invariant 4 -> q(A) = 1: 2 -> q(F_Sigma) = 0: 1 picks a UNIQUE q*, Arf type 1, sigma-preserved); iota = the parity lift onto the even five-slot code C_even(5) (bijective; beta = iota.iota == hbar on all 256 cells; q_wt = wt(iota)/2 mod 2 IS q*; sigma acts on the FIVE carrier slots as the 3+2 split -- message register and carrier register intertwined). DERIVED COUNTING from the one object: |Sp(4,2)| = 720 by full 65536-matrix census; |Stab(q*)| = 120 acting FAITHFULLY as S5 on the five, orbits [1, 5, 10]; 16 = 1 + 5 + 10 and 15 = 5 + 10; the PASCAL READING (iota-weight census (1, 10, 5) = (C(5,0), C(5,2), C(5,4))); the PROJECTIVE READING (PG(3,2): 15 points, 35 lines, 15 isotropic = the doily; the 5bar is an OVOID meeting every isotropic line exactly once); the BUDGET against the actual lattice objects: 240 = 16 x (5+10), 60 lines = 4 x 15, 30 = 2 x (5+10) = h(E8) (MESSAGE.LADDER.01 rung 1), rank 8 (HNF index 16), 248 = 16 x (5+10) + 8. THE MANDATORY NEGATIVE (built in, v775 re-anchored): the exact counting bound -- every D5(+)A3 c E8 has 128 spinor-side roots, so at most floor(128/16) = 8 all-spinor classes and >= 7 of the 15 classes ALWAYS contain adjoint-side roots (the 15 mailboxes canNOT be 15 pure matter multiplets); the concrete doubled-model census SATURATES both bounds (8 spinor-pure + 7 adjoint-side, ZERO both-type classes -- the J-quotient respects the D8 grading); the WEIGHT-level kill at the canonical split D = (0,2,4,6,7): each spinor-pure class restricts to 8 DISTINCT half-spinor weight pairs, never 1 -- v775 ROOTCLASS-MIXED [E] re-verified, NOT overturned. WHAT THE NORMAL FORM DOES NOT REMOVE (P2 typed, no upgrade): all 30 placements are Type II [8,4] doubly-even self-dual codes (enumerator 1 + 14 z^4 + z^8) -- the v799 TYPEII-FORCED echo; C_fin does NOT derive the three seam axioms A1-A3, the OPE-linearity scope note, or the v776 selection of ONE copy -- P2 (g_car = 5 = the iota slot count) is NARROWED to the v799 residual R1' ([C] class), not eliminated. Controls fire: the non-alternating dot form admits 0 refinements; the trivial deck leaves 4 selector candidates (q* NOT unique without the family 3-cycle); the wrong lift breaks C_even membership and beta == hbar. SPEC v2 declared (v836 precedent; the v1 wording of S7(b) mis-typed the v775 bound as 'mixed classes'; the factual census is the clean 8 + 7 split along the D8 grading; no numeric result changed). Exact integer/sympy; no floats, no RNG, no fit. Python-only per GATE.WOLFRAM.02.

Recorded status: Exact identity-grade normal form [Identity] with the built-in checked NEGATIVE (v775 ROOTCLASS-MIXED re-anchored via the 128-spinor counting bound + the saturated 8 + 7 census) and the typed P2 bridge: C_fin does NOT derive A1-A3 / the linearity scope / the v776 uniqueness selection -- P2 NARROWED to the v799 residual R1' ([C]), NOT eliminated; SPEC v2 declared (v836 precedent, no numeric result changed); no marker move (Python-only per GATE.WOLFRAM.02)