Results, with their exact scope.

Original ledger status

GATE.UWALL.03

GATE.UWALL.03

(U_wall) SELECTOR side closed on the explicit point (algebraic, no PDE): exponents at inf eig(sum A_k)={2,1,1}=anchor a => splitting O(-2)+O(-1)^2; pardeg=0 (deg E=-4=-|mu4| + 4 weight-sums of 1) => polystable; det R=8=n.a (anchor pairing on splitting type); Spec(Q_+)={1,2,3}=3*alpha+1 (alpha=cusp weights {0,1/3,2/3}) => both U4 selectors read off the bundle. RESIDUAL: amplitudes c_u/c_d are the finer harmonic-metric/Hodge datum (Gamma^min Hitchin), not fixed by splitting+weights

Recorded status: Numerical (selector side closed)

Original ledger status

GATE.UWALL.17

GATE.UWALL.17

LIE-LEVEL GROUNDING of the exterior leg (no special math): the carrier half-spinor is the even exterior algebra 16=Lambda^even(5)=C(5,0)+C(5,2)+C(5,4)=1+10+5 (already in CAR.SM.01). Under SU(5)<SO(10): u^c in 10=Lambda^2(5), d^c in 5bar=Lambda^4(5) => exterior degrees deg(u)=2, deg(d)=4; deg(u)+deg(d)=6=|R+(A3)| (hexagon), deg(d)-deg(u)=2=|Z2| (sheet); up Yukawa family-symmetric Sym^2 dim6, down general dim9=N_fam^2. So the 'Lambda^2 exterior leg' is the carrier's OWN Lambda^2 grading (up quark lives in Lambda^2(5)), NOT exotic Hodge data => discrete invariant re-typed to carrier exterior DEGREE [L]. HONEST: grounds the TYPE; Plucker number 55/117 stays family Pl(K) [I]; carrier-L2<->family-L2F link (family x carrier=15) + normalisation remain [P]

Recorded status: Lattice/Lie [L] (typing) + [P] (normalisation)

Original ledger status

GATE.UWALL.16

GATE.UWALL.16

Lambda^2 F bridge test (honest): (1) POSITIVE - for SU(3) the exterior square is the conjugate rep, Lambda^2(M~)=conj(M~) exactly => the exterior leg IS the 3-bar (confirms the Exterior Leg typing); (2) anchor bivector 1^a=(0,1,1), Pl(K)=(-1,6,4) is its image under the 2nd compound of K^T (integer, combinatorial); (3) HONEST NEGATIVE - continuous Lambda^2(M~).(1^a) (moduli ~0.93,0.75,0.75) != integer Pl(K) nor its direction. CONCLUSION: the bridge rho*->Pl(K) is the DISCRETE non-abelian-Hodge invariant (integer R; det R=8 & Spec(Q+)={1,2,3} confirmed v39), NOT the continuous exterior-square action = exactly the H2 equivalence, pinned from the Lambda^2 F side. No c_u/c_d fabricated

Recorded status: Numerical (typing confirmed + honest negative)

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

CAR.COUNT.01

CAR.COUNT.01

SELF-COUNTING CHANNEL (the Pascal closure is an IDENTITY of the certified channel, not an extra requirement; closes the state half of the QBL analytic core at the structural level): (1) CANONICAL BIJECTION [I] - for odd g negation w -> -w maps the S+ weights bijectively into the opposite sheet (the CAR.PAIR.01 parity stated as a bijection), so the Cartan-neutral kernels of the certified channel S+xS- are exactly the pairs (w,-w), ONE PER CODE STATE: # neutral certified kernels = dim S+ = 2^{g-1} exactly (4, 16, 64 for g = 3, 5, 7) - the channel counts the code. (2) PASCAL PARTITION [I]: the same neutral set graded by pair-degree m (zero modes of Lambda^{2m}(C^{2g})) has sizes C(g,m) for m = 0..K = (g-1)/2; g=5 -> (1,5,10). (3) THE CLOSURE IS AN IDENTITY [I]: two countings of one set give 2^{g-1} = sum_{m<=K} C(g,m), symbolically verified for all odd g = 3..13 - the closure CONDITION of CAR.LADDER.01 is the channel counting itself. (4) HODGE FOLD [I]: the code's minus-sign grading C(g,2j) equals the folded binomial C(g, min(2j, g-2j)) with min(2j,g-2j) <= K - every code sector appears as a pair-degree <= K sector (g=5: code (1,10,5) vs channel (1,5,10), same multiset, folded order). (5) LADDER GENERICITY: the identity holds for every odd g and carries NO g-selection - the selection stays with rank-8 (g + N_fam = 8, CAR.RANK/v14) and ladder integrality (CAR.LADDER.01). (6) RESIDUE RE-TYPE [P] (recorded, not claimed): QBL clause (b') loses its closure burden - given the sheet-odd kernel (CAR.PAIR.02a) the channel automatically contains exactly one neutral kernel per code state at pair-degree <= K, so (b') is the definition of the channel; the remaining physical input is 'the seam certifies through a single scalar 2-point kernel' (for the established free c=8 seam net Wick extends the single kernel to the entire even tower by determinants); the carrier selection itself is carried by rank-8 + integrality - QBL is the structural consistency frame, no longer a missing selector

Recorded status: Identity [I] (bijection + exact multiset gradings + symbolic binomial identity) + Physical [P] (residue re-type)

Original ledger status

GRAV.ENTROPIC.ACTION.01

GRAV.ENTROPIC.ACTION.01

THE ENTROPIC-ACTION BRIDGE (2026-07-13) -- Bianconi's 'Gravity from entropy' (Phys. Rev. D 111, 066001 (2025); arXiv:2408.14391; DOI 10.1103/PhysRevD.111.066001) quantified as an EXTERNAL CANDIDATE for the local volume action behind TFPT's parameter-free Einstein equation; closes NOTHING, adds ZERO knobs. Context: v359 derives G_ab + Lambda g_ab = (1/c3) T_ab thermodynamically and is honestly typed 'equation of state, not a from-action quantisation' [O]; Bianconi's S_B = -Tr_F ln(G~ g~^-1) (quantum relative entropy between the spacetime metric and the matter-induced metric; Araki-related via Delta^{1/2} = G~ g~^-1, her Appendix C) is a local covariant action whose variation yields Einstein + emergent nonnegative Lambda at low coupling -- exactly the missing variational layer IF the bridge identities hold. [E] CARRIER HODGE COUNT: her Omega^{0,1,2} matter fiber, placed on the five-slot carrier C^5 (NOT on 4d spacetime, where the fiber is 1+4+6 = 11), counts 1+5+10 = 16 = 2^(g_car-1) = dim S+_{D5}, Hodge-star folded (L^1 ~= L^4) onto L^even C^5 -- the carrier half-spinor's own Pascal row (v2/v44); vector-space identity ONLY. [E] COEFFICIENT MATCH (the one quantitative pin): her weak-coupling Lagrangian L = 3 beta R - alpha L_matter (her Eq. (45); the 3 = the three form blocks) matched to the TFPT EH normalisation 1/(16 pi G) = c3/(2G) fixes her free constant EXACTLY: beta'_B = c3/6 = 1/(48 pi); the 6 = 3 blocks x 2 (EH 1/2) = p2 = |R+(A3)| is recorded as a structural echo, NOT a proof. [E] ENTROPY POTENTIAL = QUADRATIC LAMBDA: Lambda_G = (1/2 beta) Tr_F(G - I - ln G) with s(x) = x - 1 - ln x >= 0, s''(1) = 1 (exact series) -- Lambda nonnegative and QUADRATIC around the entropic fixed point; with the TFPT displacement eps_* = e^{-1/alpha} Q this reproduces the closed v60 branch rho_Lambda/Mbar^4 = (3/4 pi^2) e^{-2/alpha}, and the prefactor match (beta' = c3/6, l_P^2 Mbar^2 = c3) yields the EXACT normalisation target Tr_F Q^2 = 32 c3^4 = 1/(128 pi^4) = (2/3) delta_top -- a closed algebraic target for the two-form (pair) block (binom(5,2) = 10), [C] until an explicit Q is constructed. [E] FRULLANI: -ln A = int_0^inf dt/t (e^{-tA} - e^{-t}) -- her log action is an all-scale-integrated heat-kernel action, the correct genre to compare with the TFPT relative spectral action S_rel,chi (KMS cutoff). [E] R^2 KILL TEST QUANTIFIED (pre-registered): the raw log expansion carries a curvature-squared coefficient of order (c3/6)^2; the TFPT Starobinsky sector needs Mbar^2/(12 M_scal^2) = 1/(12 c3^7) (M_scal^2 = c3^7 Mbar^2, v36); the EXACT mismatch is 3 c3^{-9} = 3 (8 pi)^9 ~ 1.20e13 -- THIRTEEN orders: a direct identification of the raw entropic action with the TFPT R^2 sector is FALSE without a mechanism (scalaron = light trace mode of her G-field generating c3^{-9}, or KMS-spectral renormalisation of the log action). [C] COMPRESSION CONJECTURE (the bridge proper, recorded not proven): P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} -- the physical Calderon compression of her relative metric operator equals the TFPT seam modular operator; if proven, S_B is a local volume representation of the TFPT relative spectral action; open together with the operator-level Hodge items (Clifford action, Dirac intertwining, leaf involution, Calderon polarisation). Lorentzian positivity of G~ g~^-1 is NOT automatic (timelike gradients); TFPT's OS/reflection-positive route is the sturdier construction. [O] NOT CLOSED: the v359 equation-of-state status stays [O]; adopting S_B would close it ONLY after the compression conjecture AND an R^2 mechanism; no status marker moves anywhere

Recorded status: Formal/Numerical [E] (five exact identities + quantified gap) + [C] (compression conjecture, Q-target) + [O] (variational completion open)

Original ledger status

GRAV.ENTROPIC.HODGE.01

GRAV.ENTROPIC.HODGE.01

THE OPERATOR LEVEL OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; work packages 1 + 4-algebra of GRAV.ENTROPIC.ACTION.01/v473) -- upgrades the v473 vector-space identity to the operator level on the finite carrier Fock space and DECIDES the Q-block conjecture by exact enumeration; closes NO gate. [E] CAR => CLIFFORD: on Lambda^* C^5 (dim 2^g_car = 32) the ten Gamma_{2k} = a_k^dag + a_k, Gamma_{2k+1} = i(a_k^dag - a_k) satisfy {Gamma_a,Gamma_b} = 2 delta_ab exactly (all 100 pairs, integer matrices) -- the so(10) = D5 Clifford algebra acts on the carrier exterior algebra. [E] S+ = LAMBDA^EVEN AS D5-MODULE: the 45 bilinears [Gamma_a,Gamma_b]/4 commute with gamma = (-1)^N, span exactly 45 = dim so(10), and leak nothing from the 16-dim even subspace -- the carrier half-spinor IS the even exterior algebra WITH its spinor structure (the operator level of v2/v44/v197). [E] HODGE-DIRAC SYMBOL = CLIFFORD MULTIPLICATION: c(xi) = a^dag(xi) - a(xi), the principal symbol of Bianconi's D = d + delta, satisfies c(xi)^2 = -|xi|^2 Id exactly for symbolic real xi. [E] THE FOLD IS THE 5 -> 5bar CONJUGATION: the traceless u(5) weights of Lambda^4 are exactly the negatives of those of Lambda^1, so the Hodge fold *: Lambda^1 ~ Lambda^4 conjugates the vector block -- Bianconi's fiber 1+5+10 becomes the GUT half-spinor 16 = 1 + 5bar + 10, the SM carrier decomposition (not a mere dimension match). [E] HONEST OBSTRUCTION: Omega^{<=2} is NOT Clifford-invariant (c(xi) maps Lambda^2 into Lambda^3), while Lambda^even is invariant under the even Clifford algebra -- the bridge identification must go through the Hodge fold, never through naive degree truncation (a recorded constraint on the AP2 compression). [E] Q-TARGET ENUMERATION (the v473 [C] conjecture DECIDED at the algebraic level): the uniform-block ansatz Q = k c3^2 P_d solves Tr Q^2 = d k^2 c3^4 = 32 c3^4 with integer k for EXACTLY three supports (d,k) = (2,4), (8,2), (32,1) -- the sheet pair |Z2|, the rank g_car + N_fam = rank E8 = 8, and the full Fock space 2^g_car = 32 = dim Lambda^* C^5 (minimal uniform solution: eps_* = e^{-1/alpha} c3^2 per Fock mode, q = c3^2 = 1/(64 pi^2)); the two-form block d = 10 (the problem-note 'pair sector' conjecture) and the fiber d = 16 require irrational multipliers sqrt(16/5) and sqrt(2) -- the naive pair-block reading is KILLED; bonus identity Tr Q^2 / delta_top = 32/48 = 2/3 = dim Lambda^*/Omega_adm (the 2/3 in the v473 target is a mode-count ratio). [C] the physical choice among the three admissible supports (uniform-32 is minimal, not derived). [O] UNCHANGED: the AP2 compression P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} and the R^2 mechanism (v473 pre-registered kill test) stay open; no status marker moves anywhere

Recorded status: Formal/Identity [E] (operator-level Clifford/weights/enumeration, exact) + [C] (physical Q choice) + [O] (AP2 + R^2 unchanged)

Original ledger status

QGEO.COVER.01

QGEO.COVER.01

Strategy I/II, first executed step -- the mu3-cover of P^1 branched at mu4 (the curve y^3 = x^4 - 1) carries FOUR independent compiler fingerprints (one module, 6 checks, <1 s; discovery probe mu4_cover_model_probe.py 6/6, verdict FINGERPRINTS-HIT): (1) genus 3 with five cusps (mu4 + infinity; monodromies (w,w,w,w,w^2) multiplying to 1) and Chevalley-Weil H^{1,0} = (1,2) per character -- H_1 is a RANK-3 module over the Eisenstein integers, the compiler's 3-dimensional integral frame; (2) the D4 rotation closes PROJECTIVELY VIA THE DECK: in the reduced Burau model at t = omega (braid relations machine-verified) the rotation braid satisfies r^4 = omega x identity and r^12 = 1; (3) THE TORSION: the cusp-class group is (Z/3)^5 modulo one global relation = (Z/3)^4 of order 81 = 3^4 -- the v566 saturation index with exactly the per-mark Z/3 mechanism the self-code program asked for (four free local classes); det_Z(1 - deck) = 27 complementarily; (4) THE SIGNATURE: the invariant hermitian form of the Burau module is unique up to scale with eigenvalues {-2, 2-2sqrt2, 2+2sqrt2} -- LORENTZ SIGNATURE (1,2), matching the Chevalley-Weil Hodge numbers AND the rank-3 polarization signature of Paper II: Strategy II's first prediction (RP positivity as Hodge/Riemann positivity) confirmed at the form level. First operator pass: the single-cusp Gram has all-integer spectrum {0,0,2}; the compiler V-spectrum {0,1,2} is not yet located -- the operator dictionary (which canonical homology operator is Q/V) and the real-structure/RP map to the v572 Gram data are the NAMED NEXT STEPS. Kill criteria live (Smith/spectral mismatch for I; signature mismatch for II -- not triggered). GATE.QGEO does NOT move; no marker changes. [2026-08-01 XI, the dictionary's first entry: U IS LOCATED -- the cusp Gram (1-T2)(1-T2)^dagger has characteristic polynomial x^2(x-3) = char(U) exactly (the winding operator as a cusp Gram; fifth fingerprint); additional exact structure: opposite-twist products (1-T1)(1-T3) are NILPOTENT (char x^3), and G1+G3 has spectrum {0,1,3}; V/H/Q not located in the scanned canonical ranges (products/sums/omega-twists of length <= 2) -- the V-dictionary likely requires the parabolic flag (the seam-cusp filtration), named as the next step.] [2026-08-01 XIII: dictionary round 2 -- v598: THREE entries now exact (U ~ G2; Q_- ~ G1-G3, the sheet coupling as the diamond-diagonal Gram difference; the binary ladder ~ 1+G1+G3); the grading half absent from 7296 canonical combinations; see QGEO.DICT.01.] [2026-08-01 XV: the real structure is constructed and the invariant form is pinned -- v599: R unique with R^2 = +1, det J = 8 (a sheet-diamond determinant list member), Lorentz (1,2) confirmed exactly; see QGEO.REAL.01.] [2026-08-01 XVI: the operator dictionary lands at ALGEBRA level -- v600: the compiler pair (U, V) realized on the cover with the exact v566 saturation structure (index 81, divisors (1,1,1,3,3,3,3)); the [P]-model of GATE.QGEO has an explicit construction; canonicity and the two-demand derivation stay open; see QGEO.EMBED.01.] [2026-08-01 XXI: the torsion picture completes -- v604: 81 = 3^4 localizes at the finite marks (divisor classes; per-frame word order) and 27 = 3^3 at infinity (monodromy-defect cokernel = equivariant-order cokernel = det(1-deck)); see QGEO.EQORDER.01.] [2026-08-01 XXVIII: the curve's wider landscape is mapped -- v610: the Jacobian splits into a Q(zeta_12)-CM surface (the field of deck AND marks) times the Q(omega)-CM elliptic quotient, point-count-confirmed; the integer-spectrum census honestly relativizes single-char hits (the anchor class is the most frequent nontrivial family; the joint structure is the distinction); see QGEO.LANDSCAPE.01.] [2026-08-01 XXXV: the bedrock direction reverses -- v617: the conformal seam axioms FORCE the mu3-cover (Z4-Mobius rigidity, uniform weights j in {1,2}, genus 3, seam reflections = the v599 pattern); the QGEO.SYM.01 residue narrows to the cyclic-3 choice and the physical<->conformal seam identification; see QGEO.SEAMFORCE.01.]

Recorded status: Exact [E] fingerprints + typed candidate [C] (Burau convention validated; Q-identification open)

Original ledger status

QGEO.REAL.01

QGEO.REAL.01

The real structure on the mu3-cover -- the v598 suspect constructed (one module, 18 checks, ~3 s; discovery probe real_structure_probe.py 18/18, verdict REAL-STRUCTURE-LANDED). EXISTENCE AND UNIQUENESS [E]: complex conjugation on the base fixes the punctures {1,-1} and swaps {i,-i} (an orientation-reversing D4 reflection); it must act semi-linearly, R(v) = M conj(v), sending each cusp twist to the INVERSE of its reflected partner, R T_k R^-1 = T_(5-k)^-1. The intertwining system has a 1-DIMENSIONAL solution space: R exists and is unique up to scale; the canonical normalization satisfies R^2 = +1 exactly (a genuine real structure, not quaternionic); all four twist intertwinings hold exactly. THE GRADING FUNCTOR [E]: Gamma(A) = M conj(A) M^-1 (the curve analogue of the compiler grading A -> Sigma A Sigma) closes the three v598 dictionary gaps at char level: V ~ G2 + G3 - Gamma(G3) with char x(x-1)(x-2); H ~ 1 + G2 - Gamma(G3) with char (x-1)^2(x-2) -- THE ANCHOR SPECTRUM (1,1,2); Q_+ ~ 1 + G2 + G3 - Gamma(G3) with char (x-1)(x-2)(x-3). THE CONSTANTS [E]: the Gamma-even/odd cusp elements E_k = G_k + Gamma(G_k), O_k = G_k - Gamma(G_k) carry the compiler constants: E2 spectrum {0,1,5} (the carrier g_car = 5), E4 {0,3,9} = {0, N_fam, N_fam^2}, O3 nilpotent (char x^3), O2 char x(x^2-5), O4 char x(x^2-27). THE JOINT NEGATIVE [X, quantified]: char-matching is NOT yet an algebra embedding -- in the +-1 Gamma-extended integer lattice (8 generators) there are exactly 4 U-candidates and 6 V-candidates, and ZERO pairs satisfy the joint compiler relations tr(UV) = 3 and char(U+V) = (x-1)(x^2-5x+3); the natural pair (G2, G2+G3-Gamma(G3)) gives char x(x^2-6x+4), an exact documented mismatch. The flag-adapted simultaneous-conjugation search remains the named next step; the Strategy-I kill criterion is NOT triggered. THE LORENTZ FORM [E, Strategy II]: the invariant hermitian form J (T_k^dagger J T_k = J for all four twists) is unique up to real scale, integral over Z[omega], with det J = 8 -- A MEMBER OF THE SHEET-DIAMOND DETERMINANT LIST (3,4,8,14,20,32) -- and char (x+2)(x^2-4x-4): LORENTZ SIGNATURE (1,2), matching the Paper-II rank-3 polarization signature AND the Chevalley-Weil Hodge prediction (h^{1,0}, h^{0,1}) of Strategy II simultaneously. GATE.QGEO does not move; no marker changes. [2026-08-01 XVI: the degree-2 Gamma-even sector closes the joint gap -- v600: the compiler pair (U, V) is realized (GL3-conjugation solved exactly, saturation index 81 on the R-fixed lattice); the v599 joint negative is thereby located precisely at degree 1; see QGEO.EMBED.01.]

Recorded status: Exact [E] existence/uniqueness/char identities + [X] quantified joint negative + [C] the TFPT readings (exact sympy; numeric prefilter with symbolic confirmation for the scan)

Original ledger status

QGEO.CANON.01

QGEO.CANON.01

The canonical equivariant realization (one module, 14 checks, ~12 s; discovery probe equivariant_dual_probe.py 14/14, verdict EQUIVARIANT-DUAL-LANDED). THE REPAIR: the naive cusp-Gram presentation of v600 is not D4-equivariant because the standard hermitian dagger is basis-dependent -- the Burau image preserves J; the canonical building blocks are the J-ADAPTED cusp projectors GJ_k = (1-T_k) J^-1 (1-T_k)^dagger J. THE PROJECTORS [E]: all four are idempotent rank-1 J-self-adjoint projectors (char x^2(x-1)) rotating exactly under D4 (r GJ_k r^-1 = GJ_(k+1)); opposite cusps J-orthogonal (tr(GJ1 GJ3) = 0), neighbors overlap with trace 1. THE EQUIVARIANT PAIR [E]: U_k = 3 GJ_k, V_k = 1 + GJ_(k-1) - GJ_k GJ_(k+1) -- Z[omega]-integral, compiler spectra, compiler joint data (char(U+V) = charQ, tr(UV) = 3, tr(UV^2) = 6), seven-word rank 7, and D4-EQUIVARIANT BY CONSTRUCTION (r^k-conjugation = mark shift, verified for all marks): the mark choice IS the compiler's winding choice. THE DUALITY [E]: the equivariant pair is NOT conjugate to (U_c, V_c) (the intertwiner space is 1-dimensional with identically singular solutions -- must-fail documented) but EXACTLY conjugate to the transposed pair (U_c^T, V_c^T): the equivariant presentation realizes the DUAL (plane-stabilizer) parabolic, the v600 Gamma-even pair the line-stabilizer -- the cover carries BOTH parabolic types, exchanged by duality. THE KLEIN GROUP ON THE CURVE [E]: all eight spectral-sign involutions of V_1 are Z[omega]-integral (v590 rigidity reproduced); the two v590 demands select the explicit integral S_0 (trace -1, commuting with V), and A B = S_0 closes the Klein four-group on the cover. HONEST OPENS [C]: the naive J-word Gram has inertia (3,2,2), not the corpus v572 RP pattern (4,0,3) -- the Hodge/RP comparison needs the Weil-twisted polarization (named); the seam-line geometric identification stays open. GATE.QGEO does not move; no marker changes. [2026-08-01 XVIII: the duality mechanism is identified -- v602: the J-adjoint exchanges the two parabolic types exactly ((U1, V1^J) ~ compiler pair); the dictionary completes in the J-projector language; Weil positivity holds exactly; see QGEO.DUALFORMS.01.]

Recorded status: Exact [E] equivariance/duality/Klein identities + [C] typed opens (exact sympy; one numeric inertia count of an exact hermitian matrix)

Original ledger status

QGEO.DUALFORMS.01

QGEO.DUALFORMS.01

The duality-and-forms round (one module, 17 checks, ~8 s; discovery probe duality_forms_probe.py 17/17, verdict DUALITY-FORMS-LANDED). THE DUALITY MECHANISM [E]: U_1 = 3 GJ_1 is J-self-adjoint; the J-adjoint of V_1 = 1 + GJ_4 - GJ_1 GJ_2 is the exact product reversal 1 + GJ_4 - GJ_2 GJ_1; and (U_1, V_1^J) is EXACTLY conjugate to the compiler line-stabilizer pair while (U_1, V_1) is conjugate to the transposed pair (v601) -- the J-adjoint exchanges the two parabolic types inside ONE equivariant structure: the cover's realization is J-self-dual as a pair of pairs. DICTIONARY COMPLETION [E]: H = 1 + V(V-1)/2 has char (x-1)^2(x-2) -- the corpus anchor formula holds VERBATIM on the curve; Q = U + V has char (x-1)(x^2-5x+3); Q_+ ~ 2 - GJ_1 + GJ_2 GJ_3 has char (x-1)(x-2)(x-3); the binary AnchorLadder ~ 2 - GJ_1 + 2 GJ_2 GJ_3 has char (x-1)(x-2)(x-4); the S_0-grading halves of Q are computed and documented. WEIL POSITIVITY [E, Strategy II at module level]: the Weil operator C from J's spectral split (Hodge numbers (1,2)) gives a POSITIVE DEFINITE twisted form JC with exact eigenvalues {2sqrt2-2, 2, 2+2sqrt2} = |Spec J| -- the Riemann bilinear positivity statement holds on the omega-sheet. THE CANONICAL BILINEAR FORM [E]: B(v,w) = (Rv)^dagger J w has matrix B = M^dagger J, SYMMETRIC and nondegenerate with det B = 8 -- the curve's complex orthogonal structure combining the real structure with the polarization. HONEST NEGATIVES [X/O, quantified]: (i) g^T g is NOT proportional to B -- the corpus transpose structure is not the (R,J)-bilinear form (corpus RP word-Gram inertia (4,3,0) documented); (ii) the naive mark-local cokernel map fails (U_2, U_3 lie outside the seven-word algebra, U_4 inside with integer coefficients) -- the carrier of the mark-local Z/3 mechanism stays open; (iii) the seam-plane normal is explicit but NOT a deck eigenvector -- the seam identification stays open. COMPANION LEAN CERTIFICATE (FORM.QGEO.04): the v600 integer core is certified in TfptCarrier/CoverEmbedding.lean -- compiler relations of (C_U, C_V), the word factorization W = T*S, the Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P (P*Pinv = 1), and S-primitivity via a unimodular minor: determinant-free, kernel decide, lake build green (3375 jobs). GATE.QGEO does not move; no marker changes. [2026-08-01 XIX: the seam question advances -- v603: the seam line is a REAL line of the cover (R-fixed), S_0 reverses it geometrically; the infinity cusp separates in the unreduced module; the first per-mark Z/3 witness lands (E_1, order 3); see QGEO.SEAMMARKS.01.] [2026-08-01 XXII: the corpus-transpose translator question is RESOLVED -- v605: theta_corpus = Jadj o inner(c) exactly, c unique up to scale, 14c integral; c's independent geometric identity is the remaining open; see QGEO.TRANSLATOR.01.]

Recorded status: Exact [E] duality/dictionary/positivity identities + [X/O] quantified negatives and typed opens (exact sympy; two numeric inertia counts of exact matrices) + Lean certificate FORM.QGEO.04

Original ledger status

QGEO.EQORDER.01

QGEO.EQORDER.01

The equivariant order and the infinity bridge (one module, 12 checks, ~4 s; discovery probe equivariant_order_probe.py 12/12, verdict EQUIVARIANT-ORDER-LANDED). CORRECTION + THE INTEGRAL FORM [E]: R IS an anti-isometry of the polarization -- the correct compatibility equation M^dagger J M = J^T holds EXACTLY (a v599-diary side remark had tested the wrong equation M conj(J) M^dagger vs J); consequently J restricts to a REAL SYMMETRIC INTEGER form on the saturated fixed lattice: J_fix = [[16,2,4],[2,-2,2],[4,2,-2]], det 72 = 8 x 9 (the polarization determinant times the (1-omega)-covolume factor). THE EQUIVARIANT ORDER IS FULL [E]: the Z-lattice generated by the seven words of ALL FOUR mark frames (the r^k-conjugates of the v601/v602 equivariant pair; all 28 words Z[omega]-integral) has FULL rank 18 in M_3(Z[omega]) with Smith divisors (1^15, 3, 3, 3) -- saturation index 27 = 3^3: one frame gives the 7-dim parabolic of index 81 = 3^4 (v600), all four give the full matrix algebra of index 27. THE 3-TORSION LOCALIZES AT INFINITY [E]: on the unreduced integral module Z[omega]^4, coker(1 - T_inf) = Z^2 + (Z/3)^3 (Smith divisors (1,1,1,3,3,3,0,0), torsion 27) while each FINITE cusp twist is torsion-free (divisors (1,1,0^6)) -- three objects share (Z/3)^3 = 27: the infinity-monodromy cokernel, the equivariant-order cokernel, and det_Z(1-deck) (v597). THE READING [C]: the two compiler constants localize on the cover -- 81 = 3^4 at the FINITE marks (cusp divisor classes, v597; per-frame word order, v600) and 27 = 3^3 at INFINITY (monodromy defect; full equivariant order): the integral infinity <-> order-torsion bridge. TRANSLATOR NEGATIVES EXTENDED [X, quantified]: the corpus RP metric pulled back to the integral fixed lattice is NOT proportional to J_fix or its Klein twists (6 tested), and neither J_fix^-1 F2 nor F2 J_fix^-1 lies in the word algebra -- after the (B, Gamma, Klein, J_fix) sweep the corpus transpose structure resists all single-sheet constructions; the conjugate character sheet (omega-bar) is the named next candidate. GATE.QGEO does not move; no marker changes. [2026-08-01 XXII: the translator is FOUND -- v605: the corpus RP anti-automorphism is the J-adjoint twisted by an explicit inner element c (14c integral; the omega-bar detour was unnecessary); the seam is 3-torsion at infinity; the 81 survives frame-joining; see QGEO.TRANSLATOR.01.] [2026-08-02 XXXIX: the fixed-lattice form J_fix gains an arithmetic identity -- v624: P^T J_det P = J_fix EXACTLY with det P = -6 (J_det the prime-front determinant form): the two Lorentz lattices are the same rational quadratic form, the cover an index-6 sublattice (72/2 = 6^2, 6 = p_2(a)); P-canonicity and the Hodge-chamber question preregistered; see EXTREV.LATTICE.01.]

Recorded status: Exact [E] anti-isometry/order/localization identities + [C] typed localization reading + [X] quantified translator negatives (exact sympy; Smith normal forms over Z)

Original ledger status

QGEO.LANDSCAPE.01

QGEO.LANDSCAPE.01

The curve landscape round -- what else the mu3-cover carries (one module, 9 checks, ~1 s; discovery probe curve_landscape_probe.py 9/9, verdict LANDSCAPE-MAPPED). THE JACOBIAN SPLITS [E chars + C reading]: the three holomorphic differentials of C: y^3 = x^4 - 1 carry (deck, rotation) characters of combined orders (12, 12, 6) (dx/y ~ (w^-1, i), dx/y^2 ~ (w, i), x dx/y^2 ~ (w, -1)); by standard CM theory the Jacobian splits up to isogeny as Jac(C) ~ A_2(CM Q(zeta_12)) x E(CM Q(omega)) -- with Q(zeta_12) = Q(i, sqrt3) THE FIELD GENERATED BY THE DECK AND THE MARK ROTATION TOGETHER, and E: y^3 = u^2 - 1 the x -> -x quotient curve. THE SPLITTING CONFIRMED BY POINT COUNTS [E]: a_p(C) = a_p(E) EXACTLY for all ten tested primes p != 1 mod 12 (5,7,11,17,19,23,29,31,41,43) -- the zeta_12 surface is supersingular there, as CM theory demands; the surface contributes nonzero traces exactly at p = 1 mod 12 ((13, 37, 61, 73) -> (-8, 4, -20, -32)); and a_p(C) = 0 for every tested p = 2 mod 3 (inert in Q(omega)). THE INTEGER-SPECTRUM CENSUS [E + honest relativization]: in the declared degree <= 2 J-projector pool, integer characteristic polynomials are COMMON (7894 entries) -- single spectra are CHEAP on the curve; the ANCHOR CLASS ((x+1)^2(x+2), i.e. {1,1,2} up to negation/shift) is the MOST FREQUENT nontrivial family (286 = the maximum), and non-compiler spectra exist ({1,1,3}: 254 entries): the compiler's distinction is the JOINT algebra structure (v600: rank 7, joint traces, saturation 81), not individual characteristic polynomials -- the v598/v599 char-level hits were correctly typed candidates. NEW NAMED RESOURCES [C]: the zeta_12 CM surface and the omega CM elliptic curve are two canonical objects of the cover not yet used by the program; their periods are classical Beta values -- the named resource for future RP/Hodge rounds. GATE.QGEO does not move; no marker changes. [2026-08-01 XXIX: the period resource is made concrete -- v611: all five basic periods are exact Beta values on the zeta_12 grid, the rotation acts with the exact characters, Omega_E is the classical omega-CM period, and the two CM factors are period-independent (PSLQ null), as the splitting demands; see QGEO.PERIODS.01.]

Recorded status: Exact [E] characters/point-counts/census + [C] CM-theory reading and named resources (deterministic integer computations; no floats beyond the census prefilter)

Original ledger status

QGEO.PERIODS.01

QGEO.PERIODS.01

The periods round -- the analytic layer of the cover is classical (one module, 6 checks, ~1 s; discovery probe periods_probe.py 6/6, verdict PERIODS-CLASSICAL). EXACT BETA IDENTITIES [E, symbolic]: the five basic period integrals I_{k,j} = int_0^1 x^k (1-x^4)^{-j/3} dx equal (1/4) B((k+1)/4, 1-j/3) EXACTLY; the Gamma denominators (Gamma(11/12), Gamma(7/12), Gamma(5/12), Gamma(1/12)) all live on the ZETA_12 GRID -- the analytic layer carries the same 12 = |mu4| x |mu3| structure as the algebra. THE ROTATION ACTS EXACTLY [E, 40-digit certificates]: substituting x -> ix maps the [0,1]-period to the [0,i]-period and multiplies by EXACTLY i^(k+1) -- the v610 differential characters act on actual periods (three cases). THE OMEGA-CM PERIOD [E, symbolic]: the quotient curve E: y^3 = u^2 - 1 has real period Omega_E = B(1/3,1/6)/2 = Gamma(1/6)Gamma(1/3)/(2 sqrt pi) exactly -- the classical CM period of the omega curve. CROSS-FACTOR INDEPENDENCE [E, must-fail consistency]: the ratio I_{1,2}/I_{0,2} -- periods of differentials on DIFFERENT CM factors (v610) -- admits NO integer polynomial relation of degree <= 4 with coefficients up to 10^6 at 40 digits (PSLQ null): the zeta_12 surface and the omega curve are period-independent, exactly as the splitting demands. THE PERIOD DICTIONARY [C]: every cover period is a Beta/Gamma monomial on the 12-grid -- the named analytic resource for the future Hodge/RP period comparisons. GATE.QGEO does not move; no marker changes. [2026-08-01 XXX: the form-level bridge lands -- v612: J diagonalizes exactly in the rotation eigenbasis and the unique positive (Hodge) direction is the primitive zeta_12 eigenspace; the canonical period normalization (Beta weights) is the named remaining step; see QGEO.POLFRAME.01.] [2026-08-01 XXXII: the canonical period normalization lands -- v613: r = deck o rotation exactly (char(r) = char(omega*M)), the period rows realize the rotation eigenframe, the Hodge weights are explicit Gamma monomials, and J = -h per character (Riemann-Hodge positivity realized by the compiler polarization); v612's named step B5 executed; see QGEO.PERNORM.01.]

Recorded status: Exact [E] symbolic Beta identities + 40-digit certificates + PSLQ-null consistency + [C] the period-dictionary reading

Original ledger status

QGEO.POLFRAME.01

QGEO.POLFRAME.01

The polarization frame round (one module, 9 checks, ~2 s; discovery probe polarization_frame_probe.py 9/9, verdict POLARIZATION-FRAMED). THE ROTATION SPECTRUM [E]: the braid rotation r has three SIMPLE eigenvalues = zeta_12 x {1, -1, -i} (e^{i pi/6}, e^{-i 5pi/6}, e^{-i pi/3}), det r = -1 exactly -- the twelfth roots organize the spectrum. J DIAGONALIZES IN THE ROTATION FRAME [E]: in the left-eigenvector basis W of r the polarization becomes EXACTLY diagonal, (W^dag)^-1 J W^-1 = diag((1+sqrt3)/2, (1-sqrt3)/2, -1) (canonical normalization). Honest typing: the diagonality is the standard consequence of r being J-unitary with simple unimodular eigenvalues (eigenspaces to distinct eigenvalues are J-orthogonal); the CONTENT is the exact values and the signature distribution. THE HODGE DIRECTION IS THE ZETA_12 EIGENSPACE [E]: the unique POSITIVE direction of J (signature (1,2), v597/v599) is EXACTLY the eigenspace of the primitive twelfth root e^{i pi/6} -- the h^{1,0} = 1 Hodge line of the omega-sheet sits on the zeta_12 eigenvector of the rotation: the analytic Hodge structure and the finite rotation agree on where positivity lives. CONSISTENCY [E]: r^dag J r = J re-verified; the product of the diagonal values equals det J / |det W|^2 = 1/2 exactly. THE READING [C]: with v611 (periods = Beta values on the zeta_12 grid) the analytic bridge is one step from closing -- the canonical period normalization of the eigenvectors (fixing the diagonal weights as explicit Beta monomials) is the named next step. GATE.QGEO does not move; no marker changes. [2026-08-01 XXXII: the canonical period normalization lands -- v613: r = deck o rotation exactly (char(r) = char(omega*M)), the period rows realize the rotation eigenframe, the Hodge weights are explicit Gamma monomials, and J = -h per character (Riemann-Hodge positivity realized by the compiler polarization); v612's named step B5 executed; see QGEO.PERNORM.01.]

Recorded status: Exact [E] spectrum/diagonalization/signature identities + [C] the bridge reading (exact sympy)

Original ledger status

QGEO.PERNORM.01

QGEO.PERNORM.01

The canonical period normalization -- the analytic bridge closes (one module, 19 checks, ~6 s; discovery probe canonical_periods_probe.py 19/19, verdict PERIOD-NORMALIZED; executes v612's named step B5). DECK-FREE ROTATION [E, 40-digit certificates]: the quarter rotation x -> ix maps the four branch-point segments of y^3 = x^4 - 1 cyclically with NO deck correction (all 12 wrap factors exactly 1); the single omega-sheet homology relation is seg1+seg2+seg3+seg4 = 0; a MUST-FAIL control documents that the raw period rows are NOT r-eigenvectors (naive ansatz falsified, residual 0.42). THE EXACT DICTIONARY r = deck o rotation [E, exact]: char(r) = char(omega*M) EXACTLY for the induced companion rotation M (M^4 = 1) -- the reduced Burau rotation at t = omega is GL3-conjugate (explicit conjugator G) to the geometric quarter rotation composed with EXACTLY ONE deck step; explains v597's r^4 = omega structurally; the deck power is UNIQUE among the four scalar twists c*M with (cM)^4 = omega. THE EIGENFRAME REALIZED [E]: the period rows (hol dx/y^2, hol x dx/y^2, ANTIHOL conj(dx/y)) are left eigenvectors of M with characters (i, -1, -i); transported through G they are r-eigenvectors with eigenvalues omega*(i,-1,-i) = spec(r); character match: antihol <-> zeta_12. THE HODGE WEIGHTS ARE GAMMA MONOMIALS [E]: h0 = (3/4) pi G(1/4)G(1/3)G(5/12)/(G(3/4)G(2/3)G(7/12)), h1 = (3 sqrt3/16 pi) G(1/3)^2 G(1/6)^2, h2 = (3/4) pi G(1/4)G(2/3)G(1/12)/(G(3/4)G(1/3)G(11/12)) (complex Beta/Gelfand-KLT formula, verified to <= 1e-25); omega-sheet signature (2,1), unique NEGATIVE = the antiholomorphic line conj(dx/y). THE SIGN BRIDGE [E]: all three character-matched sign products are negative -- J = -h PER CHARACTER: the compiler polarization is the NEGATIVE of the analytic Hodge form up to per-line positive scales; J's unique positive line (zeta_12, v612) = h's unique negative line (antihol): Riemann-Hodge positivity is realized by J. THE RESIDUE [C]: the canonical fork-basis identification (basepoint bookkeeping for G), named, not claimed. GATE.QGEO does not move; no marker changes. [2026-08-01 XXXIII: the residue F1 is closed at the transport level -- v614: the deck step is DERIVED as the boundary monodromy t^4 = omega (Gauss-Manin transport along the rotation loop; transported periods factorize exactly, the canonical transport matrix is omega*M): the dictionary r ~ omega*M is canonical; see QGEO.TRANSPORT.01.]

Recorded status: Exact [E] dictionary/uniqueness identities + 40-digit certificates + Gamma-monomial closed forms + [C] the fork-basis residue (sympy + mpmath)

Original ledger status

EXTREV.LATTICE.01

EXTREV.LATTICE.01

The external-review lattice audit (one module, 14 checks, <1 s; discovery probe external_lattice_audit_probe.py 14/14, verdict LATTICE-AUDIT-PASSED; external review 2026-08-02, three claimed derivations machine-checked, one follow-up retyped honestly). THE FOUR-MARK CLOSURE [E, conditional]: generalizing the glue to D_{d+1} + A_{d-1}, THREE independent selectors each force d = 4 -- the disc-group census (Z_d = Z_4 with cyclic D-side; negative controls d = 3, 5, 8), the even-glue norm equation (d+1)/4 + (d-1)/d = 2 (unique positive solution; no other even target admits an integer d), and the Pythagorean selector (d-1)^2 + d^2 = (d+1)^2 (the (3,4,5) = (cycles, marks, carrier) triple); (1,1,2) is the unique 3-part partition of 4 and c_3 = 1/(2 e_1 pi) = 1/(8 pi): under the named architecture the two-input presentation reduces to ONE boundary degree d = 4 plus pi; honest: conditional on the glue architecture, the residue moves to why the boundary realizes the minimal cyclically-glueable genus-0 divisor. THE ANCHOR BYTECODE [E]: chi_a = (t-1)^2(t-2) carries (4, 5, 2); p_n = 2 + 2^n with p_{n+2} = 3p_{n+1} - 2p_n; Newton recovers the e_k; p_1 p_2 p_3 = 240, p_4 - p_3 = 8, 240 + 8 = 248 (H-realization v602). THE LORENTZ CONGRUENCE [E + X]: P^T J_det P = J_fix EXACTLY with det P = -6 -- the prime-front determinant form and the v604 cover polarization lattice are the SAME rational quadratic form, the cover an index-6 sublattice (36 = 6^2, 6 = p_2(a)); the review's follow-up (integer R to B = M^dagger J) is ILL-POSED as stated (B COMPLEX symmetric, det 8 verified); partial positive: the third symmetric-Gauss ratio is an exact Q(omega) square; P-canonicity and the Hodge-chamber question are the named opens. CITATIONS VERIFIED: Suzuki arXiv 2606.09096 (screw-function Weil framework, Yoshida odd-criterion) and arXiv 2607.24830 (numerical realization) -- the basis of the PRIME.WEIL.OPERATOR contract. No marker moves; AX.P2.01 keeps its typing; no RH statement. [2026-08-02 XL: the prime-shadow audit lands -- v625: the 'primes appear AFTER E8' reading is exact at the theta level (Theta_E8 = E_4, shell counts 240 sigma_3(n) multiplicative, commuting Hecke channels with the theta as simultaneous eigenvector, L(E_4,s) = zeta(s) zeta(s-3)); the direction of explanation (geometry first, primes as readout) is fixed within the narrative; see PRIME.SHADOW.01.] [2026-08-02 XLI: the Lorentz congruence P^T J_det P = J_fix is additionally machine-proved in Lean (experiments/lean4-carrier-rigidity/TfptCarrier/LorentzCongruence.lean, kernel decide on the explicit integer matrices, lake build green, no sorry); P-canonicity and the operator construction of P executed by v635/v636 (see PRIME.PCANON.01 / PRIME.PCONSTRUCT.01).] [2026-08-02 XLV: look-elsewhere quantification -- v664: the observation 6 = Lorentz-congruence index = p_2(a) carries p_max = 0.169 under the most conservative null model: retyped as an observation WITHOUT statistical weight (the exact congruence P^T J_det P = J_fix itself is [E] and untouched).] [2026-08-28 CXL: v993 lifts the architecture conditionality at the census level -- full rank-8 ADE with cyclic Z4 both sides selects (D5, A3) uniquely; the D+A split is an OUTPUT. EXTREV.LATTICE.01's own v624 checks and typing UNCHANGED (they remain the architecture-conditional selectors).]

Recorded status: Exact [E] closure/bytecode/congruence identities + [X] the ill-posed retype + [C] conditional typing (sympy; citations web-verified)

Original ledger status

DOILY.HODGE.FACTOR.01

DOILY.HODGE.FACTOR.01

v880 (part 3) -- the doily IS the multiplication table Lambda^2 E x Lambda^2 E -> Lambda^4 E of the five-slot carrier: via the v845 parity lift the three iota-supports of every doily line partition {1..5} as {i} u {j,k} u {l,m}; b_jk ^ b_lm = +-f_i EXACT (integer exterior algebra); the map line -> (i; {j,k},{l,m}) is a BIJECTION onto all 5 x 3 = 15 configurations (three perfect matchings per carrier slot); CONVERSE: the wedge is nonzero EXACTLY on doily lines (external lines: all pairwise products vanish; secants carry <= 1 pair point). Probe S3.

Recorded status: Exact identity-grade theorem [Identity]: the incidence geometry IS the Clifford multiplication; no marker move

Original ledger status

DOILY.PLUCKER.GR25.01

DOILY.PLUCKER.GR25.01

THE GR(2,5) PLUCKER READING + THE S6 SPECHT CHANNELS + THE HADAMARD ANCHOR + THE VACUUM SELECTION (closed 2026-08-09, round CV; promoted round CVIII in v888; probe s6_plucker_hadamard_probe.py 27/27 S6PLUCKER-CLOSED, exact integer/sympy): (1) the 15 doily lines ARE the 15 monomials of the five Plucker quadrics of Gr(2,5) -- bijection 15/15 with PERFECT sign match (every quadric reads (+,-,+) = the v880 Hodge table; converse warded: secants/externals produce no monomial); (2) the S6 duad representation decomposes 15 = 1 + 5 + 9 (Specht S(6) + S(5,1) + S(4,2)) and the normalized doily operator acts by the scalars (1, 0, 2/3) EXACTLY, with 2/3 = e_3/(e_2 - e_3) and 15 = 1 + e_2 + (e_2 - e_3)^2 at the anchor (4, 5, 2); (3) the q* zero set is a (16, 6, 2) Hadamard difference set with the anchor signature (v, k, lambda, v-k) = (2^{e_1}, p_2, e_3, C(e_2, 2)) = (16, 6, 2, 10), bent ward |W| = 4 on all 16 characters; (4) THE VACUUM-SELECTION THEOREM: Sp(4,2) = S6 acts transitively on the 6 Arf-1 refinements; Stab(q*) has order 120 acting faithfully as S_5 on the remaining five; its orbits on the 16 messages are [1, 5, 10] EQUAL AS SETS to (Lambda^0, Lambda^4, Lambda^2) -- the five-slot carrier is the residual symmetry after choosing one of six equivalent vacua. Controls: Arf-0 has census (16,10,6); a secant breaks the bijection; the wrong scalar breaks the channel law. All exact; no marker moves.

Recorded status: Closed finite theorems [E]: four exact statements; the physical vacuum-selection mechanism stays [O]; no marker move

Original ledger status

DOILY.PFAFFIAN.WICK.01

DOILY.PFAFFIAN.WICK.01

THE FERMIONIC PFAFFIAN SIGNATURE OF THE FINITE COMPILER (closed 2026-08-09, round CVII; promoted round CVIII in v888; probe k6_pfaffian_selfhosting_probe.py 42/42 K6PFAFFIAN-CLOSED, exact integer/sympy, first frozen run): (1) THE K6 MOTIF THEOREM (CFIN.K6.MOTIF.01): under the corpus duad labeling the 35 lines of PG(3,2) are EXACTLY 15 perfect matchings of K6 (= the doily) + 10 triangles through the vacuum vertex (= the secants, weights {2,2,1} = the wrong anchor) + 10 triangles avoiding it (= the external lines); q*(e) = 0 iff e touches the vacuum -- the referee bit of relations; (2) THE SIGN THEOREM: the fermionic Pfaffian signs sgn(M) of the symbolic 6x6 expansion match the corpus Hodge table under the CANONICAL global character chi(i) = (-1)^{i+1} on the five vacuum edges -- triple-identified as the vacuum-row Laplace sign (Pf(A) = sum_i (-1)^{i+1} a_0i Pf(A_0i-hat) symbolic, each 4x4 minor Pfaffian = the Gr(2,5) Plucker quadric), the Lambda^4/Lambda^5 volume-form conversion (verified on all 15), and the C6 crossing character (Pf(PAP^T) = det(P) Pf(A) symbolic; sgn(pi(M)) = det(pi) c_pi(M) sgn(M) on all 15) -- a global gauge, NOT a line-dependent fudge (the flipped-chi control produces exactly the 3 expected mismatches); (3) THE SELF-HOSTING COUNTING THEOREM (P2.SELFHOSTING.PAIRING.01): C(g+1, 2) = g!! has g = 5 as its ONLY nontrivial solution (g = 1 trivial; separation for g >= 7 symbolic via g^2 - 3 > 0), and (g-2)!! = 3 = N_fam -- the [E] theorem behind the review's self-hosting pair-compiler reading; THE PHYSICAL PREMISE (the boundary grammar IS a self-hosting Wick pair compiler) stays [O] -- P2 narrowed to ONE premise, NOT eliminated, no marker move; (4) ANCHOR.MOD30.CLOCK.01: p_n = 2 + 2^n with p_{n+1} = 2 p_n - 2 has the mod-30 orbit 4 -> 6 -> 10 -> 18 -> 4 (period 4 = |mu_4|); anti-numerology census: the purely period-4 moduli below 100 are exactly {5, 10, 15, 30}, all dividing 30, lcm 30 -- CLOCK-DISTINGUISHED (30 is the maximal/universal clock), with the honest caveat that 5/10/15 already realize period 4; (5) the 6 one-factorizations of K6 (each matching in exactly 2; 30 = 6 x 5 = 2 x 15) carry the exceptional-outer-automorphism signature ((1,1,1,1,2) on vertices vs (2,2,2) on factorizations); NO canonical vertex-factorization bijection (typed). The operative-Coxeter-30 reading is separately DEAD (coxeter_schedule_probe.py 28/28 SCHEDULE-DEAD, sandbox: no root orbit realizes the 15 classes x2 -- best 12/15; the K6 side allows at most 3/15 separation-15 pairings; w^15 = -1 makes the 15-shift automatic; the counting identity 30 = 2 N_fam g_car stands). All controls fire; no marker moves.

Recorded status: Closed finite theorems [E] + one honest kill (the operative-30 reading); the Wick-compiler premise stays [O]; no marker move