# LBL3VP — vacuum-polarisation-dressed QED light-by-light box (3 loops) **Closed in fully-analytic final form, >= 37.70 digits live** (three fixed-eps gates vs the held-out 3-loop oracle: 37.70 / 38.65 / 40.08 d at eps = 2^-6 / 2^-9 / 2^-13). Kinematics: s = -1, t = -1/3, m^2 = 1. Top integral: family `lbl3vp`, nu = [1,1,1,2,1,2,1,1] (8 props, eps^-2 pole). ## The result (final form — every factor analytic, exact in d) I_VP(eps) = (1/pi) * Integral_{1}^{inf} rho_VP(w'; eps) K_P(w'; eps) dw' The convolution variable w' is the internal photon mass^2 (the VP-channel virtuality). Every factor is a closed form or a one-fold iterated integral over closed-form kernels — no numeric seeds, no node caches: **Two-body density** (all w' > 1, exact in d, elementary; verified >= 161 d against 2-loop AMFlow probes): rho_2b(w')/pi = -G(eps) [ -(1-2eps)(w'+1)/(w'-1) + eps(w'-1)/6 ] (w'-1)^{-2eps} w'^{eps-1}, G(eps) = Gamma(1+eps)Gamma(1-eps)/(eps Gamma(2-2eps)). **Three-body density** (w' > 9) — NEW in round 2: the sunrise-elliptic Gamma_1(6) piece as an exact-in-d period one-fold, rho_3b(w') = pi c1(eps)^2 w'^{eps-1} Integral_4^{(sqrt(w')-1)^2} dq (q-4)^{1/2-eps} q^{-5/2} [ ((sqrt(w')-1)^2 - q)((sqrt(w')+1)^2 - q) ]^{1/2-eps}, c1(eps) = Gamma(1-eps)/Gamma(2-2eps). Derivation: disperse the VP sub-bubble B(q^2) inside Sigma[1,2,1,1] (family sigvp: D1=(P-q)^2-1, D2=q^2, D3=k^2-1, D4=(q-k)^2-1) and partial-fraction 1/((q^2)^2(q^2-sigma)); the resulting bubble cuts are elementary powers exact in d. The 2-body formula above is re-obtained, and the 3-body piece collapses to the one-fold shown (the sigma-power is exactly -5/2, all eps-dependence in the three half-integer exponents). The branch points {0, 4, (sqrt(w')-+1)^2} are the equal-mass-sunrise quartic — the Gamma_1(6) curve (Bloch–Vanhove, arXiv:1309.5865); at eps=0 the integrand reduces to this work's R_2 elliptic integrand (SIGVP_CLOSED_FORM), which PSLQ-refused every weight-2 polylog basis — this one-fold IS its closed form. Validated per eps-order (eps^0..eps^4) at 58-108 d against banked 2-loop AMFlow probes at w = 12, 50 (/lbl3vp/r2-eichler/v1b_orders.py). This replaces round 1's DE-transported density. **Kernel** K(w';eps) — the 1-loop box with photon-line mass^2 = w', exact in eps and seed-free. Its defining analytic form is the exact 2-fold Feynman-parametric integral (one parametric integration done analytically; u = -s-t): K(w') = Gamma(2+eps) Int_0^1 dx1 Int_0^{1-x1} dx2 [ A^{-1-eps} - (A+B X)^{-1-eps} ] / ((1+eps) B), A = w' x1 + (1-x1) - u x2 (1-x1-x2), B = -s x1 + u x2, X = 1-x1-x2. The script evaluates this once per eps (anchor at w0 = 200) and carries it to every quadrature node with the exact rational IBP connection A(w',d) (46 exact-rational Kira samples, 0 mismatches) by Taylor-series stepping; threshold Taylor coefficients K_n(eps) come from an arc-VoP Cauchy circle |w'-1| = 3/8 (loop-closure defect ~1e-60, printed live). The kernel is additionally RE-DERIVED each run from the w'->infinity VACUUM boundary (K -> -Tri_u(eps)/w') via the variation-of-parameters one-fold K(w') = H(w') [ K(W)/H(W) + Int_W^{w'} (sum_j A_0j M_j)(v)/H(v) dv ], H(w') = (1+w'^2)^{-(1+2eps)/2} = exp Int A_00, with all subsector masters M_j Gamma closed forms or elementary one-folds exact in eps; the two constructions are compared live (29.9-32.0 d at the demo settings — the vacuum fold's quadrature carries a small ~1/eps-scaling numeric residual, so the parametric anchor is primary). K_P is the twice- subtracted kernel (K - K_0 - K_1 (w'-1))/(w'-1)^2. Round 1's AMFlow inputs (2-loop Sigma_VP boundary seeds, 1-loop box seeds, shipped threshold series) are ALL retired from the computation; the 1-loop blocks remain in lbl3vp-data.json as live cross-checks only (printed each run: K(5) at 44.1-44.7 d, K_2(eps) at 40.2-40.3 d). ## Analytic pole layers (closed forms; live) Sigma_VP^{(-2)}(w) = 1/(1-w) => I_VP^{(-2)} = -K_2^{(0)} I_VP^{(-2)} = -0.0454061588299285527973574028434332658 (live gate 45.12 d) Sigma_VP^{(-1)}(w) = -2 log(1-w)/(1-w) - log(1-w)/w + 2 gamma_E/(w-1) I_VP^{(-1)} = -0.0153448052040345547653673238694313064 (live gate 39.58 d; K_2^{(1)} is the single shipped 1-loop input constant left in this artifact, used only in this layer demo — the fixed-eps gate is seed-free) ## The gate (certification) Fully-analytic I_VP(eps) vs the independent held-out 8-propagator AMFlow solve at fixed eps (grid eps = 2^-k, k = 4..18, working_pre 340; grid byte-traced to the banked raw solve at 319-321 d): | eps | this work (live) | agreement | |---|---|---| | 2^-6 | -187.1117649803697229476251766007856893737 | 37.70 d | | 2^-9 | -11910.95516971382221897201394125793392711 | 38.65 d | | 2^-13 | -3047281.588924805665645691404195430424708 | 40.08 d | This certifies the integral as a function of eps, not one Laurent coefficient. Precision is arbitrary: every internal order scales with --dps; round 1's caps (eps^20 seed truncation, shipped threshold series) are gone because there are no seeds. ## How it was computed Per eps point (~6.5 min wall at dps 60 on the build host, 3 points run as forked workers): 2-fold parametric anchor (seconds), Taylor stepping of the exact rational connection to ~2700 quadrature nodes (~2 min), arc-VoP Cauchy circle for the threshold patch (~20 s), closed-form densities with the Gamma_1(6) one-fold evaluated by log-split tanh-sinh per node (memoized), threshold-peel assembly + Richardson (round-1 panel algebra), plus the independent vacuum-boundary VoP cross-check (~1 min). The oracle is an independent AMFlow evaluation of the full 3-loop integral, never used as input. AMFlow: Liu & Ma, arXiv:2201.11669 (Comput. Phys. Commun. 283 (2023) 108565). Sunrise/Gamma_1(6): Bloch & Vanhove, arXiv:1309.5865.