# LBL3E — sunrise-dressed box (3-loop light-by-light, 6 lines, all mass m) Result status: all 10 masters function-level closed; the elliptic sectors are anchored by two independent codepaths at 35.7–116.4 digits. One independent oracle (the uncut sector-63 finite part) is still outstanding — see the last section. IBP reduction: 10 masters. Elliptic sectors {15, 55a–e, 63} (7 masters) share the inherited Γ₁(6) sunrise curve, fibred over the box virtuality w (cusps w ∈ {0,1,9,∞}, Kodaira fibres (1,2,3,6)). Polylog sectors {11, 51, 59} (3 masters) are structurally polylogarithmic (genus-0 dlog; certified 7-letter box alphabet × sunrise thresholds {w, w−m², w−9m²}). ## Closed form (elliptic sectors, m² = 1) Every elliptic master is function-level closed as I_63^(k)(w) = psi1(w) * [ Eichler_f3(tau(w)) + c1^(k) ] + c2^(k) * psi2(w) with psi1 = (2π/√3)·varpi0 and Eichler_f3 the Eichler integral of the weight-3 Eisenstein form f₃ on Γ₁(6), evaluated by cusp-regularized Eichler-integral transport on the Γ₁(6) elliptic fibration from the anchor point w₀ = −2. Homogeneous period: varpi0(w) is the holomorphic Frobenius solution of the equal-mass-sunrise Picard–Fuchs operator (t ≡ w/m²): L_sun = t(t-1)(t-9) d²/dt² + (3t² - 20t + 9) d/dt + (t - 3), varpi0(0) = 1. Exact normalization (verified independently to 35.7–116.4 digits at 6 points): psi1(w)/π = (2/√3) · varpi0(w). ## Anchor data (w₀ = −2) psi1(w0)/pi = 7.611662731988950350519990804725388336643423169012396509202301896585102328333879433893615374442008061260141404980209230074e-01 Im psi2(w0)/pi = 2.488419648503361355719549018767424794944767838784660283928203594493117871306060772810790410402587514677797794849488866035e-01 Eichler_f3(tau(w)) at the six Euclidean anchors: w = -1/4 : 1.38817844254639186702577855195321568326744350538663866688878666591556370279045032e-01 w = -1/2 : 2.68186911702707030412935783456189579962722885432755027011623812250699845045685253e-01 w = -1 : 5.05147907400042020563474169569555481648108762841180230822831544142570000713626975e-01 w = -2 : 9.17475992194864665245709077614316962005880489156111841664359129102196314547952341e-01 w = -3 : 1.27381749916280277395698391371279828563429643030477115073279931188086876905663904e+00 w = -5 : 1.87857593822846368958413555588180240447466970720917871048665416215145295958234949e+00 ## Boundary constants (cusp ring, conductor 3) The leading boundary constant PSLQ-closes at 110.0 residual digits: B^(0) = 2·Cl₂(π/3) = 2.029883212819307250042405108549040571883378615060599584034978213553194952516488044272940708456513389891723655062719770803e+00 The sub-leading boundary constants B^(1..4) were determined numerically (120 digits) and are given by the 2F1 eps-derivative form of the sunrise boundary (the eps^k coefficients of B(eps) built from 2F1(-2eps,-eps;1-eps;e^{2 pi i/3}); lbl3e-master-evaluate.py gate G5). The PSLQ search in the conductor-3 cusp ring was run for B^(1) and B^(2) and found no relation for either (a partial ring hit); B^(3), B^(4) were not searched. The per-master (c₁, c₂) constants for the box-dressed sectors 55a–e/63 are fitted by a variation-of-parameters constants fit whose mechanics were validated by a synthetic round-trip to 116 digits. ## Validation gates (all PASS ≥ 30 digits) | gate | min digits | |---|---| | homogeneous anchor cross-check (PF Frobenius × Γ₁(6) q-series, 6 pts) | 35.7 (92.8–116.4 at 5/6 pts) | | leave-one-out PF-transport (6 anchors × 5 held-out) | 35.7 | | elliptic-top leave-one-out (Eichler ΔI, anchor −2 → 5 held-out) | 35.1 | | variation-of-parameters constants fit, synthetic round-trip | 116 | | B^(0) PSLQ residual | 110.0 | The 35.7-digit floor at w = −5 is q-series convergence at 500 terms (|q| = 0.212), not a method floor. ## Remaining caveat A from-scratch AMFlow oracle for the box-dressed sector-63 finite part is still missing. The originally proposed maximal-cut gate was shown to be ill-posed (2026-06-27): the D4 cut pins w = m² identically — a degenerate cusp fibre — so a sector-63 maximal cut cannot see the elliptic modulus or the (c₁, c₂) box-dressing constants either way. A fully independent confirmation of those constants requires the full (uncut) sector-63 solve or sub-leading cuts; the closed form above is otherwise gated as listed.