# Three-loop equal-mass banana (K3) — closed form at ε⁰ Top master m₁ = I₁₁₁₁ in d = 2 − 2ε, m² = 1, one ratio t = p²/m², MUM disk |t| < 4 (Euclidean −7/2 ≤ t < 0 and subthreshold 0 < t ≤ 7/2). ## The closed form m1_eps0(t) = 7 ζ₃ · ϖ₀(t) − Part_reg(t) All four word coefficients in the MUM basis {ϖ₀, ϖ₁ᴿ, ϖ₂ᴿ, Part_reg} are exact: (c₀, c₁, c₂, c_P) = (7ζ₃, 0, 0, −1), PSLQ heights (7, 0, 0, 1), confirmed at 107.9 / 108.4 / 109.1 / 108.7 digits respectively. ## The two pieces (both pure power series, toric variable x = t/64) **ϖ₀** — holomorphic K3 period = Domb series (OEIS A002895): ϖ₀(x) = Σ_{n≥0} D_n x^n = 1 + 4x + 28x² + 256x³ + 2716x⁴ + … (n+1)³ D_{n+1} = 2(2n+1)(5n² + 5n + 2) D_n − 64 n³ D_{n−1}, D₀ = 1. **Part_reg** — the unique MUM-regularized holomorphic particular solution of the same order-3 Picard–Fuchs (Domb) operator with toric source 24x exactly (physical source S_phys = 24/(t²(t−4)(t−16)) at ε⁰, sub-banana boundary m₀^{ε⁻³} = −1): Part_reg(t) = Σ_{k≥1} b_k (t/64)^k, b₁ = 24, n³ b_n = P(n) b_{n−1} − 64 (n−1)³ b_{n−2} where P(n) is the Domb recursion polynomial shifted n → n−1, i.e. P(n) = 2(2n−1)(5n² − 5n + 2). Series head: b = 0, 24, 216, 18944/9, 204440/9, 297757216/1125, 407026816/125, 1786176726016/42875, … No log resonance: the source starts at x¹, so Part_reg is a plain power series. ## Boundary / arithmetic data - 7ζ₃ = Broadhurst's d = 2 three-loop banana value I₁₁₁₁(d=2, p²=0), recognized by integer-relation search at 107.9 digits. NOTE: the literature (PWW 2207.12893) C_{2,3} = 4ζ₃/3 belongs to the UT-canonical I₂ = ε³(π²/ϖ₀)m₁ — a different normalization, not a discrepancy. - K3 = Sym²(sunrise Γ₁(6) curve), Picard rank 19; transcendental motive = weight-3 level-15 CM newform f₃ (LMFDB 15.3.d.a, CM by ℚ(√−15)). - Chowla–Selberg period: P_num = Γ(1/15)Γ(2/15)Γ(4/15)Γ(8/15); L(f₃,2) = P_num/(π·8·3^{3/2}·5) = 0.88045982535822981044968910894132568513898932… (AFE vs closed form: 70.0 d). K3 max-cut CM period = (P_num)² up to an algebraic·π normalization not yet pinned (K3_EICHLER_LIFT.md); the banana's own deeper-layer constant has not been matched against it. ## Verification - Closed form vs 18 held-out points from an independent AMFlow evaluation (t = −7/2, −3 and t = 1/8 … 7/2): **min 110.4 digits**, never used in the fit. - Discovery gate: leave-one-out cross-validation min 67.13 d over 13 in-disk points; negative control (monomial-only, same dim) floors at 5.94 d (61 d gap); positive control (ϖ₀ = sunrise ψ₁², Domb identity) residual 1.8e−71. - DE-transport cross-check: 89–110 d, two-boundary 110.6 d. ## Higher ε orders ε⁰ is the closed layer. Higher ε orders enter the remaining Γ₁(6) letters (f₂ₐ, f₂ᵦ, f₄ᵦ, f₆) through the same Dyson chain (depth ≥ 4 words) and are covered numerically by DE-transport of the full four-master system in ε-form (with the corrected m₂ row), verified at 87.7–89.3 digits against independent held-out points.