Certified p-adic Frobenius constants of the equal-mass banana tower

Candelas, de la Ossa and van Straten conjecture that the p-adic Frobenius structure constants of an order-four Calabi–Yau operator satisfy α₁ = α₂ = 0 with α₃ a rational multiple of ζₚ(3), and Beukers and Vlasenko prove this for the simplicial and hyperoctahedral families. The equal-mass banana integrals supply a tower of open cases because at ℓ loops the Newton polytope is the A_ℓ root polytope, whose facets include products of simplices. I measure the tower. At four loops, the Hulek–Verrill family, operator 34 of the Almkvist–van Enckevort–van Straten–Zudilin list (AESZ 34), I find α₁ = α₂ = 0 and α₃ ≡ −(2/3)ζ₅(3) to twelve base-5 digits, with the same rational independently at p = 7, 11 and 13, the published quintic value reproduced through the identical code path as the control. At six, eight and ten loops, at p = 5, 7, 11, 13, every certified measurement is consistent with the single closed law U = exp( Σk≥1 −(2/(2k+1)) ζₚ(2k+1) N^(2k+1) ), uniform in the loop order and the prime. The even slots are generated by the exponential, the ζₚ(3)-slot is the same p-adic number across operators, and at p = 5 the law holds through the Kubota–Leopoldt pole branch. The rationals are derived, not fitted. The four-loop ratio is −2/3 in the normalization in which the quintic reads −40, and newly computed topological data of the six-loop fivefold gives the six-loop ratios −2/3 and −2/5 classically. Every computation is exact, every reading carries its modulus, and three falsifiers run with every measurement. The four-loop row and the tower law are verified on the stated ranges and not proved.