Weighted equal-norm K3 surfaces: multiquadratic reconstruction, a Nikulin tower, and a level-2 product-Kummer moduli cover

We study the two-parameter family of K3 surfaces attached to the weighted equal-norm equations: the double covers of the product of two projective lines branched along two explicit (2,2)-curves. Uniformly in the weight, the surface recovers the equal-norm surface through a degree-eight multiquadratic cover and is the Nikulin quotient of a K3 surface with a symplectic involution. At the generic weight we determine the Néron–Severi and transcendental lattices, U⊕ E_8(-1)⊕ D_4(-1)^2 and U(2)^2, and show that the generic member is the Kummer surface of an explicit pair of non-isogenous Legendre curves, matched to the Kuwata–Shioda fibration. The squared weights form a level-two cover of the moduli of unordered pairs of elliptic curves; the modular diagonal and the level-two Hecke divisor are determined explicitly in these coordinates, with Picard number nineteen at their generic points. Preprint (submitted version, review round r31). Companion files: the proof dossier and the Lean blueprint. The verification artifact (certificates, claim ledgers, checkers, Lean 4 cores) is the separate Software record linked below and released as GitHub release v1.0.0-submission of hirokifukui/equalnorm-paper.