Arithmetic theta products and doubling at arbitrary level

We prove an arithmetic doubling identity for the standard arithmetic theta series on X₀(N) at arbitrary positive level, with constant $ψ(N)/24$ for the fixed genus-two Eisenstein series. The product comparison covers all pairs of integral indices. At rank zero, an explicit Epstein zeta correction agrees with the middle Bruhat orbit of the Eisenstein derivative. The common-component calculation combines integral CM adjunction, the cyclic difference divisor, and compact Green-current identities. It retains the different, conductor, internal contacts, and proportionality contacts at every prime, including dyadic and supersingular places. Finite intersections, curvature, and discriminant heights identify the nonpositive coefficients with an explicit modular completion. Uniform estimates for parameter derivatives give strong Fourier convergence. Green-energy estimates justify the double Petersson pairing. The full cusp lift exists as a Bochner integral and equals its Mordell--Weil projection. For Z/(2N) Z, explicit prime-power recursions determine the complete zero-component trace spectrum, including the dyadic factor. They give the scalar-to-vector Petersson norm correction and an isometric normalization on the active subspace. The Bruinier--Yang theorem then yields the scalar height formula on the specified Jacobi newform line. We also prove a double Rankin--Selberg criterion based on positive diagonal coefficients and record the paired analytic factorization with its ramified local integrals retained explicitly.