From Tate Uniformization to Rademacher Growth: Weight -12, Boundary Normalization, and a Holographic Dictionary

Abstract: We study a weight--12 modular/arithmetic model centered on the Tate curve, the discriminant, and the exact Rademacher expansion of 1/. On the Tate side, we formulate q^ Z-periodization through a restricted operator q, emphasize that two-sided periodization is not convergent for general analytic input, and isolate a concrete class of kernels for which stagewise convergence, explicit Neumann inverses, and Newton--Hensel local inversion hold on affinoid domains avoiding the poles. This yields a controlled Tate-normalized local uniformization package together with a conditional comparison between the pullback of the invariant differential and the unit-derivative criterion. On the modular side, we recall that the same normalization forces the appearance of the weight-12 cusp form, and we use the exact Rademacher formula for 1/ to obtain the dominant Hardy--Ramanujan asymptotic \ c (n) 1 2 (n+1) ^-27/4e^4+1. \ We then interpret the Bessel kernel and Kloosterman phases as a structural ``holographic'' dictionary between boundary data and bulk growth. The point is not to claim a literal AdS₃/CFT₂ duality, but to exhibit a mathematically precise weight--12 model in which Tate uniformization, boundary normalization, exact coefficient reconstruction, and residual torsion/phase data fit into a common framework. Contact & Feedback: This upload is a research preprint and part of an ongoing independent research program. Comments, corrections, questions, and discussions are highly welcome. As I pursue this work independently alongside my regular professional commitments, my replies may take some time and are typically sent during weekends or holidays. Thank you for your understanding.