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.
Chihiro Yokota (2026) studied this question.