Cyclotomic CM Normalization and Schwarz–Abel Boundary Compatibility for the Centered Jacobi Theta Section

This preprint studies the arithmetic and analytic normalization of the centered Jacobi theta section arising from Romik’s hypergeometric representation. At finite primes, the quartic hypergeometric Frobenius structure is analyzed through its complete Frobenius orbit. The resulting normalization is expressed in terms of quartic Gauss and Jacobi data, and the absolute and relative factors are organized by a pair of cyclotomic divisors. The same divisor data also recover the classical archimedean constants governing the theta normalization and the Schwarz coordinate. A key point is that the finite-prime Frobenius comparison and the archimedean boundary analysis require different coordinates. The paper separates these two settings explicitly: a rational Liouville coordinate is used for the finite-prime Frobenius comparison, while the actual Schwarz coordinate determined by Romik’s hypergeometric pair is used for the theta section and Abel regularization. This removes an apparent scalar ambiguity and gives a precise compatibility between the arithmetic normalization and the centered analytic model. The Schwarz coordinate also provides an intrinsic realization of the Abel regulator. Under this parametrization, radial Abel regularization acts directly on the centered Jacobi theta section. After subtraction of the two universal boundary modes, the resulting section extends smoothly to the boundary and is identified with the previously constructed Abel-renormalized continuous dual Hahn regulator flow. Combining these results with the companion continuous dual Hahn construction yields an exact connection between the normalized centered theta section and the gamma-normalized finite part associated with the Riemann zeta function. The paper does not claim a proof of the Riemann hypothesis; its main contribution is the explicit compatibility of the quartic CM normalization, the Schwarz coordinate, and the Abel-renormalized boundary construction.