Holomorphic Rodrigues–ray–singer Envelopes on Compact Kähler Manifolds: The General-Dimensional Theory and Adelic Connections

This paper develops the general-dimensional holomorphic Rodrigues–Ray–Singer envelope on compact Kähler manifolds. The aim is not to replace the four- and six-dimensional theory, but to isolate the dimension-dependent mechanism behind holomorphic analytic torsion in arbitrary complex dimension. The main result gives a non-asymptotic linear-volume control of the holomorphic Ray–Singer torsion under finite-order bounded Kähler–Chern geometry and a reduced Dolbeault spectral gap. The proof is organized around three dimension-dependent mechanisms. The first is the critical Seeley–DeWitt coefficient in degree equal to the complex dimension, which produces the logarithmic Mellin anomaly. The second is the dimensional tail profile, expressed through the upper incomplete Gamma function at negative integers. The third is the long-time spectral decay governed by the reduced Dolbeault gap. Together these mechanisms define an effective dimension-dependent envelope constant. The paper also introduces the holomorphic anomaly density associated with the critical heat coefficient and separates the envelope constant into local, anomalous, and long-time spectral contributions. Applications are given to strict Calabi–Yau fourfolds, irreducible holomorphic symplectic manifolds, higher-dimensional Calabi–Yau towers, higher BCOV-type torsions, and spectral confinement in bounded Kähler chambers. The final section formulates the archimedean analytic regulator for compact Hermitian locally symmetric varieties and proves its linear-volume control, while keeping finite-prime arithmetic factors outside the scope of the present paper.