Holomorphic Rodrigues-Ray-Singer Envelopes for Kahler Four- and Six-Manifolds

We develop a holomorphic analogue of the Rodrigues-Ray-Singer torsion envelope in real dimensions four and six. The de Rham complex of the odd-dimensional theory is replaced by the Dolbeault complex of a Hermitian holomorphic vector bundle over a compact Kahler manifold, and the Hodge spectral gap is replaced by a uniform Dolbeault spectral gap. Under bounded Kahler geometry of finite order and uniform bounds on the Chern curvature of the bundle, we prove a non-asymptotic linear-volume estimate for the holomorphic Ray-Singer torsion: |log T_hol(X,E,omega,h)| <= B_hol Vol(X,omega). The proof follows the heat-kernel and Mellin architecture of the odd-dimensional Rodrigues-Ray-Singer envelope, but the even-dimensional setting introduces a critical local heat coefficient producing a logarithmic Mellin contribution. We isolate and control this term uniformly. The resulting estimate gives a bounded-geometry control of Quillen determinant norms in Kahler four- and six-manifolds, with applications to Calabi-Yau geometry, HSFG-type geometric flows, and BCOV-type determinant contributions in mathematical physics. More precisely, the novelty of the present paper is not the definition of holomorphic analytic torsion itself, which belongs to the Ray-Singer-Quillen-Bismut-Gillet-Soule framework, but the construction of a uniform bounded-geometry envelope for it in real dimensions four and six. The estimate is non-asymptotic, depends only on finite-order Kahler bounded geometry, Chern-curvature control, and a Dolbeault spectral gap, and yields a linear-volume control of the holomorphic torsion and Quillen norm along geometric families and flows.