Trace-Class Depth Refinement and Finite S(U(3)×U(2)) BV Pushforward: Conditional Ultraviolet Universality Criteria in the Holographic Quantum Entropic Gravity Programme

We establish a mathematically controlled ultraviolet subsector of the Holographic Quantum Entropic Gravity (HQEG) programme, integrating discrete-to-continuum spectral analysis, finite-dimensional Batalin–Vilkovisky (BV) quantization, and nonperturbative boundary-field methods. For a compact warped depth interval, we prove Mosco convergence of canonical weighted discrete path forms to a continuum Sturm–Liouville operator, together with strong-resolvent and compact spectral convergence. An exact edge-to-path domination inequality provides computable relative-form bounds for additional microscopic graph interactions. The finite gauge fiber is constructed using S(U(3)×U(2)), with its twelve-dimensional unimodular Lie algebra. Under explicit invariance and regularity hypotheses, we establish finite-dimensional BV quantum master equation consistency and derive the corresponding quadratic determinant and boundary Schur-complement pushforward, including arbitrary one-dimensional adjoint holonomy. A central result identifies the discrete fixed-boundary Green covariance exactly with a Brownian-bridge covariance sampled in resistance coordinates. Using the appropriate piecewise-linear embedding, we establish trace-norm convergence and nonperturbative continuum limits for Gaussian-normalized Euclidean boundary amplitudes with continuous, bounded-below local interactions. For strictly vanishing boundary data and nonnegative homogeneous local quartic interactions, a controlled contour deformation additionally yields a normalized Lorentzian depth-amplitude limit. We formulate an explicit sufficient condition for microscopic washed-ladder refinement universality, h(A_m + D_m W_m r_m) → 0,* where the quantities characterize conductance mismatch, path congestion, edge weights, and refinement spans. This reduces the proposed Chain Theory–HQEG correspondence to a quantitatively testable scaling problem. However, the required theorem-level serial refinement data remain unavailable in the public Chain Theory construction; statements 90.2, 94.1, and 117.1 retain unresolved conditional or open status. The results establish rigorous control within the specified compact-depth and finite-gauge subsectors, but do not constitute complete HQEG ultraviolet completion, a continuum BV quantum master equation, a four-dimensional Yang–Mills mass-gap proof, or a construction of the full gravitational functional integral. Publication status: This v1.1 manuscript is the first public release, incorporating corrections and refinements to an unpublished internal v1.0 draft. The mathematical claims remain subject to independent peer review.