Curved-Volume Quantization, Ordering Residuals, and Visibility Scaling in SU (2) ₖ Quantum 6j Geometry

This preprint synthesizes the BC-Spec L1-P04-P/N7–N25 research cycle into a single certification framework for finite-level quantum-geometric operators associated with SU (2) kSU (2) ₖSU (2) k quantum 6j6j6j data. The article first establishes an exact holonomy-to-volume reconstruction pipeline for curved tetrahedral geometry and proves that a single local trace-cubic scalar is insufficient to recover exact spherical volume. It then develops a skein- and character-compatible operator bridge, formulates ordering ambiguity as metric projection onto a registered positive cone, and introduces an operator-adapted residual-frame inequality that controls Hilbert–Schmidt error through lower-symbol data on a declared residual subspace. The numerical part is organized around sealed and prospective validation. It tests level-dependent visibility bounds, regular-variation forecasts, Richardson extrapolation, prospectively registered failure sets, matched-shape absolute-ray-scale effects, and a two-parameter resolution–deformation audit. The latter yields an identifiability firewall: within the registered parametrization, quantum-group deformation, representation scale, and normalized geometry cannot be varied independently. The main contribution is methodological rather than phenomenological. The work converts operator-ordering ambiguity and finite-level scaling analysis into an explicit, reproducible, and falsifiable certification problem. Exact theorems, finite-dimensional certificates, sealed validations, prospective predictions, post-hoc diagnostics, and deferred hypotheses are kept strictly separated. No claim is made to derive physical gravity, spacetime emergence, matter, a cosmological constant, or a universal continuum exponent. The release includes the manuscript, LaTeX sources, figures, metadata, checksums, module registry, and a reproducibility archive containing the N7–N25 computational materials.