Which Bell Causality? Operational Abelianization and a Rideout–Sorkin Theorem for Quantum Sequential Growth

Classical Rideout–Sorkin growth is constrained by Bell-type spectator independence, whose equivalent scalar formulations need not remain equivalent for noncommuting quantum transitions. We ask which comparison domains matter for the finite-cylinder predictions of causal-past-ordered quantum sequential growth. Four conventions are distinguished: own-core, direct, source-faithful, and broad. With invertible finite-dimensional transitions, an operator Markov sum rule, and strict covariance of reached vectors, direct Bell implies a parent-independent precursor factorization and a faithful commutative algebra on the reached module. This yields an algebra-valued Rideout–Sorkin reconstruction with a single exceptional junction correction. Although the four conventions have strictly different full-matrix solution classes, the operator defects in three separating examples are invisible on reached states. By contrast, an explicit all-stage rank-two own-core model violates a direct-Bell graft constraint at the level of its Gram kernel, establishing a genuine predictive separation. An all-stage broad-Bell example and a general visibility criterion show that the junction correction can also affect finite-cylinder interference. These results distinguish full operator identities from reached dynamics and coherent finite-cylinder data; they do not claim an extension to covariant infinite-history events. Contents of this record: the manuscript (PDF) and its LaTeX source, and a certificates bundle. The bundle holds ten standalone Python scripts that check every finite computation cited in the paper using exact integer and rational arithmetic, together with their result files and SHA-256 checksums. Reproducibility: the scripts use only the Python standard library. Unzip the bundle and, inside its folder, run: python3 run_all.py. It verifies the checksums, runs all ten certificates and reports ALL PASS. The all-stage theorems are proved in the manuscript; the scripts are finite checks. Status: research preprint, not peer reviewed. Third paper in a series, following T. G. Hamilton, "Noncommutative quantum sequential growth under state-level covariance: existence, fixed-seed rigidity, and geometric limits", Zenodo (2026), doi:10.5281/zenodo.23953788, and "Bell causality and operational rigidity in quantum sequential growth", Zenodo (2026), doi:10.5281/zenodo.24006143.----