Bell Causality and Operational Rigidity in Quantum Sequential Growth: What noncommuting transition operators can do on reachable causal-set histories

A quantum sequential-growth dynamics should allow interference between causal-set histories while respecting the causal and relabelling principles that constrain classical growth. Recent work on causal-past-ordered Bell causality (CPOBC) allows noncommuting transition operators, raising a physical question: can this algebraic freedom act nontrivially on histories that the dynamics actually prepares? For invertible operator-valued growth satisfying the operator Markov sum rule and strict state-vector covariance, we prove a dimension-independent answer for immediate sibling births. At every reached parent with at least two elements, the gregarious operators of its children act identically on the parent's state, and all same-parent transition commutators annihilate that state. The proof uses only source-faithful spectator reductions; it assumes neither operator-level covariance nor global factorization. An all-stage dark-sector model shows why this is not a theorem of operator commutativity. The local conclusion strengthens to the full cyclic module generated by the transitions of any fixed reached parent, ruling out an irreducible transition algebra on that parent for carrier dimension at least two. Conditional reductions and three-dimensional examples delineate the surviving possibilities, including a measure-level effect at the exceptional singleton junction. To our knowledge, this on-state rigidity consequence has not previously been isolated for this CPOBC convention. We identify, but do not resolve, the question of independent non-sibling quantum-measure effects at later births. Contents of this record: the manuscript (PDF) and its LaTeX source, and a certificates bundle. The bundle holds seven standalone Python scripts that reproduce every finite-stage computation in the paper, using exact rational or algebraic-number arithmetic. It also contains logs, result files and SHA-256 checksums. Reproducibility: the scripts use only the Python standard library. Unzip the bundle, create an empty folder, and from it run: python -I ../certificates/run_all.py. All outputs are written to the current folder. Status: research preprint, not peer reviewed. Companion to 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.