Intrinsic Classification of Quantum Sequential Growth: Gram Kernels, First-Junction Realization, and the Equivalence of Bell Conventions

In quantum sequential growth of causal sets, the coherent predictions for finite histories are summarized by the Gram kernel of the reached history vectors, which fixes all finite-cylinder interference. A formally consistent kernel need not come from any complete growth dynamics, so a realization criterion decides which candidate quantum causal processes are admissible. For finite-dimensional, causal-past-ordered growth, and building on the operational descent and Bell-convention results of a preceding paper, we answer two questions. Realizability: a kernel arises from a complete direct-Bell model if and only if it satisfies five intrinsic conditions. Four of them (normalization, positivity and refinement; invertible graft graphs; commutation; Markov invertibility) reconstruct the unique minimal commutative late dynamics, and a first-junction rank condition decides whether that dynamics lifts. Sufficiency is proved by an injective-envelope dilation, which yields a full broad-Bell model on a carrier of dimension at most 2·dim(U)·rank(K), also when the relevant algebra is nonsemisimple. Consequently direct, source-faithful and broad Bell causality have identical kernel classes, despite their different full-matrix model classes. Observational completeness: apart from the empty/singleton pair, two finite causal sets are identified by every such model exactly when they have equal size and equal augmented past-profile multisets. Explicit examples show that a realizable kernel can require a carrier larger than its reached space, and that a consistent reduced kernel can fail the first-junction condition. We separate these finite-cylinder results from own-core classification, infinite-history extension and finite-prefix recognition. Contents of this record: the manuscript (PDF) and its LaTeX source, and a certificates bundle. The bundle holds twelve standalone Python programs 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 programs use only the Python standard library. Unzip the bundle and, inside its folder, run: python3 run_all.py. It verifies the checksums, runs all twelve programs and reports ALL 12 PROGRAMS PASSED. The all-stage theorems are proved in the manuscript; the programs are finite checks. Status: research preprint, not peer reviewed. Fourth paper in a series; it builds directly on T. G. Hamilton, "Which Bell causality? Operational abelianization and a Rideout–Sorkin theorem for quantum sequential growth", Zenodo (2026), doi:10.5281/zenodo.24083829, which in turn follows doi:10.5281/zenodo.24006143 and doi:10.5281/zenodo.23953788.