Associative finite-capacity aggregation: classification, optimal quantization, and parallel balancing

This preprint studies associative preprocessing on the finite capped-addition monoid LN={0,…,N}L_N=\{0,…,N\}LN={0,…,N}. It classifies every normalized map ΦΦΦ for which x⋆Φy=min⁡{N,x+Φ(y)}x_Φ y=min\{N,x+Φ(y)\}x⋆Φy=min{N,x+Φ(y)} is associative, using a unique residue-seed representation, and derives the exact finite census. The order-preserving subclass consists precisely of cap-admissible threshold quantizers. The paper establishes sharp worst-case error formulas, an exact communication–distortion frontier, and minimax-optimal scalar quantizers. It proves exact hierarchical mergeability for scalar and coordinatewise finite summaries across arbitrary partitions and reduction trees. The nonzero contractive subclass consists exactly of divisor-step floor quantizers. These quantizers generate feasible capacity-constrained balancing processes on parallel graph matchings, with exact conservation, Lyapunov descent, finite transfer and scheduling bounds, an exact stopping region, and pathwise and spectral terminal-imbalance estimates. The resulting formulas provide direct communication–accuracy–execution design rules. The framework terms Axiom L, Identity–Number Algebra (INA), and Causal Identity Graph (CIG) are retained as mathematical provenance and add no additional assumptions to the stated results. This is the author preprint, version 1.0, prepared for submission to Applied Mathematics Letters.