ACES: Axiomatic Class–Equivalence Sharding for Deterministic and Adaptive Distributed Computation

We present ACES (Axiomatic Class–Equivalence Sharding), a mathematically-grounded distributed algorithm that achieves deterministic, adaptive, and locality-preserving workload partitioning. ACES formalizes three foundational principles: (1) axiomatic subgroups derived from class invariants, (2) equivalence-based partitioning through canonical hashing, and (3) similarity-driven sub-sharding regulated by a continuous threshold parameter τ. The resulting system guarantees deterministic routing, monotonic refinement under threshold tuning, and stability through a PID-based auto-controller. We provide rigorous proofs of partition consistency, order-independence, monotone refinement properties, and stable rebalancing guarantees. This framework provides a new approach to provably-consistent and self-balanced distributed computation without centralized coordination, with applications in encrypted databases and privacy-preserving identity systems.