REWA: Witness-Overlap Theory -- Foundations for Composable Binary Similarity Systems

REWA introduces a general theory of similarity based on witness-overlap structures. We show that whenever similarity between concepts can be expressed as monotone witness overlap -- whether arising from graph neighborhoods, causal relations, temporal structure, topological features, symbolic patterns, or embedding-based neighborhoods -- it admits a reduction to compact encodings with provable ranking preservation guarantees. REWA systems consist of: (1) finite witness sets $W(v)$, (2) semi-random bit assignments generated from each witness, and (3) monotonicity of expected similarity in the overlap $Δ(u, v) = |W(u) \cap W(v)|$. We prove that under an overlap-gap condition on the final witness sets -- independent of how they were constructed -- top-$k$ rankings are preserved using $m = O(\log(|V|/δ))$ bits. The witness-set formulation is compositional: any sequence of structural, temporal, causal, topological, information-theoretic, or learned transformations can be combined into pipelines that terminate in discrete witness sets. The theory applies to the final witness overlap, enabling modular construction of similarity systems from reusable primitives. This yields a vast design space: millions of composable similarity definitions inherit logarithmic encoding complexity. REWA subsumes and unifies Bloom filters, minhash, LSH bitmaps, random projections, sketches, and hierarchical filters as special cases. It provides a principled foundation for similarity systems whose behavior is governed by witness overlap rather than hash-function engineering. This manuscript presents the axioms, the main reducibility theorem, complete proofs with explicit constants, and a detailed discussion of compositional design, limitations, and future extensions including multi-bit encodings, weighted witnesses, and non-set representations.