Zero-Incoherence Capacity of Interactive Encoding Systems: Achievability, Converse, and Side Information Bounds

We introduce the zero-incoherence capacity for interactive multi-location encoding systems: the maximum encoding rate that guarantees exactly zero probability of disagreement among replicated encodings. Our main information-theoretic results are compact and self-contained: an exact capacity theorem (C₀ = 1), a tight side-information lower bound for resolution ( ≥ log₂k bits for k-way incoherence), and a rate–complexity separation (modification cost O(1) at capacity vs Ω(n) above).

The paper frames encoding locations as terminals in a multi-terminal source-coding model. Derivation (automatic deterministic dependence) is interpreted as perfect correlation that reduces effective rate; only complete derivation (one independent source) achieves zero incoherence. We give concise achievability and converse proofs in IT style, formalize the confusability/incoherence graph connection, and present an explicit mutual-information argument for the side-information bound.

Theoretical contributions are supplemented by constructive instantiations (programming-language patterns and a software case study). Detailed language evaluation, extended code examples, and the full Lean proof corpus are provided in the supplementary material; the main text contains concise instantiations. Core theorems (capacity, realizability, bounds) are machine-checked in Lean 4 ; entropy arguments apply standard Fano-inequality techniques.

Lean 4 artifact: 3469 lines, 185 theorems/lemmas across 25 files (0 sorry placeholders).