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

Abstract

Single-Source Coherence Theorem. We prove that among all possible degrees of freedom (DOF = number of independent encoding locations), exactly one value (DOF = 1) guarantees coherence. DOF = 0 fails (no fact encoded). DOF $\geq$ 2 fails (permits explicit construction of inconsistency). Only DOF = 1 satisfies both requirements. Proof Sketch. Case analysis on $\mathbb{N}$: For DOF = 1, any two queries return the single location's value; transitivity of equality forces agreement. For DOF $\geq$ 2, construct two locations with values $v$ and $v' \neq v$; queries return different answers. This witness construction works uniformly for all DOF $\geq$ 2. By trichotomy of naturals, DOF = 1 is the unique solution. We introduce the zero-incoherence capacity: the maximum rate guaranteeing zero disagreement among replicated encodings. Main results: exact capacity ($C_0=1$), tight side-information bound ($\geq\log_2 k$ bits for $k$-way incoherence), and rate-complexity separation ($O(1)$ at capacity vs $Ω(n)$ above). Encoding locations are terminals in multi-terminal source coding. Derivation is perfect correlation reducing effective rate; only complete derivation achieves zero incoherence. We give achievability and converse proofs, formalize confusability/incoherence graphs, and present the mutual-information side-information bound.

Theorems & Definitions (136)

• Theorem 1.1: Resolution Impossibility, informal
• Theorem 1.2: Encoder Realizability, informal
• Definition 2.1: Encoding System
• Definition 2.2: Coherence
• Definition 2.3: Incoherence
• Theorem 2.4: Oracle Arbitrariness
• proof
• Definition 2.5: Degrees of Freedom
• Theorem 2.6: DOF = 1 Guarantees Coherence
• proof