Formal Foundations for the Single Source of Truth Principle

Any system encoding facts faces a fundamental constraint: when multiple independent locations encode the same fact, truth becomes indeterminate. We prove that DOF = 1 (Single Source of Truth) is the unique representation guaranteeing coherence, the impossibility of disagreement among encodings.

The Core Theorem (Oracle Arbitrariness). In any incoherent encoding system (DOF > 1 with divergent values), no resolution is principled: for ANY oracle claiming to identify the "true" value, there exists an equally-present value that disagrees. This is not about inconvenience. It is about the determinacy of truth.

Software as Instance. Programming languages instantiate this epistemic structure:

Encoding systems → Codebases
Facts → Structural specifications (class existence, method signatures)
Coherence → Consistency across encoding locations
DOF = 1 → Single Source of Truth (DRY principle)
We prove that achieving DOF = 1 for structural facts requires specific language features: definition-time hooks AND introspectable derivation. Most mainstream languages (Java, C++, Rust, Go, TypeScript, etc.) lack these features and cannot achieve coherence for structural facts regardless of programmer effort.

Four Theorems:

Coherence Forcing: DOF = 1 is the unique value guaranteeing coherence. DOF = 0 means the fact is unrepresented; DOF > 1 permits incoherent states.
Oracle Arbitrariness: Under incoherence, any resolution is arbitrary, no oracle is justified by the encodings alone.
Language Requirements: For structural facts in software, DOF = 1 requires definition-time hooks AND introspection. These are logically forced.
Strict Dominance: The coherence restoration complexity gap is unbounded: O(1) vs Ω(n).
Theoretical Foundation. The derivation theory (independence, derivability, axis collapse) is established in previous work (https://zenodo.org/records/18123532). This paper proves the coherence consequences and instantiates them to programming languages.

All theorems machine-checked in Lean 4 (2,104 lines, 119 theorems). Practical demonstration via OpenHCS PR #44: migration from 47 scattered checks to 1 ABC (DOF 47 → 1).