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.

翻译：单源相干定理。我们证明在所有可能的自由度（DOF = 独立编码位置数量）中，恰好只有一个值（DOF = 1）能保证相干性。DOF = 0 失败（未编码任何事实）。DOF ≥ 2 失败（允许显式构造不一致性）。仅 DOF = 1 同时满足两项要求。证明概要。对自然数进行情形分析：当 DOF = 1 时，任意两次查询均返回该唯一位置的值；等式的传递性迫使结果一致。当 DOF ≥ 2 时，构造两个分别取值 v 与 v' ≠ v 的位置；查询将返回不同答案。该见证构造对所有 DOF ≥ 2 情形均一致成立。根据自然数的三分性，DOF = 1 是唯一解。我们引入零非相干容量：指在复制编码中保证零分歧的最大速率。主要成果包括：精确容量（C₀=1）、紧致边信息界（k 路非相干需 ≥ log₂ k 比特）、以及速率-复杂度分离（容量处为 O(1)，超容量时为 Ω(n)）。编码位置即多端信源编码中的终端。推导过程通过完全相关性降低有效速率；仅当推导完全时才能实现零非相干。我们给出了可达性证明与逆定理证明，形式化了混淆/非相干图，并提出了基于互信息的边信息界。