Symbolic systems operate over precise identities: variables denote specific objects, pointers target precise memory locations, and database keys refer to singular records. Neural embeddings generalize by compressing away semantic detail, but this compression creates collision ambiguity: multiple distinct entities can share the same representation value. We characterize how much additional information must be supplied to recover precise identity from such representations. The answer is controlled by a single combinatorial object: the collision-fiber geometry of the representation map $π$. Let $A_π=\max_u |π^{-1}(u)|$ be the largest collision fiber. We prove a tight fixed-length converse $L \ge \log_2 A_π$, an exact finite-block scaling law, a pointwise adaptive budget $\lceil \log_2 |π^{-1}(u)|\rceil$, and an exact fiberwise rate-distortion law for arbitrary finite sources via recoverable-mass decomposition across representation fibers. The uniform single-block formula $D^\star(L)=\max(0,1-2^L/a)$ appears as a closed-form special case when all mass lies on one collision block, where $a = A_π$ is the collision block size. The same fiber geometry determines query complexity and canonical structure for distinguishing families. Because this residual ambiguity is structural rather than representation-specific, symbolic identity mechanisms (handles, keys, pointers, nominal tags) are the necessary system-level complement to any non-injective semantic representation. All main results are machine-checked in Lean 4.

翻译：符号系统在精确身份上运行：变量指代特定对象，指针指向精确内存位置，数据库键引用唯一记录。神经嵌入通过压缩消除语义细节实现泛化，但这种压缩引发了碰撞歧义：多个不同实体可能共享相同的表示值。我们刻画了从这类表示中恢复精确身份所需补充的信息量。答案由单一组合对象控制：表示映射$π$的碰撞-纤维几何结构。设$A_π=\max_u |π^{-1}(u)|$为最大碰撞纤维。我们证明了紧的定长反向下界$L \ge \log_2 A_π$、精确的有限块标度律、逐点自适应预算$\lceil \log_2 |π^{-1}(u)|\rceil$，并通过表示纤维上的可恢复质量分解，为任意有限信源导出了精确的逐纤维率失真律。当所有质量集中在一个碰撞块上时，均匀单块公式$D^\star(L)=\max(0,1-2^L/a)$作为闭式特例出现，其中$a = A_π$为碰撞块大小。相同的纤维几何结构决定了区分族时的查询复杂度与规范结构。由于这种残余歧义是结构性的而非表示特定的，符号身份机制（句柄、键、指针、名义标签）是任何非注入式语义表示所必需的系统级补充。所有主要结果均在Lean 4中经过机器验证。