Semantic Identity Compression: Zero-Error Laws, Rate-Distortion, and Neurosymbolic Necessity

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. Exact identity recovery requires additional information precisely when representation fibers have size greater than one.

The residual cost is controlled by a single combinatorial object: the collision-fiber geometry of the representation map π. Let A_π = maxᵤ|π⁻¹(u)| be the largest collision fiber. The finite laws include a tight fixed-length converse L ≥ log₂A_π, an exact finite-block scaling law, a pointwise adaptive budget ⌈log₂|π⁻¹(u)|⌉, and an exact fiberwise rate-distortion law for arbitrary finite sources via recoverable-mass decomposition across representation fibers. The uniform single-block formula D^⋆(L) = max (0, 1 − 2ᴸ/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.