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.

Residual ambiguity depends only on the fiber structure of the representation. Symbolic identity mechanisms, including handles, keys, pointers, and nominal tags, supply the system-level complement required by any non-injective semantic representation. All main results are machine-checked in Lean 4.