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

When a semantic representation maps distinct identities to the same value, exact recovery requires auxiliary information. For a finite class set C and representation map π, the largest collision fiber A_π = maxᵤ|π⁻¹(u)| governs that cost. Because U = π(C) is determined by the encoder's input, this is the encoder-informed branch of Witsenhausen's zero-error side-information problem. On the induced cluster graph, the classical fixed-length law gives the one-shot budget ⌈log₂A_π⌉ and exact finite-block scaling; conditional prefix coding gives the pointwise budget ⌈log₂|π⁻¹(u)|⌉.

Beyond these classical specializations, optimal recoverable mass decomposes exactly across fibers for arbitrary finite sources. A fixed family of binary queries has minimum size d ≥ ⌈log₂A_π⌉, while adaptive depth can be strictly smaller. Inclusion-minimal distinguishing families are the bases of a matroid exactly when distinguishing is the corresponding spanning predicate; orthogonality alone fails by a finite counterexample.

Collision-freedom is unstable under open-world extension, and barrier-freedom is uncomputable for arbitrary future-generating code. Bits, queries, and distortion are three currencies for the same residual ambiguity; symbolic handles supply the missing identity information. Lean 4 checks the cited finite coding laws, query criteria, counterexamples, and reduction cores.