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

We prove that no fixed semantic representation, however sophisticated, can achieve zero-error identity recovery without explicit auxiliary information. Any non-injective representation creates an irreducible identity debt: the worst-case auxiliary description requires at least log₂A_π bits, where A_π is the maximum collision-fiber size. If those bits are withheld, the distortion floor on the worst fiber is at least 1 − 1/A_π.

This paper shows that a single structural object, the collision fiber geometry, governs all operational costs for exact identity: the bit cost (fixed-length and adaptive), the distortion floor, and the query complexity for operational disambiguation. On the query side, the paper proves that complete derivability-aware query systems reduce to orthogonal minimal cores, so the query cost is structural rather than presentation-dependent. These are alternate currencies for the same residual ambiguity; none yields a free way to restore identity.

All main results are machine-checked in Lean 4. The formalization both verifies correctness and demonstrates that the fiber-geometry framework is conceptually clean enough for full mechanization.