Semantic Identity Compression: Zero-Error Laws, Rate-Distortion, and Neurosymbolic Necessity
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914522781974528 |
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| author | Simas, Tristan |
| author_facet | Simas, Tristan |
| contents | 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 $A_π=\max_u |π^{-1}(u)|$ be the largest collision fiber. The finite laws include 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14252 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semantic Identity Compression: Zero-Error Laws, Rate-Distortion, and Neurosymbolic Necessity Simas, Tristan Information Theory Programming Languages 94A15, 94A24, 05B35 E.4; G.2.1 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 $A_π=\max_u |π^{-1}(u)|$ be the largest collision fiber. The finite laws include 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. |
| title | Semantic Identity Compression: Zero-Error Laws, Rate-Distortion, and Neurosymbolic Necessity |
| topic | Information Theory Programming Languages 94A15, 94A24, 05B35 E.4; G.2.1 |
| url | https://arxiv.org/abs/2601.14252 |