The Optimizer Quotient and the Certification Trilemma
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915903967330304 |
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| author | Simas, Tristan |
| author_facet | Simas, Tristan |
| contents | The optimizer quotient is the canonical object for exact decision-relevant information: it is the coarsest exact decision-preserving abstraction (Theorem 2.15). This paper proves that exact certification of this object's coordinate structure is subject to an impossibility trilemma: under $\mathrm{P} \neq \mathrm{coNP}$, no certifier can be simultaneously sound, complete on all in-scope instances, and polynomial-budgeted (Theorem 7.1). The cost of this impossibility varies by regime: coNP (static), PP-hard (stochastic decisiveness), PSPACE-complete (sequential). Six structural restrictions collapse certification to polynomial time. The finite reduction and verification core is mechanized in Lean 4. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14689 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Optimizer Quotient and the Certification Trilemma Simas, Tristan Computational Complexity 68Q15, 68Q30, 03D15 F.2.2; I.2.7 The optimizer quotient is the canonical object for exact decision-relevant information: it is the coarsest exact decision-preserving abstraction (Theorem 2.15). This paper proves that exact certification of this object's coordinate structure is subject to an impossibility trilemma: under $\mathrm{P} \neq \mathrm{coNP}$, no certifier can be simultaneously sound, complete on all in-scope instances, and polynomial-budgeted (Theorem 7.1). The cost of this impossibility varies by regime: coNP (static), PP-hard (stochastic decisiveness), PSPACE-complete (sequential). Six structural restrictions collapse certification to polynomial time. The finite reduction and verification core is mechanized in Lean 4. |
| title | The Optimizer Quotient and the Certification Trilemma |
| topic | Computational Complexity 68Q15, 68Q30, 03D15 F.2.2; I.2.7 |
| url | https://arxiv.org/abs/2603.14689 |