Refining Hardness after ETH: Task-Relevant Distinctions and the Cost of Exact Counting
Year: 2026
Type: preprint
Abstract: The reported subexponential algorithms for satisfiability leave a strong successor question: can exact counting gain a fixed power in its exponent? We examine that question through the distinctions a task requires, the operations that resolve them jointly, and what those operations cost, with the public hypotheses #SH and Weak #ETH as the quantitative setting. After comparing the existing method-specific barriers, we give a fully formalized finite analysis of the coordinate-swap construction behind one interpolation algorithm. We construct the measurement rows it selects, prove their fixing-kernel property and their exact row count, and connect them to a uniform tradeoff among measurement, degree and enumeration. Under the stated row and dimension budgets, the complete clipped substitution enumeration has size at least 2^(n log₂log₂n/(4√log₂n)) for n ≥ 2^16, and eventually at least 2^(n^(1−ε)) for every fixed ε > 0. This bounds the enumeration as completed; an algorithm that evaluates its aggregate by other operations calls for a separate analysis. The supporting mathematical implications and the constructed-measurement application are checked in Lean 4, and the unrestricted counting hypothesis stays open. Files. refining_hardness_after_eth.pdf is the 17-page preprint. what_remains_hard_after_eth.pdf is a 4-page explanatory note. lean_proofs.zip is the Lean 4 development (Lean and Mathlib v4.28.0; 709 audited declarations; axioms limited to propext, Classical.choice and Quot.sound; builds with lake exe cache get && lake build), distributed under the MIT License. editable_source_bundle.zip contains the editable sources, both TeX source archives with the Lean sources as ancillary files, and the verification records. SUBMISSION_VERIFICATION.json records the clean-extraction rebuild of both documents. Source repository: https://github.com/rohanvinaik/refining-hardness-after-eth
Author: Vinaik, Rohan
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225455962