The klam value revisited: finite hardness after galactic algorithms
Year: 2026
Type: article
Abstract: Two preprints of October 2026, by OpenAI and by Anthropic, claim to refute the Exponential Time Hypothesis with galactic algorithms, which are exhaustive search on every input that fits into a computer. We show that a hardness hypothesis that no galactic algorithm can refute has to be finite, a bound at specified input sizes with explicit constants. We state one for sparse 3-SAT (2^(0.1n) gates for 143 ≤ n ≤ 2^40) and one for k×k Clique (k^(k/2) gates for 8 ≤ k ≤ 4096), and we compare them with algorithms and proved lower bounds at the budget 10^20, using the klam value of Downey, Fellows, and Stege. Through the standard reduction, a finite 3-SAT hypothesis gives no bound above the input length for k×k Clique below k = 85, even with the best known exponent, so we state the Clique hypothesis directly. Lower bounds for single tensor networks and for tree-like resolution exceed 10^20 for k ≥ 22 and k ≥ 29, and we know no way to apply either to the galactic algorithms. The tree-like bound is checked in Lean 4. Files: finiteness_note.pdf (the note, 11 pages) and klam_code.zip (scripts, data, the Lean project, and klam_numbers.py, which recomputes the numbers of the note).
Author: Szeider, Stefan
Institution: TU Wien
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225406391