OpenAI Problem #109 Tightened Again: κ Reaches 2⁻³⁴, a 48M-fold Improvement
RexDouglass · x · 2026-10-08
Researchers published another tightening of OpenAI problem #109 (integer multiplication):
- The new bound is κ = 2⁻³⁴ with an exact witness of 8.3 × 10⁻¹¹ — roughly a 48-million-fold improvement over their previous result (κ = 2⁻⁵⁹) and a 2¹⁴⁸-fold improvement over OpenAI's original κ = 2⁻¹⁸².
- The latest gain came from removing the spacing penalty behind the quadratic bottleneck, by moving compact control bits instead of entire windows.
- The prior round achieved κ = 2⁻⁵⁹ by redesigning the finite network to share intermediate computations and scratch space, then tightening the recursion and Gaussian estimates.
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