With OpenAI Codex, Doug Colkitt tightens OpenAI’s sub-n log n integer multiplication exponent from 2^-182 to 2^-59 (conditional)

OpenAI’s 6 Oct release (see OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems) included family #109, a manuscript claiming that two n-bit integers can be multiplied in time O(n (log n)^(1−κ)) with κ = 2^-182. That would beat the O(n log n) bound of Harvey and van der Hoeven (2019), widely believed to be optimal, though only by a tiny power of log n.

Doug Colkitt (X @0xdoug; X bio: founder of SLX, founding contributor to Fogo, former quant) called integer multiplication “probably the most shocking result” of the release at 00:35 UTC on 7 Oct. Under two hours later he pushed github.com/CrocSwap/integer-mult-bounds, “Research draft by Douglas Colkitt — conditional on the underlying manuscript”. It keeps OpenAI’s files unchanged under upstream/ and ships each improvement as a patch, with Python certificates for the arithmetic:

• 02:22 UTC: tuned parameters and a smaller finite network (h = 46) give κ = 5.8×10^-33, “a nearly 2^75 fold increase”. He also proved a ceiling of κ < 5.838×10^-33 for that network family.
• 13:35–13:48 UTC: direct nonadjacent axis swaps, which the manuscript already allowed, route around the cubic bottleneck behind that ceiling: κ = 2^-78.
• 19:27 UTC: paired-block circuits that share intermediate sums and scratch space, an exact early-stopping guard width and a smaller Gaussian width: κ = 2^-59 (minimum margin about 0.4%). The README calls this “a 2^123-fold increase in exponent saving over the original”.

The README states the limits plainly: the upstream theorem “remains an assumption”, the scripts “do not constitute a formal proof of the complete algorithm”, and there is no Lean formalization. Disclosure, verbatim: “The research, implementation, and drafting were performed with assistance from OpenAI Codex. This assistance is not independent review or endorsement by OpenAI. No priority claim is made.”

On X, @bubbleboi’s quote post (~25k views) said: “This guy has 6.1 Astra running in a loop and is breaking the record for integer multiplication algorithms every few hours lmaooooo.” The model attribution is bubbleboi’s claim; the repository credits OpenAI Codex and names no model. We found no press or Hacker News coverage by the evening of 7 Oct.

It is an early example of what the release unlocked: outsiders with AI coding agents building on AI-written manuscripts within hours. The improvements are only as good as OpenAI’s unverified theorem, and a smaller κ changes nothing in practice. Still, finding a 2^123-fold larger exponent saving in a day suggests the original constants were far from tight, and that AI agents can now do this kind of parameter and construction search quickly.