---
id: "2026-10-07-colkitt-codex-integer-multiplication-exponent"
url: "https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/"
as_of: "2026-10-07T23:43:00+02:00"
date: "2026-10-07"
date_precision: day
category: science
importance: 3
confidence: high
status: [Event confirmed, Awaiting review]
verification: Expert-checked
sources: 7
editor: Adam Bicz
human_review: null
version: "2026-10-07"
---

As of: 2026-10-07 23:43 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz.
Canonical page: https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/

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

Within a day of OpenAI's 722-manuscript math release, Doug Colkitt (@0xdoug, founder of SLX) published a GitHub repository, CrocSwap/integer-mult-bounds, that improves the parameters of OpenAI's family #109, "Integer multiplication below n log n", which claims T(n) = O(n (log n)^(1−κ)) with κ = 2^-182. In three releases on 7 Oct (UTC) he raised the exponent saving to 5.8×10^-33, then 2^-78, then 2^-59, a 2^123-fold larger saving than the original. The work is conditional on OpenAI's unverified manuscript, has no Lean proof, and was "performed with assistance from OpenAI Codex". The gains are asymptotic exponent comparisons, not practical speedups.

## Key facts
- Upstream: OpenAI family #109, 'Integer multiplication below n log n' (manuscript dated 23 Sep 2026, released 6 Oct), claims T(n) = O(n (log n)^(1−κ)) with κ = 2^-182 on a fixed multitape Turing machine; Colkitt pins it at openai/math commit adc7f12
- Repo github.com/CrocSwap/integer-mult-bounds, four commits by Douglas Colkitt: 7 Oct 02:22 UTC κ = 29/(5·10^33) = 5.8×10^-33 (parameter and network refinements, between 2^-108 and 2^-107); 13:35–13:48 UTC κ = 2^-78 (direct nonadjacent axis routing); 19:27 UTC κ = 2^-59 (paired-block circuits that share intermediate sums, plus sharper downstream estimates)
- First release also proved a ceiling κ < 5.838×10^-33 for that network family and cost inequalities; the later results got past it by changing the network and routing (nonadjacent axis swaps, already allowed by the manuscript, cut layout routing from O(d²) to O(d) swaps)
- README: 'The upstream theorem remains an assumption'; 'These are asymptotic exponent comparisons, not measured practical speedups'; the repo has 'no full multiplication-machine implementation or Lean formalization of the claimed bound'. Arithmetic is checked by Python certificates (`make verify`), and each change is a patch to the pinned manuscript
- AI disclosure (README): '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.'
- X reach (fxtwitter, 7 Oct evening): 02:24 UTC post ~105k views, 13:56 UTC post (2^-78) ~135k, 19:35 UTC post (2^-59) ~42k. A quote post by @bubbleboi (~25k) said Colkitt has '6.1 Astra running in a loop'. That model name is bubbleboi's claim; the repository names only OpenAI Codex

## What happened

OpenAI's 6 Oct release (see [2026-10-06-openai-math-release-722-manuscripts](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)) 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.

## Why it matters

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.

## What is disputed or not yet verified

- Verification: Unverified preprint-style draft, conditional on OpenAI's unverified manuscript; arithmetic checked by Python certificates, no Lean formalization, no independent review

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 160 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 99 days after its cutoff
- Gemini 3.8 Flash (training cutoff March 2026): 190 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 129 days after its cutoff

## Sources

1. [GitHub: CrocSwap/integer-mult-bounds (Douglas Colkitt)](https://github.com/CrocSwap/integer-mult-bounds) (github.com, code)
2. [Proof note for the 2^-59 bound (PDF)](https://github.com/CrocSwap/integer-mult-bounds/blob/main/artifacts/paired-note.pdf) (github.com, paper)
3. [OpenAI family #109: Integer multiplication below n log n (pinned commit)](https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026) (github.com, paper)
4. [@0xdoug on X: κ = 5.8×10^-33 (first result)](https://x.com/0xdoug/status/2107658319632433524) (x.com, discussion)
5. [@0xdoug on X: κ = 2^-78](https://x.com/0xdoug/status/2107832360527913445) (x.com, discussion)
6. [@0xdoug on X: κ = 2^-59](https://x.com/0xdoug/status/2107917690568327225) (x.com, discussion)
7. [@bubbleboi on X: '6.1 Astra running in a loop'](https://x.com/bubbleboi/status/2107934679294935317) (x.com, discussion)

## Changes

- 2026-10-07 (filed): Created from the repository README and git history, and @0xdoug's and @bubbleboi's X posts (fxtwitter)

## Related

- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)
- Archived: 5 posts, listed in https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/index.json