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/p/2026-10-07-0xdoug-integer-multiplication-kappa-2-78/

# Doug Colkitt: OpenAI problem #109 tightened further to κ = 2^-78

Doug Colkitt (@0xdoug), X, 2026-10-07.
Source: https://x.com/0xdoug/status/2107832360527913445

## Why it matters

Highest-reach post of the series (~135k views): second conditional tightening, to κ = 2^-78, via nonadjacent axis swaps.

## Summary

Posted 13:56 UTC. Claims κ = 2^-78 (a 2^104-fold improvement over OpenAI's 2^-182) by using nonadjacent axis swaps, which the original manuscript already supported, to cut layout routing from O(d²) to O(d) swaps; the earlier ceiling no longer applies to the new witness. Quotes the 02:24 UTC post.

## Archived text

> We are publishing an update to OpenAI Problem #109 (integer multiplication) with a substantial further tightening:
> 
> T(n) = O(n (log n)^(1 − κ)),
> 
> With κ = 2⁻⁷⁸   (tightened from κ = 2⁻¹⁸²)
> 
> Approximately 570 million fold improvement over our previous result and a 2¹⁰⁴ fold improvement over original OAI result.
> 
> Our earlier ceiling applied to a cubic bottleneck in the network. The new witness scales quadratically; we haven't established a new ceiling.
> 
> The key was using nonadjacent axis swaps to route around the cubic bottleneck. The original manuscript already supported nonadjacent axis swaps. Using them directly reduces layout routing from O(d²) to O(d) swaps.
> 
> > Quoting @0xdoug: We're publishing a result demonstrating a substantial tightening to the results from OpenAI Problem #109 (integer multiplication).

Archived 2026-10-07 via fxtwitter (unofficial).
Counts: 134,750 views, 1,536 likes, 83 reposts, 40 replies (at fetch time)

## Cited in

- 2026-10-07: [With OpenAI Codex, Doug Colkitt tightens OpenAI's sub-n log n integer multiplication exponent from 2^-182 to 2^-59 (conditional)](https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/)