As of: 2026-10-08 23:45 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz.
Canonical page: https://postcutoff.com/p/2026-10-08-sj-swapnil-jain-integer-multiplication-kappa-2-15-lean/

# Swapnil Jain: OpenAI problem #109 past 2^-15, certificates and assembly checked in Lean

Swapnil Jain (@SJ_Swapnil_Jain), X, 2026-10-08.
Source: https://x.com/SJ_Swapnil_Jain/status/2108196538568851574

## Why it matters

Sixth update (~102k views), reaching 2^-15 about 90 minutes before Colkitt's community repo and claiming Lean kernel checks.

## Summary

Posted 14:03 UTC on 8 Oct. The Lean files (core Lean 4, `decide`) check histogram rank sums, moment bounds and the κ assembly inequalities; per the README, analytic premises, the upstream theorem and independent review are not covered.

## Archived text

> Sixth update to OpenAI problem #109 (integer multiplication): we are now past 2^-15.
> 
> κ > 2⁻¹⁵  (tightened from κ = 2⁻¹⁸²)
> 
> The exact witness is 3.667 × 10⁻⁵, about 2.4 fold over our previous 1.548 × 10⁻⁵, and a 2¹⁶⁷ fold improvement over the original OAI result. 
> 
> Both interchange certificates and the full assembly are checked in Lean's kernel.
> 
> > Quoting @SJ_Swapnil_Jain: Fifth update to OpenAI problem #109 (integer multiplication): we are now past 2^-16.

Archived 2026-10-08 via fxtwitter (unofficial).
Counts: 101,807 views, 923 likes, 28 reposts, 34 replies (at fetch time)

## Cited in

- 2026-10-07: [Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to above 2^-14 (conditional)](https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/)