The Auto Research system provides a structured process for advancing the mathematical frontier of integer multiplication, targeting improvements in the exponent saving parameter κ in the time complexity formula T(n) = O(n (log n)^(1−κ)), where larger κ indicates better asymptotic performance. Contributors follow a five‑step protocol: establishing the current frontier by reviewing recent code and documentation, reproducing a chosen baseline’s verified results, conducting an evidence‑driven search for promising hypotheses, independently certifying any candidate improvement, and packaging the outcome for potential submission. This workflow is designed to ensure that each step yields a genuine, reproducible advance rather than a mere literature summary or unverified proposal.
The most recent verified contribution, recorded in the CrocSwap/integer‑mult‑bounds repository, achieves κ = 1661634653669567 / 10^18 ≈ 0.001661634653669567, which the project reports is approximately 0.026005 % above the previously selected exponent saving. This result builds on Jeremy Kothe’s pull request #799, which extended Chafik Boukhalfa’s #741 frame construction and the earlier source‑gauge approach, while incorporating DreamingOfClouds’ #589 bit‑level improvement on eumemic’s #555 supplier. The gain is attributed to four paired read exchanges that refine the recursive rank profile while preserving helper endpoints and physical owners, alongside cumulative techniques such as icekylinx’s source gauge and Rohan Arun’s recurrence correction.
This image corresponds to the repository where the latest verified improvement in the integer multiplication exponent saving parameter κ was achieved, as discussed in the article.
Despite the numerical advance, the project emphasizes that the finite verification checks performed do not constitute a complete proof in the Lean theorem prover and do not translate to practical runtime speedups. The documentation explicitly states that “These finite checks are not a complete Lean multiplication theorem” and that “Finite replay and written review do not prove the complete asymptotic theorem in Lean.” All advances remain conditional on the assumed OpenAI #109 framework, with properties such as all‑size compilation, precision/tape analysis, and analytic interfaces depending on specific assumptions. Consequently, the work is presented as a step in comparing exponent savings rather than as a claim of measurable performance improvement in real‑world applications.