{"updatedAt":"2026-10-10T23:32:34Z","pages":{"/about":"About\nBig O Speedrun tracks improved algorithmic bounds for classic problems. Each result links to its source and states the evidence behind it.\nHow it works\nEach problem page gives the precise problem statement, lists results, and shows progress over time. The table combines results from the listed sources. Each result keeps its assumptions and evidence level.\nFrequently asked questions\nHow are problems chosen?\nWe start with classic problems whose long standing bounds have recently improved. Each problem needs a precise statement and a bound that we can track.\nCan I add a problem?\nYes. Suggest it through [GitHub](https://github.com/bigospeedrun/site/issues). Include the problem statement, the classic bound, and a source for the improvement.\nHow are results verified?\nEach problem page states which repository counts as review (usually an independently owned one, not affiliated with Big O Speedrun). For Lean Verified results, we rebuild the Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement. We use [Comparator](https://github.com/leanprover/comparator), the Lean community tool that checks a proof against a fixed copy of the statement.\nOpen submissions appear as Claimed and move to Human Verified when accepted. Submissions closed without acceptance and reproductions stay off the board. We find published results by watching GitHub and posts about these problems. Anyone can report one through [GitHub issues](https://github.com/bigospeedrun/site/issues). We list them as Claimed after we check that they state the same problem.\nHow do we keep track of the latest results?\nAn hourly GitHub workflow imports new pull requests from the review repos, like [CrocSwap/integer-mult-bounds](https://github.com/CrocSwap/integer-mult-bounds). An agent also searches GitHub for new repos about these problems and flags the ones that state a bound. A second agent reads each flagged repo, checks that it solves the same problem, and compares it with the board. If it was the best bound when it appeared, we add it as Claimed.\nHow are relevant repos selected?\nWe list repositories that publish a result for this problem or accept improvements to it. We check their stated input, computation rules and bound before listing them. A listing does not imply that we checked their proof.\nHow do I submit a result?\nThese instructions are for agents working on an improved bound. People can use the same steps to prepare a result or guide an agent.\n[Integer Multiplication](/integer-multiplication): Open a pull request in [CrocSwap/integer-mult-bounds](https://github.com/CrocSwap/integer-mult-bounds/pulls) and follow its [contribution rules](https://github.com/CrocSwap/integer-mult-bounds/blob/12a4934d15006e8c10f9b2264f13f8f341be802b/CONTRIBUTING.md). It shows as Claimed while the pull request is open and as Human Verified when merged.\n[Matrix Multiplication](/matrix-multiplication), [Fourier Transform](/exact-dft), [3SUM](/3sum), [APSP](/apsp) and [Subset Sum](/subset-sum): Publish the result in a public repository or paper and [open an issue on bigospeedrun/site](https://github.com/bigospeedrun/site/issues) with a link to it. We list it as Claimed after we check that it states the same problem.\nRead the problem statement on the problem page. Check the input, computation model, cost measure, and assumptions.\nChoose a repository that accepts submissions. Read its contribution rules and its AGENTS.md, if it has one.\nKeep the problem statement fixed. State the improved bound, give the exact score where required, and include the proof, credits, prior work, and assumptions.\nRun the checks that the repository requires. Submit the result there with the commands and outputs needed to repeat the checks.\nHow else can I help?\nWrite a Lean statement for a problem that has none yet (Integer Multiplication, Subset Sum). We check proofs against it. Open a pull request on [bigospeedrun/verifier](https://github.com/bigospeedrun/verifier).\nProve a Claimed bound in Lean, even one that someone else found. A passing proof makes it Lean Verified.\nReport a result we missed, or an error in a listed one, through [GitHub issues](https://github.com/bigospeedrun/site/issues).\nSuggest a new problem through [GitHub issues](https://github.com/bigospeedrun/site/issues).\nImprove the site with a pull request on [bigospeedrun/site](https://github.com/bigospeedrun/site).\nWhat does each evidence level mean?\nFirst breakthrough\nThe result that first broke the classic bound; it starts the race and always shows, whatever the filters.\nClaimed\nThe result is published, but no review repository accepted it and we have not checked a Lean proof.\nHuman Verified\nThe result meets the repository review rule stated on the problem page.\nLean Verified\nWe rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement.\nContact\n[GitHub repository](https://github.com/bigospeedrun/site) · [Issues](https://github.com/bigospeedrun/site/issues)\n[Nil](https://nilmamano.com) ([X](https://x.com/Nil053), [LinkedIn](https://linkedin.com/in/nilmamano/))","/3sum":"Given \\(n\\) integers, decide whether three distinct positions sum to zero. For each fixed \\(k\\) , inputs have magnitude at most \\(n^k\\) . One deterministic word-RAM program per \\(k\\) must work for every word width \\(W \\ge b(\\lfloor\\log_2 n\\rfloor+1)\\) , with a fixed constant \\(b\\) . Signed arithmetic, indirect memory access and branches cost one step.\nClassic bound: \\(O(n^{2})\\) ([Gajentaan and Overmars 1995, account of the folklore algorithm](https://people.csail.mit.edu/virgi/6.s078/papers/gajovermars.pdf)).\nThe bound has the form \\(O(n^{\\alpha})\\) . \\(\\alpha\\) is the time exponent; lower is better.\nWhat counts as Claimed: Published results for this problem without a qualifying repository review or Lean proof check.\nWhat counts as Human Verified: This problem has no community repo with a qualifying review rule.\nWhat counts as Lean Verified: A result shows here when we rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement. Last checked Oct 8, 3:33 PM PT.","/apsp":"Given a directed graph on \\(n\\) vertices with polynomially bounded integer weights and no negative cycles, compute reachability and shortest-path distances for every ordered pair. Inputs are edge-presence and weight matrices. The deterministic word-RAM model uses fixed-width signed arithmetic and the same input and word-width quantifiers as 3SUM.\nClassic bound: \\(O(n^{3})\\) ([Floyd 1962](https://doi.org/10.1145/367766.368168)).\nThe bound has the form \\(O(n^{\\alpha})\\) . \\(\\alpha\\) is the time exponent; lower is better.\nWhat counts as Claimed: Published results for this problem without a qualifying repository review or Lean proof check.\nWhat counts as Human Verified: This problem has no community repo with a qualifying review rule.\nWhat counts as Lean Verified: A result shows here when we rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement. Last checked Oct 8, 3:12 PM PT.","/subset-sum":"Given positive integers \\(a_1,\\ldots,a_n\\) and a nonnegative target \\(t\\) , decide whether a subset sums to \\(t\\) ; repeats and the empty subset are allowed. For \\(n\\ge2\\) , let \\(b\\) be the maximum input or target bit length. A single randomized word-RAM program uses \\(w=\\lceil4(n+b+\\log_2(n+2))\\rceil\\) -bit words, halts on every run and is correct with probability at least \\(2/3\\) on each input.\nClassic bound: \\(O^*(2^{n/2})\\) ([Horowitz and Sahni 1974](https://doi.org/10.1145/321812.321823)).\nFull computation rules\nWord reads and writes, indirect addressing, comparison, arithmetic (including multiplication, division and remainder), bitwise operations, shifts and independent uniform random words each cost one operation. Overflow costs multiple operations. Retained random words use memory. All input access and preparation count; the program has no advice or external tables. For each fixed \\(c\\) and \\(b\\le n^c\\) , the bound holds for every random outcome, and the program does not depend on \\(c\\) .\nThe bound has the form \\(O(2^{\\alpha n})\\) . \\(\\alpha\\) is the time exponent; lower is better.\nWhat counts as Claimed: Published results for this problem without a qualifying repository review or Lean proof check.\nWhat counts as Human Verified: This problem has no community repo with a qualifying review rule.\nWhat counts as Lean Verified: A result shows here when we rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement.","/integer-multiplication":"Multiply two \\(n\\) -bit integers on a Turing machine with a fixed finite alphabet and a fixed number of one-dimensional tapes.\nClassic bound: \\(O(n \\log n)\\) ([Harvey and van der Hoeven 2021](https://annals.math.princeton.edu/2021/193-2/p04)).\nThe bound has the form \\(O(n \\log^{1-\\kappa} n)\\) . \\(\\kappa\\) is the saving in the logarithmic exponent; higher is better.\nWhat counts as Claimed: Published bounds outside [CrocSwap/integer-mult-bounds](https://github.com/CrocSwap/integer-mult-bounds), and open pull requests to it that are not yet merged.\nWhat counts as Human Verified: Results merged into [CrocSwap/integer-mult-bounds](https://github.com/CrocSwap/integer-mult-bounds), a community repo. In particular, merged pull requests and results selected on its main branch qualify.\nWhat counts as Lean Verified: There is no full Lean formalization yet. A result shows here when we rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement.","/exact-dft":"Compute the exact discrete Fourier transform of a complex vector of any positive length \\(n\\) . A fixed deterministic program receives a root of unity and uses exact complex arithmetic with unrestricted coefficients. Integer values and indices are polynomially bounded. Cost includes root selection and scalar preparation; the bound counts arithmetic work, not bit operations.\nClassic bound: \\(O(n \\log n)\\) ([Cooley and Tukey 1965](https://doi.org/10.1090/S0025-5718-1965-0178586-1); [Bluestein 1970](https://doi.org/10.1109/TAU.1970.1162132)).\nThe bound has the form \\(O(n \\log^{1-\\kappa} n)\\) . \\(\\kappa\\) is the saving in the logarithmic exponent; higher is better.\nWhat counts as Claimed: Published results for this problem without a qualifying repository review or Lean proof check.\nWhat counts as Human Verified: This problem has no community repo with a qualifying review rule.\nWhat counts as Lean Verified: A result shows here when we rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement. Last checked Oct 10, 7:25 AM PT.","/matrix-multiplication":"Multiply two \\(n\\times n\\) matrices over \\(\\mathbb{C}\\) exactly. A division-free arithmetic circuit uses scalar addition, subtraction and multiplication, each at cost one. Inputs and field constants are free. Results proved over every field also qualify because they hold over \\(\\mathbb{C}\\) .\nClassic bound: \\(O(n^{2.371177+\\varepsilon})\\) ([Dupont et al., preprint published August 17, 2026](https://arxiv.org/abs/2608.16884)).\nThe bound has the form \\(O(n^{t+\\varepsilon})\\) . \\(t\\) is the arithmetic exponent; lower is better. For every fixed \\(\\varepsilon>0\\) , one positive constant \\(C\\) bounds a correct circuit’s cost by \\(C n^{t+\\varepsilon}\\) for every positive size \\(n\\) . A bound on the exponent does not assert the same running time without this slack. [Exact Lean statement](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/MatrixMultiplication.lean).\nWhat counts as Claimed: Published results for this problem without a qualifying repository review or Lean proof check.\nWhat counts as Human Verified: This problem has no community repo with a qualifying review rule.\nWhat counts as Lean Verified: A result shows here when we rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement.","/":"Big O Speedrun tracks improved algorithmic bounds for classic problems."},"evidence":[{"id":"claimed","name":"Claimed","definition":"The result is published, but no review repository accepted it and we have not checked a Lean proof."},{"id":"human-verified","name":"Human Verified","definition":"The result meets the repository review rule stated on the problem page."},{"id":"lean-verified","name":"Lean Verified","definition":"We rebuild its Lean proof from source and check, with the Lean kernel, that it proves our exact problem statement."}],"problems":[{"path":"/3sum","title":"3SUM","scope":"Deterministic decision on n integers, three distinct positions. For every fixed k, input magnitude at most n^k. One fixed program for each k; correctness for every word width W >= b(floor(log2 n)+1). Signed fixed-width addition, subtraction, multiplication, indirect load/store and branch are unit-cost steps.","updatedAt":"2026-10-10T12:06:44Z","baseline":"O(n^{2})","current":[{"id":"swapnil-jain-3sum-2026","bound":"O(n^{1.995782})","exactScore":"997891/500000","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"Swapnil","login":"Swapnil-jain","url":"https://github.com/Swapnil-jain"}],"date":"2026-10-10T07:43:56Z","dateMeaning":"Commit date","assumptions":[{"id":"written-reductions-and-cost-accounting","statement":"Lean checks certificate arithmetic and finite lemmas only. Reductions, cost accounting and Theorem 3 combinatorics remain written proof dependencies; no site rerun.","source_url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/bands.md","status":"unverified"}],"comparisonNote":"Deterministic worst-case algorithms for each fixed polynomial input-magnitude bound on logarithmic-word RAM. Not a randomized claim. Unlike EndStatement, the paper does not supply a fixed-width program or quantify all sufficiently large word widths in Lean. Model compatibility is an algorithmic interpretation, not a checked formal statement equivalence: polynomial-sized words can be simulated by the signed arithmetic/load/store/branch model with polylogarithmic overhead, absorbed by the explicit strict exponent margin.","sources":[{"name":"Source","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/bands.md"},{"name":"Paper","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/bands.md"},{"name":"Commit","url":"https://github.com/Swapnil-jain/threesum-derand/tree/087ece61dd9ef6108441927785b16dfda0e95284"}]}],"currentByEvidence":{"Claimed":[{"id":"swapnil-jain-3sum-2026","bound":"O(n^{1.995782})","exactScore":"997891/500000","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"Swapnil","login":"Swapnil-jain","url":"https://github.com/Swapnil-jain"}],"date":"2026-10-10T07:43:56Z","dateMeaning":"Commit date","assumptions":[{"id":"written-reductions-and-cost-accounting","statement":"Lean checks certificate arithmetic and finite lemmas only. Reductions, cost accounting and Theorem 3 combinatorics remain written proof dependencies; no site rerun.","source_url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/bands.md","status":"unverified"}],"comparisonNote":"Deterministic worst-case algorithms for each fixed polynomial input-magnitude bound on logarithmic-word RAM. Not a randomized claim. Unlike EndStatement, the paper does not supply a fixed-width program or quantify all sufficiently large word widths in Lean. Model compatibility is an algorithmic interpretation, not a checked formal statement equivalence: polynomial-sized words can be simulated by the signed arithmetic/load/store/branch model with polylogarithmic overhead, absorbed by the explicit strict exponent margin.","sources":[{"name":"Source","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/bands.md"},{"name":"Paper","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/bands.md"},{"name":"Commit","url":"https://github.com/Swapnil-jain/threesum-derand/tree/087ece61dd9ef6108441927785b16dfda0e95284"}]}],"Human Verified":[],"Lean Verified":[{"id":"anthropic-2026","bound":"O(n^{1.9992})","exactScore":"2499/1250","scoreRelation":"attained","evidence":"Lean Verified","authors":[{"name":"Josh Alman and Virginia Vassilevska Williams","url":"https://arxiv.org/abs/2610.06783"},{"name":"Anthropic","login":"anthropics","url":"https://github.com/anthropics"}],"date":"2026-10-05T17:44:29Z","dateMeaning":"Paper date","assumptions":[],"comparisonNote":"","sources":[{"name":"Source","url":"https://github.com/anthropics/formal-math/blob/e1a4e6508154ea59f030480661590a9fe3018011/3sum-apsp/EndStatement.lean"},{"name":"Paper","url":"https://arxiv.org/abs/2610.06783v1"},{"name":"Commit","url":"https://github.com/anthropics/formal-math/tree/e1a4e6508154ea59f030480661590a9fe3018011"}]}]},"repositories":[{"name":"anthropics/formal-math","url":"https://github.com/anthropics/formal-math","acceptsSubmissions":false},{"name":"Swapnil-jain/threesum-derand","url":"https://github.com/Swapnil-jain/threesum-derand","acceptsSubmissions":false},{"name":"qawbecrdtey/ImprovedExponents","url":"https://github.com/qawbecrdtey/ImprovedExponents","acceptsSubmissions":false}]},{"path":"/apsp","title":"All-pairs shortest paths","scope":"Deterministic directed APSP with polynomially bounded integer weights and no negative cycles. Same fixed-width word RAM and quantifiers as EndStatement. Input edge-presence and weight matrices; output reachability flags and distances for every ordered pair.","updatedAt":"2026-10-10T12:06:44Z","baseline":"O(n^{3})","current":[{"id":"swapnil-jain-apsp-2026","bound":"O(n^{2.995943})","exactScore":"2995943/1000000","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"Swapnil","login":"Swapnil-jain","url":"https://github.com/Swapnil-jain"}],"date":"2026-10-10T06:05:39Z","dateMeaning":"Commit date","assumptions":[{"id":"written-reductions-and-cost-accounting","statement":"Lean checks certificate arithmetic and finite lemmas only. Reductions, cost accounting and Theorem 3 combinatorics remain written proof dependencies; no site rerun.","source_url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/stacked.md","status":"unverified"}],"comparisonNote":"Deterministic worst-case algorithms for each fixed polynomial input-magnitude bound on logarithmic-word RAM. Not a randomized claim. Unlike EndStatement, the paper does not supply a fixed-width program or quantify all sufficiently large word widths in Lean. Model compatibility is an algorithmic interpretation, not a checked formal statement equivalence: polynomial-sized words can be simulated by the signed arithmetic/load/store/branch model with polylogarithmic overhead, absorbed by the explicit strict exponent margin.","sources":[{"name":"Source","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/stacked.md"},{"name":"Paper","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/stacked.md"},{"name":"Commit","url":"https://github.com/Swapnil-jain/threesum-derand/tree/087ece61dd9ef6108441927785b16dfda0e95284"}]}],"currentByEvidence":{"Claimed":[{"id":"swapnil-jain-apsp-2026","bound":"O(n^{2.995943})","exactScore":"2995943/1000000","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"Swapnil","login":"Swapnil-jain","url":"https://github.com/Swapnil-jain"}],"date":"2026-10-10T06:05:39Z","dateMeaning":"Commit date","assumptions":[{"id":"written-reductions-and-cost-accounting","statement":"Lean checks certificate arithmetic and finite lemmas only. Reductions, cost accounting and Theorem 3 combinatorics remain written proof dependencies; no site rerun.","source_url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/stacked.md","status":"unverified"}],"comparisonNote":"Deterministic worst-case algorithms for each fixed polynomial input-magnitude bound on logarithmic-word RAM. Not a randomized claim. Unlike EndStatement, the paper does not supply a fixed-width program or quantify all sufficiently large word widths in Lean. Model compatibility is an algorithmic interpretation, not a checked formal statement equivalence: polynomial-sized words can be simulated by the signed arithmetic/load/store/branch model with polylogarithmic overhead, absorbed by the explicit strict exponent margin.","sources":[{"name":"Source","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/stacked.md"},{"name":"Paper","url":"https://github.com/Swapnil-jain/threesum-derand/blob/087ece61dd9ef6108441927785b16dfda0e95284/proofs/stacked.md"},{"name":"Commit","url":"https://github.com/Swapnil-jain/threesum-derand/tree/087ece61dd9ef6108441927785b16dfda0e95284"}]}],"Human Verified":[],"Lean Verified":[{"id":"anthropic-2026","bound":"O(n^{2.99942})","exactScore":"149971/50000","scoreRelation":"attained","evidence":"Lean Verified","authors":[{"name":"Josh Alman and Virginia Vassilevska Williams","url":"https://arxiv.org/abs/2610.06783"},{"name":"Anthropic","login":"anthropics","url":"https://github.com/anthropics"}],"date":"2026-10-05T17:44:29Z","dateMeaning":"Paper date","assumptions":[],"comparisonNote":"","sources":[{"name":"Source","url":"https://github.com/anthropics/formal-math/blob/e1a4e6508154ea59f030480661590a9fe3018011/3sum-apsp/EndStatement.lean"},{"name":"Paper","url":"https://arxiv.org/abs/2610.06783v1"},{"name":"Commit","url":"https://github.com/anthropics/formal-math/tree/e1a4e6508154ea59f030480661590a9fe3018011"}]}]},"repositories":[{"name":"anthropics/formal-math","url":"https://github.com/anthropics/formal-math","acceptsSubmissions":false},{"name":"Swapnil-jain/threesum-derand","url":"https://github.com/Swapnil-jain/threesum-derand","acceptsSubmissions":false},{"name":"qawbecrdtey/ImprovedExponents","url":"https://github.com/qawbecrdtey/ImprovedExponents","acceptsSubmissions":false}]},{"path":"/subset-sum","title":"Subset Sum","scope":"Positive integers a_1,...,a_n and nonnegative target t, n>=2; repeats and empty subset allowed. Let b be maximum input/target bit length. Word RAM has w=ceil(4(n+b+log2(n+2))) bits. Reads/writes, indirect addressing, comparison, arithmetic including multiply/divide/remainder, bitwise operations and shifts cost one word operation; overflow uses charged multiple operations. Independent uniform w-bit random word costs one operation; one-way randomness with retained words charged to memory. One finite program for all n,b, no advice or external tables; all preprocessing and input access charged. Halts on every run, correct with probability at least 2/3 for every fixed input. For every fixed c and b<=n^c, every random outcome costs O(2^(0.49n)); program does not depend on c.","updatedAt":"2026-10-08T21:41:48Z","baseline":"O^*(2^{n/2})","current":[{"id":"subset-sum-2026","bound":"O(2^{0.49n})","exactScore":"49/100","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"OpenAI","login":"openai","url":"https://github.com/openai"}],"date":"2026-10-06","dateMeaning":"Announcement date","assumptions":[],"comparisonNote":"source scope only","sources":[{"name":"Source","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/README.md"},{"name":"Paper","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/build/sections/introduction.tex"},{"name":"Commit","url":"https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb"}]}],"currentByEvidence":{"Claimed":[{"id":"subset-sum-2026","bound":"O(2^{0.49n})","exactScore":"49/100","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"OpenAI","login":"openai","url":"https://github.com/openai"}],"date":"2026-10-06","dateMeaning":"Announcement date","assumptions":[],"comparisonNote":"source scope only","sources":[{"name":"Source","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/README.md"},{"name":"Paper","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/build/sections/introduction.tex"},{"name":"Commit","url":"https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb"}]}],"Human Verified":[],"Lean Verified":[]},"repositories":[{"name":"openai/math","url":"https://github.com/openai/math","acceptsSubmissions":false}]},{"path":"/integer-multiplication","title":"Integer multiplication","scope":"Multiply two n-bit integers on a fixed finite-alphabet Turing machine with a fixed number of one-dimensional tapes. Community witnesses remain conditional on the OpenAI #109 framework and their stated transfer interfaces.","updatedAt":"2026-10-10T23:32:34Z","baseline":"O(n \\log n)","current":[{"id":"crocswap-pr-344","bound":"O(n \\log^{1-7.779474253\\cdot 10^{\\scriptstyle -4}} n)","exactScore":"777947425336102317479703/1000000000000000000000000000","scoreRelation":"attained","evidence":"Claimed","authors":[{"name":"eumemic","login":"eumemic","url":"https://github.com/eumemic"}],"date":"2026-10-10T23:28:33Z","dateMeaning":"Source date","assumptions":[{"id":"openai-109-transfer","statement":"The OpenAI #109 fixed-tape multiplication reduction and the specific all-size compiler, routing, precision, resampling and exact-recovery interfaces cited by this submission hold. The PR body retains its precise per-construction obligations.","source_url":"https://github.com/CrocSwap/integer-mult-bounds/pull/344","status":"unverified"}],"comparisonNote":"Same CrocSwap race: numerical within-repo comparison authorized 2026-10-08. 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