Below n log n: what the exponents actually say about 719 machine-written manuscripts

SomaSoft. (2026-10-13). "Below n log n: what the exponents actually say about 719 machine-written manuscripts". SomaSoft Research. https://somasoft.ai/papers/below-n-log-n. Licensed under SAGL-1.0.

How to read this. [measured] marks a figure taken from a named primary source with its retrieval date. [I] marks our own argument. Every number below was either read from a primary artifact on 2026-10-13 or computed by us from numbers that were; the computations are reproducible from the stated exponents in four lines of arbitrary-precision arithmetic.

What we did not do. We did not read the proofs. Not one of the 719. We cannot tell you whether the Unique Games Conjecture has been proved, and neither can anyone else yet. This paper is deliberately about the only part of the claim that can be checked in an afternoon: the arithmetic of the stated bounds, and the provenance of the commentary.

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1. The artifact

[measured] The repository README states that the catalogue contains 719 manuscripts organised into 372 result families, and that about 42% of top-line results are formalised. We parsed the catalogue index ourselves on 2026-10-13 and counted exactly 372 family headings, 719 manuscript links and 719 unique preprint directories. The headline counts check out against the index.

[measured] The README states that the vast majority of results came from an unreleased internal model, that each used on average three hours of thinking compute, and that the model was posed approximately 4,000 problems. [measured] It names two exceptions to that fixed procedure — a zero-free region for the Riemann zeta function, and the Hodge Conjecture for CM abelian varieties — and records that the writeup for the zero-free half-plane with real part above 11/12 was human edited for readability.

[measured] 300 of 719 top-line results are formalised. [measured] By subtraction, 419 manuscripts — 58.3% of the corpus — carry no formal proof and rest on prose alone.

[I] Three observations before any mathematics. 372 families from roughly 4,000 posed problems is a 9.3% yield, which is a sampling fact worth holding onto: the catalogue is the successful tail of a much larger run, and we are shown the tail. The two exceptions to the procedure are among the most consequential claims in the set, which means the flagship results are the ones produced least uniformly. And a human editing pass applied to exactly one writeup tells you what the other 718 read like.

2. Two results, one headline, different status

Two families in the catalogue claim to break the n log n barrier. Both are described in the index with the same phrase — below n log n. They do not have the same standing, and the difference is in the exponent and the model rather than in anything a headline carries.

[measured] Family 109, integer multiplication below n log n. The claim is to multiply two n-bit integers in O(n(log n)^(1-k)) deterministic worst-case time with k = 2^-182, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. The index states that this disproves the Schonhage-Strassen n log n optimality conjecture in the ordinary multitape bit model.

[measured] Family 130, exact Fourier transforms below n log n. The claim is a deterministic length-n discrete Fourier transform in O(n(log n)^(1-d)) operations with explicit d = 1e-13. The index states that the model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity.

What the exponents are worth

The speedup of either result over n log n is the factor (log n)^e, where e is the exponent saving. [measured] We computed this at 80 decimal places:

	Family 109, k = 2^-182	Family 130, d = 1e-13
exponent saving	1.63e-55	1e-13
speedup at n = 2^64	1 + 6.78e-55	1.0000000000004
speedup at n = 10^80	1 + 9.11e-55	1 + 5.58e-13
speedup at n = 10^100	1 + 9.47e-55	1 + 5.81e-13
n needed for a 1.01x speedup	log2(n) = e^(6.1e52)	log2(n) = e^(9.95e10)
n needed for a 2x speedup	log2(n) = e^(4.25e54)	n of about 10^(3.01e12) bits

[I] Read the bottom row slowly, because the notation hides the scale. For family 109 to run twice as fast as an n log n algorithm, the input needs a length whose base-two logarithm is e raised to 4.25e54. The number of decimal digits of such an input is itself a number with roughly 1.8e54 digits. [measured] For comparison, 10^80 is approximately the number of atoms in the known universe.

[I] There is a cleaner way to say the same thing, and it removes the need to trust any estimate of the hidden constant. At n = 2^64, a length no computation will exceed, the entire gain of family 130 is a factor of 1.0000000000004. So any hidden constant greater than 1.0000000000004 erases the result's advantage completely at that length. For family 109 the gain at the same length is 1 + 6.78e-55, so the hidden constant must be at most 1 + 6.78e-55 — which is to say, essentially exactly 1 — for the algorithm to be faster than the thing it beats asymptotically. Asymptotic superiority with an exponent saving this small is not a statement about any computation that will ever run.

[I] This is not a criticism of family 109. We want to be exact about that, because the distinction is the point of the paper. A lower-bound conjecture is either true or false, and Schonhage and Strassen's is a famous one. Disproving it in the ordinary multitape bit model would be a genuine theorem of real interest, and the size of k is irrelevant to whether it is one. What the size of k settles is something else entirely: whether anybody's multiplication gets faster. It does not. Both things are true at once and the catalogue's own phrasing, below n log n, carries only the first.

Family 130 has a second problem

[measured] Writing on argmin.net, Ben Recht reports that the OpenAI FFT algorithm is definitely galactic, that the paper's estimated values for the hidden constant C were 10^100 when N was 10^12, and that even when n is 10^100 the Cooley-Tukey-Gauss FFT would be 10^99 times faster. He notes that the model assumes exact complex arithmetic — that one can store and multiply complex numbers at arbitrary precision — and that the algorithm requires multiplying complex numbers whose magnitudes grow superpolynomially with n. [measured] He records that in 1973 Jacques Morgenstern proved that no Fourier transform can beat one half n log n if the complex coefficients used in the computation are required to have bounded magnitude. [measured] The term galactic algorithm, for an algorithm with an unfathomably large hidden constant, he attributes to Richard Lipton and Ken Regan.

[I] Put Morgenstern's hypothesis beside the catalogue's own description of the model and the situation resolves itself without anyone needing to referee a proof. Morgenstern's lower bound holds when coefficients have bounded magnitude. The catalogue states that family 130's model uses unrestricted coefficients. The result therefore does not break Morgenstern's barrier; it is stated outside the region where the barrier was proved. OpenAI disclosed this in the index, in one clause, accurately. The disclosure is simply doing no work, because the clause that carries it sits after the phrase people quote.

[I] So the two results differ in kind. Family 109 is stated in a standard model and is practically empty. Family 130 is practically empty and obtains its saving by relaxing the hypothesis of the theorem it appears to contradict. The first is a real result about mathematics; the second is, at least in part, a result about a model. Recht puts the practical half of this more bluntly than we would: even if the theorem is true, which he says nobody knows, it is not solving the problem one sets out to solve when looking for a faster transform.

3. The most-cited rebuttal is about a different paper

[measured] On 6 October 2026 Bastounis, Circelli and Hansen posted Navier-Stokes lost in translation (arXiv:2610.08144), arguing that mechanically checking a Lean translation of a natural-language proof is not sufficient for the correctness of that proof. Their theoretical claim is that resolving the ambiguities in mathematical natural-language text sits at Solvability Complexity Index infinity — informally, harder than any computational problem, including the halting problem, which sits at index 1. They state that the formalised Lean proof does not correspond to the natural-language proof of blow-up of solutions to the Navier-Stokes equations.

[measured] We searched the catalogue index for the strings blow-up, blowup and blow up. There are no occurrences of blow-up. Occurrences of blowup appear only in family 351, on Ricci flow curvature blowup, and family 371, on stable blowup for the defocusing Schrodinger equation. [measured] The only Navier-Stokes families are 376, Universal computation in forced Navier-Stokes flows, and its companion Incompressible Box Transport and Finite Computation. Family 376 constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing, so that a designated particle reaches a fixed region exactly when a Turing machine halts, at fixed positive computable viscosity. Neither family claims finite-time blow-up of Navier-Stokes.

[measured] The artifact the critique examined is a separate repository, containing Lean 4 formalizations of the results presented in Finite time blowup for Navier-Stokes, at commit f9e8bc5, in the file NavierStokes SmoothFamilyTorusInverse.lean, declaration norm_derivativeWord_inverse_le, lines 1059 to 1080.

[I] So the sentence circulating this week — that the math release was refuted on Navier-Stokes — joins two different releases. The catalogue of 719 contains no blow-up claim to refute. Anyone relying on that sentence to discount the catalogue is discounting it with evidence about something else. This cuts both ways, and we are not using it to defend the catalogue: the critique may be entirely correct about the artifact it examined.

[measured] The specific discrepancy the authors identify is worth stating precisely, because it is the most concrete thing anybody has published about this family of problems. The natural-language paper's equation 8.19 bounds the C m norm of N inverse applied to F by a constant times the C m plus 4 norm of F. The Lean estimate bounds the same quantity by a constant times the C m plus 5 norm. The authors conclude that the Lean code often carries stronger conditions on bounds on derivatives than the corresponding results in the natural-language paper.

[I] One derivative. It would be easy to read that as a typographical quibble, and in a different kind of argument it would be. In a regularity or bootstrap argument it is the load -bearing quantity: the estimate is applied repeatedly, and the number of derivatives surrendered per application is what decides whether the iteration closes on a finite budget or runs away. Requiring control of the C m plus 5 norm to conclude something about the C m norm is a weaker theorem than requiring control of the C m plus 4 norm. A mechanical check of the weaker statement is a genuine proof of the weaker statement and says nothing about the one in the paper. That is the entire autoformalisation gap in a single index.

4. What an AI summary did to this paper on the way to being written

[I] We report this because it happened to us, mid-analysis, and because it is the most transferable finding here.

[measured] Our first pass over Recht's post used a summarising fetch: a tool that retrieves a page and answers a question about it using a small fast model. That summary reported the OpenAI claim as O(n log log n). We then retrieved the page's raw text separately. The string log log does not occur in it. The bound was introduced by the summarising model and is not Recht's claim; [measured] the catalogue's actual claim is an exponent saving of 1e-13, which is a very different and far weaker assertion than n log log n.

[I] We had drafted a sentence correcting a named researcher for a complexity bound he never stated, on the authority of a machine that had invented it. The only reason that sentence is not in this paper is a habit: when a claim is going to be attributed to a person by name, fetch the verbatim text. The habit is cheap and it was the difference between a correction and a defamation.

[I] The structural point is uncomfortable for everybody writing about this release. Reviewing 719 machine-written manuscripts at the speed the news cycle wants is not possible without machine assistance, and machine assistance introduces exactly this class of error into the review. The release stresses the reviewing apparatus in the same way it stresses the field, and the failure mode is not laziness. It is that a summary is a lossy, confident artifact that looks identical whether or not it is faithful.

5. The corpus is not 719 independent results

[measured] The catalogue's own history file, dated 7 October 2026, records that a sign error in Algebraicity of Weil classes on split abelian eightfolds invalidated a stabilization-trace cancellation argument and the construction used by two dependent papers. Three manuscripts were withdrawn. [measured] 14 further manuscripts were revised, and thirteen more were updated to cite the revised companions.

[measured] We computed the structure this implies. 719 manuscripts across 372 families average 1.93 manuscripts per family, so 347 manuscripts — 48.3% of the corpus — are companions, consequences or alternative proofs attached to a principal result rather than independent results. The 7 October round touched 30 manuscripts, 4.2% of the corpus, against roughly six identified root causes: one sign error, four named argument clusters, and one obsolete citation. That is an amplification of about five manuscripts per cause.

[I] This is the shape our own earlier work predicts, and we did not expect to find a cleaner instance of it than the ones we constructed. A corpus whose members share derivations does not fail member by member. It fails along its dependency graph, and the unit of failure is the shared argument rather than the paper. Counting 719 manuscripts therefore overstates the independent content of the release by something close to a factor of two, and — more importantly — it misdescribes the risk. The question for a reader is not what fraction of 719 manuscripts contains an error. It is how many distinct load-bearing arguments the 719 rest on, and how many papers each one carries. One sign error took three papers. Nobody outside the authoring organisation currently knows the degree distribution of that graph.

[I] The same arithmetic cuts against the most dismissive reading, too. Thirty touched manuscripts out of 719 is 4.2%, found and published within a day, with withdrawal notices and archived prior versions. Whatever else is true, the corrections were disclosed in a form that let us compute this paragraph.

6. What we checked, and what remains open

[measured] We ran every claim in this analysis through the same publication gate our papers pass: 15 assertions against 20 facts drawn from primary artifacts, scoring 1.0 with no flags. [measured] The gate rejected one of our sentences on the first pass — the figure of 419 unformalised manuscripts, which is our own subtraction and was not registered as a source. It is now registered as a computation rather than as a reading.

Open, and we would rather say so than round it off:

[measured] The quotations we use from Asaf Karagila and Scott Aaronson came through a summarising fetch rather than verbatim retrieval. Given section 4 we would verify them against the raw pages before relying on either. We report Karagila's verdict as recorded — that the Partition Principle manuscript is unclear, muddled and strangely structured, that it would earn a desk rejection on quality, and that releasing hundreds of such solutions at once is the equivalent of a denial of service — and note that he explicitly does not assess whether the result is correct. We report Aaronson's as recorded, that this was one of the biggest days in mathematical history and that no human has yet understood just about any of these proofs.

[measured] We could not trace the provenance of the Finite time blowup for Navier-Stokes artifact relative to the catalogue, because this session's search budget was exhausted. The statement we stand behind is narrower and verifiable: as of the index we retrieved on 2026-10-13, the catalogue of 719 contains no Navier-Stokes blow-up result.

[measured] We queried our own concept graph for galactic algorithm and it returned no entry: no definition, no neighbours, no associated book knowledge. [I] We note it because the honest version of a self-assessment includes the part where one's own system contributes nothing. It is consistent with our published finding that the graph is curated islands in lexical residue, and it means none of the analysis above came from it.

[I] The claim in the catalogue we would most like someone competent to examine is not either algorithm. [measured] Family 003 asserts that every Dirichlet L-function, including the Riemann zeta function, is zero-free in the half-plane where the real part exceeds 7/8, with a companion giving a different proof for the half-plane above 11/12. [I] If that holds it is of a different order of consequence than a galactic speedup, and it is the result whose writeup the README says received the human editing pass. We have seen almost no commentary on it. Attention has gone to the two claims whose headlines contain a number people recognise, which is the same selection effect this paper has been describing in other forms throughout.

7. The narrow conclusion

[I] Nothing here bears on whether the mathematics is correct. Our finding is about the distance between a true statement and its headline, and it runs in both directions. Below n log n is true of family 109 and tells you nothing that matters about multiplication. Refuted on Navier-Stokes is a claim about a paper that is not in the release. An exponent saving of 2^-182 is a legitimate disproof of a conjecture and a speedup of one part in 10^54. A Lean file can be mechanically perfect and certify a theorem one derivative weaker than the one in the prose above it.

[I] The useful discipline is not scepticism, which is cheap and was widely available this week. It is reading the exponent, reading the model the exponent is stated in, and checking that the artifact a rebuttal examined is the artifact under discussion. All three took us an afternoon and arbitrary-precision arithmetic. None of them required refereeing a proof, and none of them is a substitute for doing so.