Claimed AI-assisted

Closed form for the anisotropic Watson integral of the cubic lattice

The return probability of a random walk on the cubic lattice with three unequal hopping rates, open since Watson's 1939 solution of the symmetric case, is given in closed form through periods of a genus-two curve, with a proof that no formula in elliptic integrals exists for generic rates.

Lab: Independent
Model: Claude (the announcement names Claude Fable 5)
Field: Mathematics
Date: 2026-10-01
Human collaborators: Noam Elkies, Thomas W. Grimm, Matthew D. Schwartz
Problem posed: 1939 · open 87 yrs

Sources

Original work

bootloops.aiBootLoops: WatsonW.py, the closed form as code↗

Announcement

anthropic.comAnthropic: Claude-shaped science (guest essay by Matthew Schwartz)↗bootloops.aiBootLoops: The Watson integral (authors' summary)↗

What was found

The lattice Green's function $W(\alpha_1, \alpha_2, \alpha_3; w) = \frac{1}{\pi^3}\int_{[0, \pi]^3} \frac{d^3k}{w - \alpha_1\cos k_1 - \alpha_2\cos k_2 - \alpha_3\cos k_3}$ gives the return probability $R = 1 - 1/(w^\star W)$ at the band edge $w^\star = \alpha_1 + \alpha_2 + \alpha_3$ . Closed forms were known when one rate vanishes or when two or three are equal (Watson 1939, Joyce 1973, Delves and Joyce 2001), each an algebraic function times at most two complete elliptic integrals. The authors' summary states that for generic rates $wW$ satisfies a fifth-order differential equation whose group has connected part $\mathrm{SO}(5)$ , so no such elliptic formula exists, and gives $W$ instead through a minor of the period matrix of a genus-two curve built from the rates. At rates in the ratio $1:5:7$ the walker returns with probability about $0.4394$ , against $0.3405$ at equal rates. The result is one of roughly fourteen in a guest essay by Schwartz on Anthropic's blog describing work with Claude through his BootLoops harness.

Novelty check

Read the announcement and the authors' BootLoops summary on 2026-10-06 and searched the web for closed forms of the anisotropic simple cubic lattice Green's function. The solved cases are documented: Watson (1939) for equal rates at the band edge, Joyce (1973) for equal rates at every $w$ , Glasser and Zucker (1977) in gamma functions, and Delves and Joyce (2001) for two equal rates. Zucker's survey '70+ years of the Watson integrals' (J. Stat. Phys., 2011) called the three-rate case the final problem and judged its difficulties insuperable. No earlier closed form for three unequal rates was found. The paper is not yet public, so the claim could not be checked against its own literature review.

Caveats and known objections

Claimed because there is no paper yet, only a summary that lists it as in preparation, so the proofs of the fifth-order equation, of the absence of an elliptic formula and of the closed form itself are not public. What is public is the formula as runnable code, and agreement to many digits is a numerical check rather than a proof, as the summary itself says; one further statement there, an absolute theta identity, is left as a conjecture. The published script handles generic rates only: on the walls where two or three rates are equal, the cases the classical formulas already cover, it raised an error or, at one equal-rate point tried here, returned a wrong value. That is a limit of the script, not evidence against the theorem, which the summary says holds on the walls too. Autonomy is graded ai-assisted, the weaker defensible reading: the summary says only that the authors used AI tools interactively and that the code was written by Claude under Schwartz's supervision, the essay frames the problem as one Claude found through the harness, and the division of labour on the proofs is not documented. The essay is on Anthropic's blog, Schwartz is a visiting researcher there and the code is copyright Anthropic, although the essay states that BootLoops is not an Anthropic project. The essay's other results are not entered here and need separate grading as their manuscripts appear.

Independent checks

whataifound.org (numerical comparison, mpmath): Evaluated the published closed form and compared it with the classical Bessel-function form $\int_0^\infty e^{-wt} I_0(\alpha_1 t) I_0(\alpha_2 t) I_0(\alpha_3 t)\,dt$ , computed separately by quadrature that reproduces Watson's $0.5054620197$ at equal rates. At the summary's points $(1, 5, 7; 17)$ and $(5, 6, 9; 22)$ the two agree to 24 and 23 digits, and at four further generic rate triples, two of them close to the walls, to at least 15. On the walls the script fails. Numerical agreement only, not a check of the proof. · link ↗

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Entry history (1 event)

2026-10-06AddedEntered the registry graded Claimed and AI-assisted.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded claimed for verification and ai-assisted for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Closed form for the anisotropic Watson integral of the cubic lattice. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-01-anisotropic-watson-integral

BibTeX

@misc{whataifound-independent-2026-integral,
  title        = {Closed form for the anisotropic Watson integral of the cubic lattice},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Claimed. Autonomy: AI-assisted.},
  url          = {https://whataifound.org/finding/2026-10-01-anisotropic-watson-integral}
}

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