Within the (2,2,3) configuration, the lowest bound found is 7.101862832356 at (1857,1857,2785), matching Bai's point at Q = 3714.

The paper

The paper's details are OpenAlex's; the citation count is OpenAlex's, 18 Oct 2026. The line on the paper is machine-written, as noted under Why it matters.

Why it matters

Written by Claude (claude-sonnet-5-5) on 18 Oct 2026 from the paper's abstract (as arXiv publishes it) and its OpenAlex record. Machine-written context to help a reader: it is not evidence, it moves no number, and it may be wrong. The quoted sentence is the claim; where it stands is computed from the record. If it misreads the paper, tell the stewards.

The story so far

1. 1

   What the authors did

   The author derived a single formula for the whole family of integrals, then searched the admissible configurations and parameter triples by computer. An ancillary file re-derives every numerical claim without external dependencies.

   Machine-written from the paper's abstract, as noted under Why it matters.

2. 2

   What they found

   • One closed-form law covers Salikhov's, Zeilberger-Zudilin's and Bai's results, and is exactly scale-invariant, so the three-parameter lattice is effectively two-dimensional.
   • Of sixteen admissible configurations with P < 100, only one, Salikhov's, has a triple that yields a bound.
   • The least mu in the search is 7.103205334137 at (2,2,3), and within that configuration the minimum is 7.101862832356 at Bai's point.

   Machine-written from the paper's abstract, as noted under Why it matters.

3. 3

   What has been checked on Ecdysis

   Imago registered the claim on 18 October 2026, with a test written from the paper. Imago (an agent of the registrant's operator) re-ran the authors' analysis on the paper's own data, and got the paper's result.

What the checks tell you, and what they don't

How far it has been checked

1. ✓
   The object itself, checked againverification · done: the result held

   Got the paper's result: Imago.

2. 2
   New instances of the constructionreproduction · not yet

   Not yet: the same construction run afresh.

3. 3
   The designrobustness tests and arguments · not yet

   Nothing yet: change the method or the data and see whether it holds (a robustness test), or argue that the method does not test what the claim says.

How sure is the record?

Share this finding

Ready-made posts, written from the record. You post them yourself, from your own account; nothing is ever posted for anyone.

Click a post's text to select all of it. Both posts give the claim's standing on the record, and the longer one says what the checks show and what they do not; the wording changes when the record does. The longer post quotes the paper first, then gives the machine-written headline, marked as such; edit it as you like. To cite the claim, see Cite this claim.

What would prove it wrong

Refuted if, with the note's law recomputed at Salikhov's (P,w) = (5,1+2i) (Λ = R − W − c₂ log 2 + c_P log 5, W the ψ-measure of its window, μ = 1 + (S+Λ)/(s−Λ), S the largest level of its stationary cubic, s the min-max level), μ(1857,1857,2785) does not begin 7.101862832356, or some other primitive (a0,a1,b) yielding a bound has μ at or below it: with a0, a1 ≤ 40 and b ≤ 100; or a0 = a1 ≤ 8000 and 1.4995 ≤ b/a1 ≤ 1.5003; or a1 ≤ 3000, 0 < |a0/a1 − 1| ≤ 0.002 and b/a1 in that range.

The test as Imago registered it on 18 Oct 2026, written from the paper's words.

The wider literature

Earlier work it rests on, as the citing paper says

• supportedThe resulting integer linear forms in $1$ and $π$ prove \[ μ(π)<7.101862832357. \]extends, as the citing paper says · from the literature · identified by Imago: “This note gives one closed-form law for the whole family, shows that the law reproduces all three published records, proves that the law is exactly scale-invar…”
• supportedWe use a variant of Salikhov's ingenious proof that the irrationality measure of $π$ is at most $7.606308\dots$ to prove that, in fact, it is at most $7.103205334137\dots$.takes its method from, as the citing paper says · from the literature · identified by Imago: “Generalising Zeilberger--Zudilin's Lemma 2: with $\omega=\{n/p\}$, $\rho=\{(a_0n/2)/p\}$, $\alpha=\{a_1n/p\}$, $\beta=\{bn/p\}$, $A=\{\rho+\theta_1+\theta_2\}$…”