trureturing — Discovering truth

bibkey: rivin2026permanents authors: Igor Rivin year: 2026 title: “Permanents of matrix ensembles: computation, distribution, and geometry” doi: null url: https://arxiv.org/abs/2602.10141v3 claim: “Open Problems 2 and 3: the midpoint formula and a closed form for the universal function.” strata_touched:

• D5/S3/Combinatorics/Permanental/CycleGeodesic/GaudinPermanent
• D5/S3/Combinatorics/Permanental/CycleGeodesic/CycleGeodesicProduct
• D5/S3/Combinatorics/Permanental/CycleGeodesic/CycleGeodesicMidpoint
• D5/S3/Combinatorics/Permanental/CycleGeodesic/CycleGeodesicScaling license: citation-only triage: anchor

Verified locator

Source: https://arxiv.org/abs/2602.10141v3

Source statements

Page 19, Section 8.4, Open Problem 2:

Prove the midpoint formula $\mathrm{perm}(\gamma (\frac{1}{2}))=(-1{)}^{(n-1)/2}\cdot 2e^{-n}(1+\frac{1}{3n}+O(n^{-2}))$.

Page 19, Section 8.4, Open Problem 3:

Find a closed form for the universal function $f(t)$.

Page 7, Section 4.1:

The geodesic $\gamma (t)$ is therefore a circulant matrix with explicit entries

$$\gamma (t{)}_{jl}=\frac{e^{2\pi it}-1}{n(e^{2\pi i(l-j+t)/n}-1)}.$$

Page 7, Observation 4:

Define $f(t)=-\frac{1}{n}\ln |\mathrm{perm}(\gamma (t)).|$. Then $f(t)$ converges to a universal function of $t$ as $n\to \infty$, with the properties:

The printed full stop inside the absolute value is a typographical artifact. The logarithm is taken of the complex norm of the literal permanent. The matrix indices are zero-based elements of Fin n; they are cast to real numbers before subtraction.

Page 8, Observation 5 specifies the odd-dimensional midpoint expansion and the even-dimensional permanent zero. At the midpoint the scaling limit is therefore taken along odd dimensions. The all-dimension nonmidpoint limit is on $0<t<1$, $t\ne 1/2$.

Closed form and range

With $\delta =\min (t,1-t)$, the universal function on the interior is $1-\pi \delta \cot (\pi \delta )$. Reflection symmetry follows from the definition of $\delta$; its continuous extension is zero at the endpoints, its midpoint value is one, and its onset is ${\pi }^2{t}^2/3$. The formal settling statement proves the interior rate limit and the odd-dimensional midpoint limit; those other analytic consequences are follow-up deductions rather than additional Lean assertions in the settling module.

Literature scope

The exact finite product specializes Han’s Scott identity. The connection of that identity to the literal cycle-geodesic matrix, with a second-order Stirling estimate and logarithmic integral, supplies the two answers. The bounded title, identifier and phrase searches recorded in issue #13612 report no subsequent explicit settlement; this is a bounded literature finding, not a global novelty certificate.