Rivin's cycle-geodesic midpoint formula

Igor Rivin, “Permanents of matrix ensembles: computation, distribution, and geometry”, arXiv:2602.10141v3, Section 8.4, Open Problem 2, states:

Prove the midpoint formula $\operatorname{perm}(\gamma(\frac12))=(-1)^{(n-1)/2}\cdot 2e^{-n}(1+\frac{1}{3n}+O(n^{-2}))$.

The literal Section 4.1 matrix is

$$\gamma_n(t)_{jl}=\frac{e^{2\pi it}-1}{n(e^{2\pi i(l-j+t)/n}-1)},\qquad j,l\in{0,\ldots,n-1}.$$

Observation 5 also states the even-dimensional zero clause. Dimension is positive: $n\ge1$. The empty permanent at $n=0$ is $1$.

D5/S3/Combinatorics/Permanental/CycleGeodesic/CycleGeodesicMidpoint.result2 proves that one $C\ge0$ satisfies

$$ \left|\frac{\operatorname{perm}\gamma_n(1/2)}{(-1)^{(n-1)/2}2e^{-n}} -1-\frac1{3n}\right|\le\frac C{n^2} $$

for every odd $n\ge1$, and proves $\operatorname{perm}\gamma_n(1/2)=0$ for every even $n\ge1$. The proof supplies $C=16$. Complex absolute value is encoded by the complex norm; the displayed scale uses natural subtraction and division only on positive odd dimensions.

Preregistration #13612 classifies both questions as Tier 1, quotes the published statements and gives all quantifiers. Its bounded title, identifier, phrase, Borchardt and product-formula searches report no explicit subsequent settlement. The exact finite product itself is credited to Han (2000), DOI 10.1016/S0024-3795(00)00035-5. The algebraic input is a formalization of the credited Gaudin identity, attested by Faribault–Schuricht (2012), DOI 10.1088/1751-8113/45/48/485202. Neither identity is claimed as a new literature result. Their application supplies the two preregistered answers.

GaudinPermanent.gaudin_permanent identifies the reciprocal-difference Cauchy permanent with a Gaudin determinant. Its proof differentiates Lagrange interpolation and uses a determinant polynomial. A weighted Vandermonde basis evaluates that determinant, yielding the literal cycle-geodesic product

$$ \operatorname{perm}\gamma_n(t)=n^{-n}\prod_{k=0}^{n-1}(n-k+kq), \qquad q=e^{2\pi it},\quad n\ge1,\quad 0<t<1. $$

At the midpoint the product reduces to $n^{-n}\prod_k(n-2k)$. For positive even $n$ one factor vanishes. For odd $n$, a factorial expression and the second-order Stirling estimate yield the uniform error bound. Away from the midpoint, the logarithmic rate is a left Riemann sum of $-\log|1-x+xe^{2\pi it}|$. Uniform continuity and evaluation of the quadratic-logarithm integral give $1-\pi\delta\cot(\pi\delta)$. The odd midpoint estimate supplies the remaining odd-dimensional limit.

A falsifier for Problem 2 would violate the uniform odd-dimensional remainder or the positive even-dimensional zero clause. A falsifier for Problem 3 would be an interior nonmidpoint parameter whose rate has a different limit, or failure of the odd midpoint limit. The kernel-checked conclusions exclude these for the literal matrix and their stated domains. Endpoint evaluation of the total Lean definitions and the empty permanent are outside those domains.

The four Lean modules and their Scribe mirrors are in D5/S3/Combinatorics/Permanental/CycleGeodesic/ and Blueprint/D5/S3/Combinatorics/Permanental/CycleGeodesic/. The settling declarations are CycleGeodesicMidpoint.result2 and CycleGeodesicScaling.result3; each settling Scribe node records its own OpenProblemResolutionClaim with resolution Proved. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$. No new axiom or sorry is used. Each settling module has admission_basis: open-problem-resolution (#13612; Proved). Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

• Proved: the Han product holds for every positive dimension and every interior parameter, by CycleGeodesicProduct.product_formula. Its Gaudin input is credited to Faribault–Schuricht, rather than asserted as a new identity. The product replaces an exponential permutation sum by $n$ explicit factors for this family.
• Proved: Open Problem 2 holds with the single constant $C=16$ for all positive odd dimensions; the permanent vanishes for all positive even dimensions, by CycleGeodesicMidpoint.result2. The positive-dimension convention is essential: at $n=0$ the empty permanent is $1$.
• Proved: Open Problem 3 has the interior closed form $1-\pi\delta\cot(\pi\delta)$ and odd midpoint limit $1$, by CycleGeodesicScaling.result3. All-dimension convergence excludes the midpoint, where the even permanent is zero. Endpoint value $0$ refers to the continuous extension of the interior expression: the total Lean function has universal 0 = universal 1 = 1, and result3 excludes both endpoints. Reflection symmetry, the continuous endpoint extension and the $\pi^2t^2/3$ onset remain open as separate Lean corollaries.
• Computed: the next normalized midpoint coefficient approaches $1/18$ in the five tested dimensions below. The full midpoint asymptotic series and even this next coefficient as a uniform theorem remain open; finite numerical evidence is not an asymptotic proof.
• Open: the DFT-geodesic prime-versus-composite dichotomy, Rivin's Open Problem 4, is not addressed. The two cycle-geodesic conclusions give no theorem about that distinct endpoint family.
• Proved: the exact product and its Stirling/Riemann consequences supply both answers without the source's suggested Ryser saddle-point analysis. Formalization of that alternative route remains open. The source's cycle-midpoint and interior-scaling observations become theorems in the stated domains; its other ensemble-distribution questions and Open Problem 4 remain open.

Finite next-coefficient reading

At 80 decimal digits, evaluate

$$ n^2\left(\exp(L_n)-1-\frac1{3n}\right),\qquad L_n=2\log\Gamma(n+1)-(n-1)\log2 -2\log\Gamma((n+1)/2)-(n+1)\log n+n-\log2. $$

The tested scope is exactly $n\in{101,201,401,1001,2001}$. These are numerical readings; no assertion about all odd $n$ follows.

| $n$ | Computed coefficient | | --- | --- | | 101 | 0.0551752740022157666346222726818 | | 201 | 0.0553648015644500096126274046435 | | 401 | 0.0554600269625193626369300240258 | | 1001 | 0.0555173079603667562093490438847 | | 2001 | 0.0555364257507125260616290303381 |

Command: python3 next-coefficient.py (exit $0$); SHA-256 7ac75ef7ba32af85368f8d7a6a2b6aa22e3e14484a1dd3990e88f917d81a1f35. Save the following exact source as next-coefficient.py; its dependency is mpmath.

import mpmath as mp
mp.mp.dps = 80
for n in (101, 201, 401, 1001, 2001):
    log_ratio = (2 * mp.loggamma(n + 1) - (n - 1) * mp.log(2)
                 - 2 * mp.loggamma(mp.mpf(n + 1) / 2)
                 - (n + 1) * mp.log(n) + n - mp.log(2))
    coefficient = n**2 * (mp.exp(log_ratio) - 1 - mp.mpf(1) / (3 * n))
    print(n, mp.nstr(coefficient, 30))

The literature non-settlement reading is bounded to the searches in #13612, not an exhaustive priority certificate. The next-coefficient calculation is numerical and finite. The full midpoint series, the three additional analytic corollaries as Lean declarations, the alternative saddle-point route, and Open Problem 4 remain open. The source's endpoint properties refer to continuous extension, and the dimension convention is positive.