Novaes's strong-absorption cumulant conjecture

M. Novaes, Time delay statistics for chaotic cavities with absorption, arXiv:2305.18235v1, Section 5, p. 7, states “we conjecture that, for n > 1,” followed by Eq. (58):

$$ (-1)^n k_n(\gamma) =\frac{(n-1)!}{M^{2n-2}\gamma^{2n}} -\frac{(2n-1)n(n-1)!}{M^{2n-2}\gamma^{2n+1}} +O(\gamma^{-2n-2}). $$

Here $M\geq1$ is an integer channel number and $n\geq2$ is an integer cumulant order. The symmetry class is $\beta=2$, with perfect coupling and uniform absorption. The journal publication is Journal of Statistical Physics 190, 182 (2023), DOI 10.1007/s10955-023-03183-7; its Eq. (65) is the same conjecture.

Statement echo

| Source clause | Formal counterpart | | --- | --- | | Every positive integer channel number | claim quantifies every natural M with 1 ≤ M. | | Every cumulant order greater than one | Every natural n with 2 ≤ n. | | Grabsch's inverse-rate law at $\beta=2$, $N=M$ | Foundations.weight is the literal joint density below; Foundations.law divides its measure by total mass. | | Wigner time delay in dwell-time units | Foundations.tauW is $M^{-1}\sum_i(\gamma+x_i)^{-1}$. | | The $n$th cumulant | Foundations.kappa is the $n$th derivative at zero of the observable's cumulant generating function. | | Both coefficients of Eq. (58), including $(-1)^n$ | The literal unsimplified factorial, channel and absorption powers in claim. | | $O(\gamma^{-2n-2})$ as $\gamma\to\infty$ | There exist real $C,\gamma_0$ with $\gamma_0>0$, and the absolute error is at most $C/\gamma^{2n+2}$ for every $\gamma\geq\gamma_0$. Constants may depend on $(M,n)$. |

Grabsch, arXiv:1909.01002v2, Eqs. (3)–(4), supplies the model law. Writing the inverse delay rates as $\Gamma_i=\gamma+x_i$, its unnormalized weight on $x_i\geq0$ and $0\leq y_j\leq\gamma$ is

$$ e^{-M(\sum_i x_i+\sum_j y_j)}\Delta(x)^2\Delta(y)^2 \prod_{i,j}(\gamma+x_i-y_j). $$

The five conventions weight, law, tauW, kappa, claim are literal to #16825. The probability normalization and bounded observable are proved for this law. Identification of this law with the physical random-matrix model is the cited mathematical-model premise, rather than a theorem about a physical cavity's dynamics.

D5/S3/Quantum/Transport/StrongAbsorptionCumulants.result proves claim. The conclusion determines two coefficients for every finite channel number and cumulant order, with an eventual absolute remainder estimate.

This is the named external conjecture preregistered in #16825. No prior all-order proof or refutation was found in the screened sources: Novaes's later publications and Scholar list, arXiv:2604.13193, and the author-hosted journal text. The journal's Eq. (65) matches arXiv v1's Eq. (58). Literature completeness is ASSUMED-UNVERIFIED.

The normalized inverse-rate law is a polynomial perturbation of two Laguerre weights. Finite-product remainder estimates and exponentially small cutoff tails give its first-order density expansion. Homogeneity and polynomial integration by parts identify the ordinary and weighted Laguerre moment generating functions. The observable's power remainder then gives each rescaled moment to first order.

The triangular moment–cumulant recurrence transports this estimate by strong induction in the order. Affine cumulant rescaling yields the literal coefficients and the error exponent in Eq. (58). The supporting content witnesses are Foundations.pow_linear_remainder, LaguerreMoments.cumulant_expansion_of_moments, and Density.finiteProduct_remainder; they are on result's proof path.

The formal claim would fail if, for some $M\geq1,n\geq2$, the scaled absolute error were unbounded as $\gamma\to\infty$. A numerical discrepancy alone would require checking the law, units and cumulant convention. The theorem concerns the stated normalized law and these conventions.

The four mathematical modules and their Scribe mirrors are:

• D5/S3/Quantum/Transport/StrongAbsorption/Foundations.lean;
• D5/S3/Quantum/Transport/StrongAbsorption/LaguerreMoments.lean;
• D5/S3/Quantum/Transport/StrongAbsorption/Density.lean;
• D5/S3/Quantum/Transport/StrongAbsorptionCumulants.lean.

The settling theorem is D5.S3.Quantum.Transport.StrongAbsorptionCumulants.result. The axiom closure of every public declaration is contained in ${\texttt{propext},\texttt{Classical.choice},\texttt{Quot.sound}}$.

The four-slot escape audit for all new public theorems, including result, is unfinished under #17044. The missing evidence is faithful source-family reconstruction, its realization equivalence bridge, whole-family counter-intervention, per-role sensitivity and actual observational dependence. No declared_validated registration is asserted.

Experiment

The numerical check is pinned to experiments commit 58ac049c0008f37ced35f8e2920589b15c24b574. Command: python3 check.py in that directory; exit 0, ALL_OK. Script SHA-256: a539d2588b8f5d55e06d47d34f9dd5de046fa0327bb6e73b6c3d4999c0d4b82d.

The tested scope is exactly $M=1$, 50-digit quadrature, $\gamma\in{50,100,200,400,800}$, and orders $n=2,3,4,5$. At $\gamma=800$, the error in the signed cumulant times $\gamma^{2n+2}$ is respectively $30.782$, $302.426$, $2794.118$, $26688.311$. The last three changes decrease in each tested order. For $n=3$ the signed scaled error approaches $306$, corresponding to $-306\gamma^{-8}$ in $k_3$, the $M=1$ coefficient of Eq. (56). The second-moment error against Eq. (53), times $\gamma^6$, is $47.7022$ at $\gamma=800$. These are finite numerical readings, not a proof of convergence or of a general next coefficient.

What the settlement shows

| Target item | Status and evidence | | --- | --- | | Mechanism: Laguerre perturbation and exact moments | Kernel-checked: Density.law_polynomial_firstOrder, LaguerreMoments.laguerreH_near, LaguerreMoments.laguerreHw_near, and Foundations.observable_power_remainder supply the density, weighted moments and observable estimates used by result. | | Mechanism: propagation of the $O(\gamma^{-2n-2})$ error | Kernel-checked: LaguerreMoments.cumulant_expansion_of_moments uses strong induction through the triangular recurrence; StrongAbsorptionCumulants.result gives the final exponent. | | Scope: every $M\geq1,n\geq2$ | Kernel-checked: StrongAbsorptionCumulants.result; constants may depend on both parameters. | | Consequences for Eqs. (56)–(57) | Kernel-checked for their first two coefficients and the stated remainder only: specialize result to $n=3$ or $n=4$. The finer coefficients in those equations are outside this theorem. | | Next coefficient at $n=3,M=1$ | Computed in the experiment entry: the tested signed scaled errors approach the source's value $306$. No convergence theorem or exact next coefficient is proved here. | | General $\gamma^{-2n-2}$ coefficient | Open. The remainder estimate does not identify it. | | Uniformity in $M$ | Open. The quantifier order permits constants depending on $M$. | | Symmetry classes $\beta=1,4$ | Open. Their laws and coefficients are outside this statement. | | Experiment | Computed: the pinned script, exit 0, ALL_OK, with exactly the scope and readings in the experiment entry. | | Literature | Literature reading: the screened sources contain no prior all-order settlement; completeness is ASSUMED-UNVERIFIED. |

The decisive structure is a first-order moment estimate at every order, combined with a recurrence that is triangular in that order. The proof requires neither a uniform bound in $M$ nor the next asymptotic coefficient. The source's uses of the conjectured first two coefficients have the stated conditional formula discharged for this law; other source results gain no additional coefficients or symmetry-class coverage from this theorem.

Literature-search completeness and worldwide priority remain ASSUMED-UNVERIFIED. The model-law correspondence uses Grabsch's published law and Novaes's units. Numerical quadrature covers only the finite tested scope and supplies no uniform or unbounded conclusion. Escape registration is unfinished under #17044.