An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison--Reiman Class and a Completely-$\mathcal{S}$ Class Obstruction

For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison--Reiman data with a nonsingular $M$-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular $M$-matrix assumption is structural. In the larger completely-$\mathcal{S}$ class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai--Dieker question has a positive answer in the Harrison--Reiman $M$-matrix class and a negative answer in a natural completely-$\mathcal{S}$ extension.

A Stationary-Distribution Theory for Triplet-Based Plateau Search in Random Forest Ensemble-Size Selection

The number of trees is a central computational parameter in Random Forests: increasing it reduces finite-ensemble variability but increases training and prediction cost. Plateau-based tuning adapts this parameter through local comparisons of out-of-bag scores at a geometric triplet of tree counts. After the remaining hyperparameters have stabilized, however, the central triplet point need not converge to a deterministic value; instead, it fluctuates around a stationary regime. This paper develops a stationary-distribution theory for this process. The central ensemble size $B_t$ is modeled as a birth-death Markov chain on a geometric grid, and its stationary distribution is derived through local balance. Under a leading centered folded-normal approximation, equilibrium equations are obtained for the original update rule and a symmetric modified variant, implying that the stationary center $B_*=O(\varepsilon^{-2})$ as $\varepsilon\downarrow 0$. The stationary spread is also characterized. A local Gaussian approximation and a Fokker-Planck interpretation give grid-level variance constants. After conversion to the ensemble-size scale, $σ_{B,*}=O(\varepsilon^{-2})$, while the variance is $O(\varepsilon^{-4})$. The leading relative spread is independent of $\varepsilon$ and controlled by the scale factor and update rule. These results interpret plateau-based Random Forest tuning as a stochastic process rather than a deterministic stopping rule.

Unbiased Derivative Estimation for Stationary Mean of Parameterized Markov chains

We propose a new approach to unbiased estimation of the gradients of the stationary means associated with parametrized families of Markov chains. Our estimators are particularly efficient when the Markov chains have slow mixing rate. Our approach does not require a specific parametrization except for an oracle to evaluate the transition density and its gradient at a given data point without any additional knowledge about the density function itself. It makes our estimator suitable for parametrizations associated with neural networks. The estimator can potentially achieve large improvement in terms of efficiency. Numerical experiments confirm the good performance predicted by the theory.