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

Abstract: 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.

• The paper proves that any finite signed BAR solution in the Harrison–Reiman class is a scalar multiple of the stationary measure.
• It employs a resolvent insertion mechanism with measure-level smoothing to overcome regularity challenges of SRBM test functions.
• It constructs infinite families of nontrivial solutions in the completely-𝒮 class, underscoring the necessity of nonsingular M-matrix conditions for uniqueness.

AI-Assisted Proof of the Signed BAR Uniqueness Conjecture and Its Structural Limitations

Introduction and Problem Statement

The signed BAR uniqueness problem concerns the characterization of invariant (steady-state) distributions of multidimensional semimartingale reflected Brownian motions (SRBMs) in orthants via the Basic Adjoint Relationship (BAR). Specifically, for a diffusion with reflection data of Harrison–Reiman class (nonsingular $M$-matrix reflections), the question is whether any finite signed measure solution to the BAR must be proportional to the unique stationary distribution. This represents a linearized version of the classical positive uniqueness theorem because allowance for cancellation between interior and boundary signed measures makes the analytic mechanism more subtle.

This problem has been open for over thirty-five years, as the classical BAR approach does not a priori exclude signed solutions that are not scalar multiples of the stationary distribution. Moreover, the question extends naturally to the more general completely-$\mathcal{S}$ reflection class, where, as demonstrated in this work, the uniqueness property can fail in the absence of further invertibility assumptions on the reflection matrix.

The main results are:

1. A full resolution of the signed BAR uniqueness problem for the Harrison–Reiman class with nonsingular $M$-matrix reflection: every finite signed BAR solution is a scalar multiple of the stationary law and boundary occupation measures.
2. A construction of explicit infinite-dimensional families of nontrivial signed BAR solutions (with zero interior mass) in the completely-$\mathcal{S}$ class whenever a singular proper principal block exists, demonstrating that the $M$-matrix hypothesis is crucial.

This work explicitly details the technical boundaries between the positive uniqueness regime and its structural obstruction, and provides a rigorous account of how AI collaboration shaped its proof development.

BAR Formalism and Main Theorem

Let $Z$ be an SRBM in $E=\mathbb{R}^d_+$, driven by drift $\mu$ and covariance $\Sigma$, with a reflection matrix $R$. The stationary distribution $\mathcal{S}$0 is characterized by, for all $\mathcal{S}$1,

$\mathcal{S}$2

where $\mathcal{S}$3 is the diffusion generator, the $\mathcal{S}$4 are oblique derivatives along the $\mathcal{S}$5th face, and $\mathcal{S}$6 are finite measures representing time occupation on the boundary.

A finite signed BAR tuple $\mathcal{S}$7 solves the same equation, but allows the $\mathcal{S}$8 and all $\mathcal{S}$9 to be general finite signed measures. The main result is:

Theorem:

Suppose $M$0 is a nonsingular $M$1-matrix and $M$2 (stability). Then any finite signed BAR tuple is a scalar multiple of the stationary solution: $M$3 for some $M$4.

Outline of Proof

Analytic Core: The Resolvent Insertion Mechanism

The heart of the analysis is the resolvent insertion identity: for any finite signed BAR tuple $M$5 and any $M$6,

$M$7

where $M$8.

If $M$9 were sufficiently regular (i.e., $\mathcal{S}$0 with $\mathcal{S}$1 on $\mathcal{S}$2), direct insertion into the BAR would yield the result. However, for SRBMs in orthants, the resolvent generally fails to be $\mathcal{S}$3 up to the boundary (see the appendix), a well-known technical obstacle.

The authors resolve this using a measure-level one-sided smoothing: the family $\mathcal{S}$4 defined by convolution strictly from the interior, with carefully constructed mollifiers, produces genuinely admissible BAR test functions. As $\mathcal{S}$5, $\mathcal{S}$6 in the sense required to pass to the limit inside the BAR equation, and, crucially, all boundary pairing terms vanish asymptotically with arbitrary finite signed boundary measures.

This reduction leverages two key algebraic facts:

• Invertibility of all principal blocks of $\mathcal{S}$7: This ensures that at any boundary stratum, the projection inherent to the pathwise derivative of the SRBM acts as a strict oblique normal, allowing boundary terms to be controlled.
• Directional differentiability and pathwise sensitivity (Lipshutz–Ramanan): The resolved derivative at the boundary is always tangent to the stratum under the hypotheses, so acts as an effective Neumann-type boundary condition after smoothing.

Uniqueness Follows by Laplace–Semigroup Analysis

Given the resolvent identity, integration against Laplace transforms and the boundedness/strong continuity of the SRBM’s Feller semigroup yield that $\mathcal{S}$8 must be invariant under the transition semigroup, and thus—in the presence of uniqueness and positive recurrence—must be proportional to $\mathcal{S}$9. The same logic, together with an induction on the face codimension (and invertibility of principal blocks), implies all boundary measures must be proportional as well.

Obstructions in the Completely-$M$0 Class

If some principal block $M$1 is singular, a boundary gauge construction produces nontrivial tangential derivatives along the corresponding stratum. Integration by parts on the stratum yields a nontrivial, signed, and centered interior measure $M$2 supported on the stratum $M$3. Extending $M$4 into the interior via the semigroup’s zero potential, and matching boundary occupation measures, one constructs nontrivial signed BAR solutions with zero total mass:

$M$5

By varying the supporting function $M$6 on the stratum, an explicit infinite-dimensional family is constructed, showing the failure of uniqueness.

The explicitly constructed family $M$7 in dimension three, for rational parameters satisfying certain positivity and invertibility conditions, demonstrates the effectiveness and transparency of the construction.

Numerical Results and Algebraic Claims

The proof is sharply structural, not relying on explicit formulas for the stationary measure (which are rarely available except in special cases), but rather on functional-analytic and operator-theoretic properties. The construction gives, for any smooth, compactly supported, nonnegative $M$8, the ability to check $M$9 for standard SRBM data.

In the obstruction, the parameter family for $Z$0 is chosen so that the off-diagonal entries are positive (hence ensure completely-$Z$1), but a singular $Z$2 block exists, which is the algebraic engine for the nonuniqueness phenomenon.

Implications and Future Directions

These results definitively settle the signed BAR uniqueness conjecture for the canonical Harrison–Reiman class. The necessity of the nonsingular $Z$3-matrix structure for signed uniqueness is sharp: the obstruction in the completely-$Z$4 class is unavoidable and gives strong evidence for the conjecture's natural boundary.

The analytic techniques—measure-level smoothing and the use of projected derivatives/factorization at the boundary—are likely extensible to other Skorokhod-type reflection-driven processes on polyhedral domains, as is the functional-analytic exact sequence that connects BAR solution spaces to well-posedness of the reflected generator.

The methodology marks a paradigm shift: complex measure-theoretic boundary analysis can be made tractable with current AI-assisted computation, organization, and prior literature suggestion (notably, directional differentiability theorems and homological algebraic perspectives were surfaced and explored via LLMs). However, full rigor enforcement and proof-level innovation remain strictly human-driven, as LLM systems are currently unable to genuinely orchestrate such analytic reductions without explicit supervision, as exemplified by the comprehensive "proof outline" developed iteratively and checked for mathematical validity.

Moving forward, one might seek similar analytic reductions for signed measure uniqueness in other stochastic networks, general reflected diffusions, or higher-dimensional processes with anisotropic reflection or singular boundary structure. Another avenue is the potential for invariance to fail for more general source terms in SRBMs with less structure, leading to infinite-dimensional signed solution spaces for more exotic processes.

Conclusion

This paper provides a comprehensive and definitive resolution to the signed BAR uniqueness conjecture for multidimensional SRBMs in the Harrison–Reiman class, and characterizes precisely when and how this uniqueness fails in more general completely-$Z$5 settings. At the core is a rigorous analytic mechanism—resolvent insertion—made accessible via one-sided smoothing and exploitation of the pathwise differentiability structure. The failure in the completely-$Z$6 class is traced to singular principal blocks, with explicit counterexamples and infinite-dimensional signed nullspaces constructed. The boundary of uniqueness for BAR characterizations is thus mapped precisely, with both theoretical reasoning and practical certificate construction. The work also makes clear the emerging role and limitations of AI in mathematical discovery: essential for organizing, exploring, and refining, but still dependent on human expertise for foundational mathematical validation.