Papers

For some discrete parameters, multivariate (Dunkl-)Bessel processes on Weyl chambers $C$ are projections of Brownian motions on Euclidean spaces $V$; the most prominent examples are Dyson Brownian motions. More generally, the projections of Brownian motions on $V$ with drift are Bessel processes with drifts on $C$, where again the associated transition densities can be described in terms of multivariate Bessel functions. Moreover, Dunkl processes with drift can be defined in an analogous way. In this paper we prove central limit theorems for these Bessel and Dunkl processes with drift for regular drift vectors, and arbitrary root systems and multiplicities. These results improve recent results of the author.

Let $q$ be an odd prime power. We use complements of unions of subsets of $\mathbb F_{q^2}$ as locator sets and establish a sufficient condition under which a generalized Reed–Solomon (GRS) code is Hermitian self-orthogonal. Using cosets of multiplicative subgroups and sets with prescribed trace or norm values, we construct five families of Hermitian self-orthogonal GRS codes over $\mathbb F_{q^2}$. The Hermitian construction then yields five corresponding families of $q$-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than $q/2+1$. By comparing codes of the same length, we give conditions under which our codes have strictly larger minimum distances than those obtainable from several previously known constructions based on trace maps, linear transformations, and cosets of multiplicative subgroups, either directly or via the propagation rule. We further show that such improvements occur for infinitely many values of $q$.

We determine $2,540$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range $[2,3]$, we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.

Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g_{L^2}}B_{L^2}(p,R) =\frac{2^{4g-3}\pi^{9g-9}}{(3g-3)!}\,R^{6g-6}+o(R^{6g-6}) $$ for every $p\in\M$. We also determine the leading asymptotics of Hamiltonian sublevel volumes and exponentially weighted volumes. The proof combines homogeneity of the hyperkähler volume form with metric asymptotics on the regular Hitchin locus. Symplectic reduction and the Prym polarization evaluate the coefficient, which is independently recovered by equivariant localization.

The heavy top is a classic example of Euler-Poincare reduction on a semidirect product group, demonstrating how an advected quantity-the direction of gravity-couples to rotational motion. While the description of relativistic spinning particles via Lie group configurations is well established in Hamiltonian (symplectic) mechanics, we demonstrate that an explicit Lagrangian Euler-Poincare reduction provides a direct, transparent parallel to the heavy top. Specifically, we show that the free relativistic spinning particle on the Poincare group is the precise Lorentzian analog of this system, with spacetime position being the advected quantity. This geometric unification not only illuminates the underlying symmetry of both systems but also clarifies why the introduction of external electromagnetic fields leads to a partial reduction. Finally, we establish that the transition to left-trivialized coordinates acts as a geometric selection mechanism, naturally favoring a class of interaction Lagrangians where spin-field coupling manifests as a dynamical redefinition of the particle's mass.

We study the quenched pressure of the nearest-neighbor Edwards–Anderson Ising model with free and periodic boundary conditions. First, we prove that the infinite-volume pressure is $\beta d$-Lipschitz in the coupling law for the $1$-Wasserstein distance, yielding quantitative thermodynamic limits for spatially inhomogeneous disorder. Second, for periodic volumes, we prove that almost-sure convergence under every joint coupling of the finite-volume disorder arrays is equivalent to complete convergence of the one-volume pressure laws. Third, we prove that $\mathbb{E}\vert{}J\vert{}^{1+1/d}<\infty$ guarantees this universal-coupling conclusion. We further show that this exponent is optimal among uniform power-moment assumptions: for every $1\leq q<1+1/d$, there is a centered symmetric law with finite $q$-th moment for which canonical nested volumes converge almost surely, whereas independently resampled volumes with the same fixed-volume marginals converge in probability but not almost surely. Finally, in dimension one, we prove that the first-moment condition is also necessary for a finite limiting pressure.

Let $D=2^n$ and equip $\operatorname{PU}(D)$ with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and $D^2$ in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by $D$, converges to $\pi/\sqrt3$ in probability and in $L^p$ for every $1\le p<\infty$. Quantitatively, it lies within $O(D^{-1/8}(\log D)^{1/2})$ of this limit outside a set of Haar measure at most $\exp\{-\Omega(D^{7/4}\log D)\}$. For each fixed $0<x<\pi/\sqrt3$, the ball of radius $xD$ has Haar measure $\exp\{-\Theta_x(D^2)\}$; for $x>\pi/\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\uparrow\pi/\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(\pi^2/3-x^2)^2/(16\zeta(3))$. The small-ball upper bound follows from a comparison of Jacobi determinants, obtained by rescaling the linearized geodesic equations and applying Kato transport. Weyl integration reduces the remaining integral to an Abel-regularized logarithmic-energy estimate on the circle. A centered principal logarithm and concentration of the circular unitary ensemble eigenangle second moment give the distance upper bound.

We disprove both parts of Conjecture 1.8 of Gabrielov, Novikov, and Shapiro (GNS). For the Coulomb potential, a generic rational configuration of 24 unit charges in $\mathbb{R}^3$ has at least 18 nondegenerate critical points of Morse index one but exactly 14 effective Voronoi one-cells. A separate proper-line construction has at least two nondegenerate minima but only one relatively effective zero-dimensional cell intersection. For each positive subspace dimension, product suspension gives a proper-subspace counterexample.

A symmetric representation of Minkowski spacetime is constructed by combining a normalized fourth-order Hadamard transformation with an imaginary temporal coordinate. The resulting complex coordinates $q^a$ distribute time and space equally among four components and transform the Minkowski metric into a Euclidean-form complex bilinear metric. The four Hadamard spatial sign vectors form a regular tetrahedron and appear simultaneously in the transformed Dirac and Maxwell operators. The Dirac matrices satisfy a Euclidean-form Clifford algebra and contain tetrahedral combinations of Pauli matrices, whereas the Riemann–Silberstein Maxwell matrices realize the corresponding spin-1 structure. A gauge-invariant QED Lagrangian is formulated in the same coordinates. In Fourier space, the Dirac and Maxwell systems are governed by the common invariant $\kappa_a\kappa^a$, while the complete Maxwell operator admits an exact rectangular Clifford-like factorization. The formulation is further extended to the weak-field Einstein–Dirac–Maxwell system. In harmonic gauge, the linearized gravitational field is governed by the same scalar wave operator $\Box_q$, while gravitational corrections to the Dirac and Maxwell equations enter through the perturbed tetrad, spin connection, and curvature. This common principal-operator structure suggests shared spectral kernels and scalar-wave preconditioning for coupled field solvers.

We study the structure of Gibbs states in weakly perturbed interacting fermionic systems. First, for a sparse Hamiltonian $H=H_0+V$ with a quadratic term $H_0$ and a non-quadratic perturbation $V$ of scale $\epsilon$, we show that the Gibbs state $\rho_{\beta}$ decomposes into a convex combination of Gaussian states whenever the inverse temperature satisfies $\beta \le O(\log(1/\epsilon))$. Moreover, we prove that this bound is asymptotically tight by establishing that $\beta \le \Theta(\log(1/\epsilon))$ is necessary for certain sparse Hamiltonians. This general framework applies directly to the weak-coupling (small-$\vert{}U\vert{}$) regime of the Fermi–Hubbard model with hopping $t$ and on-site interaction $U$ on any graph of maximum degree $D$. Complementarily, in the strong-coupling (small-$\vert{}t\vert{}$) regime, we show that the Gibbs state remains convex-Gaussian up to $\beta \le O\big(\vert{}U\vert{}^{-1}\log(\vert{}U\vert{}/(D\vert{}t\vert{}))\big)$, revealing a mechanism for convex-Gaussianity distinct from the weak-coupling setting.

Understanding how thermal fluctuations modify or suppress quantum-enhanced sensing is a central problem in quantum metrology. Closely related questions arise in quantum thermodynamics, particularly regarding the relation between the work induced by a signal and the information acquired by a sensor. Here, we establish general relations among the sensitivity of thermal quantum states, as quantified by different quantum Fisher information metrics, their temperature dependence, and the work induced by a unitary signal. We further show that, at high temperature, all monotone QFI metrics coincide to leading order with a universal quantity that can be expressed as the variance of a simple observable and itself defines a lesser-known QFI metric. This emergent uniqueness is reminiscent of the uniqueness of classical Fisher information in information geometry. In continuous-variable systems, the infinite-temperature limit is finite and generically non-zero for quadratic Gaussian signals, while it diverges for higher-degree signals. Remarkably, for any purely quadratic signal generator, the infinite-temperature QFI is at least twice the zero-temperature SLD QFI or, equivalently, at least eight times the ground-state variance of the generator. Consequently, the Cramér–Rao lower bound on the estimator variance is reduced by at least a factor of two. We illustrate these results using harmonic chains of bosonic modes relevant to trapped-ion platforms and quantum field-theoretic models.

We study the linearized Boltzmann collision operator for three-dimensional hard spheres and reorganize the complete operator so that its angular and radial structures are explicit before any Sonine truncation. From the Carleman representation, the collision integral becomes a superposition of orthogonal translations; in Hermite/Fock and Bargmann form these generate a single analytic coherent-state kernel for all Hermite collision brackets. Rotational covariance separates fixed angular-momentum sectors, each reducing exactly to a one-variable radial Bargmann space with an $\mathfrak{su}(1,1)$ structure. The radial operator is $\mathfrak A_\ell=\kappa\pi G(\mathcal J_\ell)-\mathfrak C_\ell$, where $\mathcal J_\ell$ is an explicit Laguerre–Jacobi operator and $\mathfrak C_\ell$ is compact. A unitary intertwiner identifies this realization with the classical Burnett/Sonine coordinate, recovering the Burnett matrix and standard stress and heat-flux corrections. For the untruncated operator, the compact gain gives only compact corrections, while the loss sets the common essential spectrum $[\nu_{\min},\infty)$. Radially squeezed $\mathfrak{su}(1,1)$ coherent packets yield threshold Weyl sequences in every fixed angular sector. With $\ell\to\infty$ and squeezing scaled as $\ell^{-1}$, they furnish full-space Weyl sequences throughout the essential band interior, with $\langle j\rangle\sim\ell^2$, revealing quadratic continuum corridors in the $(\ell,j)$ plane. At fixed $\ell$ and $j\to\infty$, the matrix tail approaches a universal Toeplitz operator whose zero at $\theta=\pi$ is the local signature of the same threshold escape. Thus the Fock reformulation provides exact radialization and constructive control of the global spectral geometry.

In this paper we prove that simple Finsler manifolds are conformally stable. Given a simple Finsler manifold and a class of conformal factors, we characterize the singularity of their induced boundary distance functions which allows us to define an appropiate $H^2$ norm. We then obtain a stability estimate with respect to this Sobolev norm and the $L^2$ norm on the class of the conformal factors. To prove the main theorems, we adapt the classical integration by parts technique employed by Mukhometov \cite{Muhometov} to the Finsler setting.

We study sample covariance matrices $K = \frac{1}{N} \sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^* \in \mathbb{R}^{n \times n}$ in the proportional regime $n \asymp N$. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance $\mathbb{E} \mathbf{x}_i \mathbf{x}_i^* = \Sigma$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \mathbf{x}_i^* A \mathbf{x}_i - \mathrm{Tr} \Sigma A | \prec \| A \|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\Sigma$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta:= \mathrm{Im}\, z \geq N^{-1 + \tau}$, \[ \big| \langle \mathbf{u} , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)\Sigma \big)^{-1} \big) \mathbf{v} \> \big| \prec \sqrt{\frac{\mathrm{Im}\, \widetilde m_0 (z)}{N\eta}} + \frac{1}{N\eta} \] for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where $\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.

Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations \([r^s]\) as determinants of \(r\times r\) and \(s\times s\) matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda \(\tau\)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.

Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras ($\mathbb Z_2$-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, i.e., it is stably bounded-spread isomorphic to a tensor product fermionic algebra.

In this paper, we study non-perturbative vortex partition functions of four-dimensional N=1 supersymmetric gauge theories on the space-time geometry $T^2 \times D^2$, and we propose that these partition functions offer a notion of quantum elliptic cohomology. For particular U(1)-gauge theories we calculate these partition functions explicitly by applying equivariant localization methods to Handsaw quiver varieties that realize for this particular class of U(1)-gauge theories the moduli spaces of the non-perturbative vortex sectors. The determined vortex partition functions are annihilated by difference operators, which are interpreted as Ward identities among N=(0,2) BPS surface defects. Compared to lower dimensional gauge theories with four supercharges, anomalies play an essential role for a consistent formulation of the four-dimensional partition functions. In our proposal towards a mathematical formulation of the vortex partition functions in terms of equivariant elliptic cohomology, the gauge theory anomalies relate to geometric properties of the Thom sheaves corresponding to the relevant quasimap moduli spaces. Motivated by the explicit computations we reflect on the existence of a `virtual structure sheaf' on the moduli space of quasimaps for a general mathematical theory of quantum elliptic cohomology.

Higher genus Gromov-Witten invariants of Calabi-Yau threefolds are encoded in a generating function which is an asymptotic series in a formal parameter $\lambda$. Using resurgence, analytic functions in this formal parameter were uncovered. In this paper we focus on the resolved conifold and study the enumerative meaning of the strong-coupling asymptotic expansion, in powers of $ 1/\lambda $, of the resurgent analytic functions. We show that this expansion contains both a closed curve-counting contribution in dual variables and a contribution governed by relative Gromov-Witten invariants, naturally interpreted in logarithmic geometry.

The McVittie metric describes a self-gravitating compact object embedded in an expanding universe. It inherits two singularities: a curvature singularity at $r \to 0$ and a cosmological singularity at $a(t) \to 0$. Motivated by the construction of black-bounce geometries, we regularize both of these singularities and propose a regularized McVittie metric. The central singularity is regularized by replacing $r \to \sqrt{r^{2}+b^{2}}$, leading to a geometry that interpolates between a cosmological black hole, a black bounce, and a traversable wormhole. Similarly, the cosmological singularity is regularized by introducing a non-vanishing scale factor $a \rightarrow \sqrt{a^{2} + a_b^{2}}$. The resulting spacetime is supported by an effective imperfect fluid with finite anisotropic stress. We investigate the formation of the apparent horizon, analyze the circular geodesic structure, and argue that the geometry can also be interpreted as a conformally evolving Morris-Thorne wormhole embedded in a regular cosmological background.

This is the first part of a program to construct hyperholomorphic families of boundary conditions for the Kapustin–Witten B-twist of the Langlands QFT, otherwise known as (BBB)-branes. We define the category of quasi-algebraic sheaves over the Deligne moduli stack, which serves as an analog of the twistor space of Hitchin's moduli stack. This allow us to construct a representation of a simplified version of the Moore–Tachikawa category which is motivated by the relative Langlands program in the sense of Ben-Zvi–Sakellaridis–Venkatesh.

We study one-dimensional spinless bosons with semirelativistic spinless-Salpeter dispersion and attractive pairwise contact interactions. Because the dispersion becomes linear at large momentum, the contact interaction is marginal by power counting and produces a logarithmic ultraviolet divergence. We construct the renormalized many-body theory using an enlarged Fock space and a Schur-complement representation of the resolvent, eliminating the bare coupling in favor of the physical zero-total-momentum two-body bound-state energy. The resulting cutoff-independent resolvent defines a self-adjoint Hamiltonian in each fixed particle-number sector. We treat the two-body problem explicitly and show, in the norm-resolvent sense, that the nonrelativistic limit reproduces the attractive Lieb–Liniger Hamiltonian. We also formulate a mean-field approximation directly within the renormalized theory. In the massless and deeply bound large-particle-number regimes, it predicts an exponentially increasing binding scale whose exponent is determined by a one-dimensional variational problem.

Nonlinear wave equations exhibit rich interplay among dispersion, nonlinearity, coupling, dissipation, and external forcing, producing diverse coherent and complex wave structures. Although exact travelling-wave solutions provide analytical benchmarks, their construction alone does not reveal underlying phase-space, bifurcations, stability, or responses to perturbations. This review presents a reduction-based methodological framework connecting nonlinear partial differential equations(PDEs) to reduced dynamical systems, invariant phase-space structures, exact waveform reconstruction, stability analysis, and verification in PDEs. Its applicability and limitations are examined across Schrödinger-type, coupled, nonparaxial, dissipative, magnetic, shallow-water, and fractional wave equations. Particular emphasis is placed on correspondence between analytical waveforms and invariant orbits. Equilibria, periodic trajectories, homoclinic orbits, and heteroclinic connections geometrically represent constant, periodic, localized, and front-like structures, respectively. This review distinguishes existence and stability of invariant structures from onset of chaos, emphasizing complementary diagnostics rather than reliance on phase-portraits or finite-time indicators alone. Major methodological gaps include incomplete parameter-space characterization, weak correspondence between reduced models and full-PDE dynamics, ambiguities in generalized and fractional formulations, limited robustness analysis, inadequate numerical reproducibility, and insufficient links to experimental observables. The framework prioritizes physical admissibility, stability, robustness, and predictive relevance over generation of additional formal solutions. It supports reliable nonlinear wave prediction, stability assessment, control, and system design in fluid, optical, plasma, and other nonlinear physical systems.

We show that the capacity of a quantum channel demarcates a phase transition: while reliable transmission below capacity is always possible, any attempt to transmit information above it fails catastrophically. Specifically, we prove exponential strong converse theorems for unassisted quantum and classical communication over arbitrary finite-dimensional memoryless quantum channels. At rates beyond the respective capacity, the entanglement-generation fidelity and the success probability for classical communication decay exponentially with the number of channel uses. This rules out transmission above capacity even when one tolerates arbitrarily large errors. Our proof follows the classical Arimoto strategy, augmented by a crucial new ingredient: integral representations of Rényi information measures that lead to asymptotic continuity bounds for Rényi capacities.

We prove that, for any given sufficiently regular one-particle density, there exists a unique external potential for which the solution to the many-body Schrödinger equation has this prescribed density. This implies in particular that one can exactly reproduce the time-dependent density of an interacting system using non-interacting electrons, which is at the heart of time-dependent density functional theory. Our result covers the Coulomb interaction and densities arising from extended nuclei.

We provide the first fully rigorous justification of time-dependent density functional theory in the continuum: Given a time-dependent density, we prove that there exists an external potential, unique up to a time-dependent constant, such that the corresponding Schrödinger equation reproduces the given density. This potential can be obtained with an iteration scheme. Our argument covers Coulomb interactions and does not need any Taylor expansion in time. It instead requires analyticity in space for the density and potential, which is compatible with extended nuclei.

Ding-Song-Sun proved the remarkable correlation inequality that the truncated two-point correlation function of ferromagnetic Ising models with arbitrary mixed-sign external field is largest when the field vanishes. This inequality has various important consequences such as log-Sobolev inequalities for Ising and $\varphi^4$ models up to and at the critical point. We extend the DSS inequality to the class of single spin measures satisfying the GHS condition. This is an important ingredient in our proof of the log-Sobolev inequality for the massless sinh-Gordon model on $\mathbb{R}^2$ in a forthcoming article.

We study the dynamics of a many-body quantum system strongly interacting with a bosonic reservoir. The coupling is given by a potential operator for the system and it is linear in the field. We show that in the limit of infinite coupling strength the system undergoes instantaneous spatial decoherence. To resolve the decoherence process in time we consider large coupling strengths $\lambda$ and short times scaling as $t\propto \lambda^{-\alpha}$, with $\alpha\ge0$, in the limit $\lambda\rightarrow\infty$. On short time scales $\alpha>1$ the dynamics is trivial while on longer ones $0\le\alpha<1$ the decoherence is instantaneous. We show that the decoherence process is resolved exactly for $\alpha=1$, defining the fine-grained time scale $\tau=\lambda t$. We construct an approximate effective evolution map of the many-body system with controlled error estimates. Generically, the effective dynamics is markovian, but not given by a dynamical semigroup. As a physical application we give a rigorous description of the phenomenon of localization of macroscopic quantum objects in position space.

We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both the level sets of the Gaussian free field and the vacant set of random interlacements undergo non-trivial percolation phase transitions at deterministic critical levels. A key aspect of the proof is the development of certain quenched controls over tree embeddings, permitting the application of a static renormalization scheme in the presence of spatial irregularities, which may be of independent interest.

We analyze the covariant phase space of Klusoň's canonical Carroll non-BPS D$_p$-brane actions. The Carroll limit is formulated as a contraction of the canonical phase space: the canonical one-form is invariant under the scaling of conjugate pairs, while the leading Hamiltonian constraint differs between the electric-like and magnetic-like sectors. This difference controls the weak closure of the electric-like constraint algebra and the stronger closure of the magnetic-like Hamiltonian brackets. We separate the generic non-BPS sector, which carries $D-1$ local phase-space degrees of freedom, from the tachyon-vacuum sector, which carries $D-2$ only after imposing second-class background conditions. We also identify the rank condition for the generic electric-like sector and its strengthening in the vacuum sector. Finally, we show that the Carroll–Weyl $\chi$ transformation, the matter-sector analogue of the null-string Carroll–Weyl symmetry, preserves the symplectic form and admits a charge, but is obstructed by the constraints and cannot be promoted to a first-class gauge generator except in a highly restricted global vacuum sector. The obstruction is algebraic in $\chi$, so no restriction of its spacetime profile can remove it. This contrasts with null string theory, where a restricted $\chi$-symmetry can be completed to an additional first-class constraint.

We establish a one-to-one correspondence between geodesics associated with the one-parameter family of $\alpha$-connections on the Gaussian statistical manifold and a class of continuous stochastic processes characterized by a time-independent noise intensity. We demonstrate that geodesics in expectation parameters naturally classify into three distinct geometric categories, among which the Boundary-connecting class allows us to construct an explicit linear stochastic realization with constant diffusion, representing a generalized bridge process. This result demonstrates how continuous stochastic processes within this Gaussian class can be extended along geometric curves. Under appropriate operational limits, this generalized bridge process reduces to fundamental stochastic dynamics, either Ornstein-Uhlenbeck (OU) relaxation or free Brownian diffusion. Crucially, the physical restoring force governing the resulting OU relaxation directly determines the underlying connection parameter $\alpha$, providing a concrete physical observable to constrain the manifold geometry. Depending on the chosen affine connection representation, this restoring force can be attributed either to scalar curvature or purely to non-metricity, establishing a direct conceptual analogy with the Geometrical Trinity of Gravity. Furthermore, applying this framework to driven stochastic thermodynamics, we show that the work-minimizing optimal protocol in the slow-driving limit coincides precisely with an expectation geodesic of the statistical manifold with non-metricity equipped with $(g, {}^{(1/2)}\Gamma, {}^{(-1/2)}\Gamma)$, highlighting the active physical role of non-metricity in information geometry.