Doubly Free-Boundary Macdonald Processes: Reflection Identities and Jack Asymptotics

We introduce a doubly free-boundary Macdonald process on rail-yard interlacings and develop a reflection calculus for its observables. Boundary Cauchy--Littlewood identities, combined with Negu\c t operators, yield exact multipoint contour formulas for arbitrary \(L/R\) words. Under the Jack scaling \[
q=t^\alpha,\qquad t=e^{-n\beta\epsilon}, \] and piecewise-periodic data, these formulas imply a Laplace-transform law of large numbers and a weak slope-measure limit shape at \(L\)-type columns.
For arbitrary piecewise-periodic \(L/R\) backgrounds and finitely many \(L\)-type marked columns, under the stated contour, branch, and normal-convergence hypotheses, the centered height-Laplace observables converge jointly to a Gaussian vector. Its covariance exhibits a boundary--deformation separation: the Jack parameter and the microscopic rail-yard data enter through the one-point spectral factors and the normalization, whereas the two-point interaction is the logarithmic derivative of an annular prime function generated by the two boundary reflections. Thus the deformation changes the spectral map while preserving the annular image geometry of the Schur specialization.
For \(\beta=1\), in the all-\(L\) sector and under explicit signed zero--pole and root-localization hypotheses, we characterize regular liquid and frozen points through the nonreal-root structure of the characteristic equation and identify nondegenerate interfaces as real double-root this http URL half-space Macdonald-process formulas are recovered in the continuous degeneration \(v\downarrow0\), which forces the right boundary partition to be empty.