Distributional Future-Direction Gravity: Constrained Dynamics and Positive-Semidefinite Vacuum Response

This record contains the submitted manuscript of: “Distributional Future-Direction Gravity: Constrained Dynamics and Positive-Semidefinite Vacuum Response” This work develops a spatially covariant gravitational effective theory in which the auxiliary response sector originates from a normalized distribution of local future/update directions. The central constitutive object is the spatial second moment M_ij = , whose distributional origin determines its trace h = tr M, the transverse response T_ij = h gamma_ij - M_ij, and the positive-semidefinite condition M >= 0. The manuscript investigates the dual role of the second-moment trace in matter/update response and spatial compatibility, constructs a minimal memoryless dynamical completion, and analyzes its constraint structure using the Dirac formalism. On a generic smooth interior branch at nonzero spatial wave number, the principal constraint structure is compatible with two local tensor degrees of freedom. The exact zero-response vacuum is treated separately because positive semidefiniteness places M = 0 at the vertex of the PSD cone, leading naturally to a KKT/variational-inequality formulation. A regular finite-source solution in spherical symmetry is also derived, exhibiting evolution from an isotropic core toward asymptotic radial collimation. This version was submitted to Physical Review D as a Regular Article on September 7, 2026. Author:Kuniyuki MoritaIndependent Researcher, Japan