Microphysical Reduction of Vacuum Kernel Tilt and Internal Precession in the Axis Model

This paper completes the microphysical foundations of vacuum anisotropy in the Axis Model by deriving, rather than assuming, the parameters that govern internal orientation and projector-weight suppression in Scenario-B normalization.

Previous work introduced an internal-precession interpretation in which reduced projector weights arise as time-averaged orientation fractions of a coherent internal degree of freedom. While that interpretation successfully explained phenomenology, several key elements were left open: the microscopic origin of the anisotropic kernel, the dynamical timescale controlling the averaging procedure, and the identification of the internal direction probed by gravity.

Here we close those gaps at quadratic order.

Starting from a generic collective-coordinate Lagrangian for the coherent branch, containing light internal coordinates and heavy locking or mediator modes, we perform a controlled quadratic reduction. Integrating out the heavy sector via a Schur-complement construction yields an effective light-sector kernel and inertia. Canonical normalization then fixes the physical meaning of the anisotropy parameters and removes basis ambiguities. We include the explicit calculation of the precession frequency and timescale validation and include the reproducible code.

This version corrects formatting error only-no change in content.