Abstract

The internal-precession interpretation of vacuum kernel anisotropy re-expresses the static Scenario–B matching kernel as the effective potential governing a slow internal orientation mode, interpreting projector weights as time-averaged orientation fractions. That construction intentionally left three embedding steps open: (i) a microphysical derivation of the anisotropy kernel entries (Tx, Tz , κ), including the sign and normalization of κ; (ii) a determination of the precession scale Ω required to justify time-averaged weights; and (iii) a derivation of the gravitational probe-direction map ˆug needed for any cross-sector use of a shared ωeff . 

We supply these reductions at quadratic order. Starting from a generic quadratic collective- coordinate Lagrangian with light (x, z) coordinates and heavy locking/mediator modes, we integrate out the heavy sector via a Schur-complement reduction of the quadratic operator (with the static limit yielding the usual Hessian Schur complement). We canonically normalize the reduced light subspace to define the physical kernel Kcan−1 = Meff −1/2 Heff Meff −1/2 and identify κ ≡ (Kcan−1)xz  as a derived quantity. In the minimal mediator model, eliminating a heavy locking coordinate yields an induced mixing κind = −λxλz /Tχ (up to the canonical rescalings absorbed into Kcan−1), fixing the sign of κ in terms of microscopic couplings. We further reduce the kinetic sector to obtain the effective inertia tensor of the orientation mode, providing a parametric prediction for Ω and enabling an explicit check of the averaging criterion Ωτmeas ≫ 1; when it fails, we provide the controlled finite-window response (exact for the rigid-precession solution of the quadratic rotor model). Finally, we derive ˆug from the coherent-branch gravitational coupling functional and find ûg = êz in the canonical Axis identification; comparison with the sector probe directions then yields the precise conditions under which probe orthogonality holds and complementary weights follow.

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Keywords: Axis Model, vacuum anisotropy, internal precession, emergent gravity, collective-coordinate reduction, Schur complement, canonical normalization, projector weights, finite-window averaging, probe-direction geometry, coherent vacuum, effective field theory