Conditional Spectral Geometry and Orientation Fractions in the Axis Model
Andrew Morton · Public preprint · Conditional spectral geometry
Separates the geometry of a quadratic kernel from the orientation averages of a specified motion.
Abstract
A supplied quadratic kernel determines spectral projectors, while a supplied orientation state or trajectory determines a geometric probe fraction. We derive these two constructions separately. For a real symmetric two-dimensional kernel, both eigenprojectors are given explicitly, their angular branches are resolved without dividing by a stiffness difference, and constrained second variations distinguish the static minimum from the maximum. Degenerate and zero-mixing limits retain their dependence on the stated branch and domain. For a prescribed orientation process, the normalized second moment gives bounded probe fractions. A fixed-cone trajectory yields exact phase-averaged and finite-window expressions, with its dynamical realization treated as a separate input. The action, symmetry and stability information required for such a realization is identified. These results provide conditional geometric interfaces for the Axis Model; physical source normalization, ultraviolet matching, gravitational response and particle selection require their originating constructions.
Scope
The paper establishes exact geometry for supplied kernels and trajectories. The lower eigenbranch minimizes the stated static energy; the upper branch is a maximum. Physical motion, source normalization and gravitational response require separate derivations.
Paper and supporting files
The manuscript source and a shared reproducibility package for the conditional calculations are available on Zenodo.
Read the current paper and supporting files on Zenodo
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