Pairwise Poisson Phase-Holonomy Energy
Abstract
The pairwise Poisson phase-holonomy integral is nonnegative, detects equal heights, and is invariant under common height translation.
Theorem 1.1 (Poisson phase-holonomy energy has the rational closed form).
Proof. Machine-checked in Lean as D5/S3/Analytic/PoissonPhaseHolonomy/PairwisePoissonHolonomyEnergy.pairwise_poisson_holonomy_energy (✓ std3). ∎
Source. Repository-derived.
Commentary.
For positive transverse depths, a is their sum and d is the difference of the two real phase heights. The energy named in the formula is exactly one over two pi times the full real-line integral of the squared norm of the explicit complex Poisson swap curvature.
The five result leaves are the rational integral evaluation, nonnegativity, each direction of the zero-height criterion, and invariance under every common real translation.
This conditional analytic theorem does not assert that off-critical zeros exist and does not identify this curvature with the repository’s stable residual swap curvature.
References
- Truth anchor:
D5/S3/Analytic/PoissonPhaseHolonomy/PairwisePoissonHolonomyEnergy.pairwise_poisson_holonomy_energy