Point-Gap Exact Inertia

Abstract

A finite point gap gives exact half-dimensional chiral inertia.

Theorem 1.1 (Point-gap exact zero-scale inertia).

$$\mathrm{posIndex}(\mathrm{localizerZero}(X,H,x,z))=\mathrm{card}(n)\wedge \mathrm{negIndex}(\mathrm{localizerZero}(X,H,x,z))=\mathrm{card}(n)$$

Proof. Machine-checked in Lean as D5/S3/SpectralTopology/PointGapExactInertia.zero_scale_localizer_inertia_of_point_gap (✓ std3). ∎

Source. Repository-derived.

Commentary.

The positive and negative inertia counts of a finite Hermitian matrix add to its rank, and a finite point gap gives the zero-scale localizer full rank on the doubled carrier.

Combined with the frozen chiral inertia balance, the positive and negative zero-scale counts therefore both equal the original carrier cardinality: under a point gap there are no zero modes, and the doubled finite spectrum splits into equally many positive and negative eigenvalues.

References

• Truth anchor: D5/S3/SpectralTopology/PointGapExactInertia.zero_scale_localizer_inertia_of_point_gap
• Dependency: D5/S3/SpectralTopology/FiniteSpectralLocalizer