APS Boundary Modes and Chiral Spinorial Sectors in the Annulus Reconstruction Domain

This paper studies the selection of chiral spinorial sectors in the SO(2)SO(2)SO(2)-invariant reduced sector of the local reconstruction framework. It first shows that the geometrically appropriate domain is not a triangle but an annulus bounded by two natural scales: the Planck scale Tmin⁡=ℓPT_{\min}=\ell_PTmin=ℓP and the Hubble scale Tmax⁡=c/H0T_{\max}=c/H_0Tmax=c/H0. Under the logarithmic change of variable ρ=ln⁡T\rho=\ln Tρ=lnT, the Fisher–Riemannian metric becomes the metric of a flat cylinder, allowing a direct Gauss–Bonnet verification. On this flat cylinder, the paper applies the Atiyah–Patodi–Singer index theorem to the Fisher–Riemannian Dirac operator. Under the periodic Ramond spin structure, the two boundary components carry zero modes of the one-dimensional boundary Dirac operator. The resulting APS index is non-zero, which forces the existence of zero modes of the interior Dirac operator. These modes are interpreted as candidates for physically admissible chiral sectors, subject to the analytic extension problem: determining which boundary zero modes extend to L2L^2L2 solutions inside the cylinder. The paper therefore proposes a three-filter selection chain: spinorial admissibility, global extensibility, and APS boundary conditions. It also outlines a conditional connection with the crossed-product programme and the Born rule through the modular flow associated with logarithmic dilations. The main result is topological and spectral: in the reduced annulus sector, global reconstruction constraints impose non-trivial chiral spinorial sectors, while their complete physical identification remains an open analytic problem.