Selection Geometry of Hilbert Space: Arrow of Time, Hawking Radiation, and Spectral Non-Closure

Fundamental physical laws are unitary and reversible, yet observed physical phenomena exhibit irreversibility, thermality, entropy increase, and horizon radiation. Reconciling this apparent contradiction remains a central problem in modern theoretical physics.

This work introduces a unified geometric framework, termed selection geometry, in which irreversibility emerges naturally from non-invertible selection acting on projective Hilbert space.

We show that physical states are naturally represented as points in projective Hilbert space, whose compactified structure introduces a geometric point at infinity analogous to conformal boundaries in spacetime and the Riemann sphere. Selection mapping acts as a non-invertible projection, compressing spectral degrees of freedom and producing a realized state manifold that is spectrally non-closed.

This spectral non-closure provides a geometric origin for thermality, entropy increase, and Hawking radiation. Thermality emerges from circular Euclidean time geometry and the KMS condition, while horizon radiation arises from horizon-induced spectral selection. Event horizons act as selection surfaces that restrict accessible spectral degrees of freedom while preserving global spectral completeness.

Within this framework, global spectral evolution remains unitary and reversible, while realized evolution becomes irreversible due to geometric quotient structure induced by selection. This dual structure resolves the black hole information paradox by distinguishing global spectral completeness from realized spectral accessibility.

Selection mapping is physically realized as completely positive trace-preserving (CPTP) evolution on density operators. Irreversibility therefore emerges not from dynamical law violation, but from geometric structure.

This work establishes selection geometry as a foundational geometric principle underlying quantum mechanics, thermodynamics, and gravitational physics.