The Physical Bidual Gap and Banach Space Non-Reflexivity: A Lean 4 Formalization of WLPO via Trace-Class Operators

We present a Lean 4 formalization establishing that the trace-class operators S₁(H) on a separable Hilbert space are not reflexive as a Banach space, and that any constructive witness of this non-reflexivity implies the Weak Limited Principle of Omniscience (WLPO). In the algebraic formulation of quantum mechanics, S₁(H) is the canonical state space of density matrices; the bidual gap (S₁(H))** \ J(S₁(H)) contains singular states not representable by any density matrix. The formalization comprises 754 lines of Lean 4 code across 8 files (Mathlib4 v4.28.0-rc1), with one interface assumption bridging a companion formalization for ℓ¹ non-reflexivity. The key technical contribution is a machine-checked proof that every closed subspace of a reflexive Banach space is reflexive, using two applications of the Hahn-Banach theorem without James's theorem. Extends Paper 2 (doi:10.5281/zenodo.18501689) from the abstract c₀/ℓ¹ setting to the physically motivated trace-class operator setting. Version 1.1 change: P7_Minimal addition, revised paper Other papers in the Constructive Reverse Mathematics Series (see Zenodo for current files): Paper 1: WithdrawnPaper 2: Bidual gap and WLPO — DOI: 10.5281/zenodo.17107493Paper 3: WithdrawnPaper 4: Quantum spectra axiom calibration — DOI: 10.5281/zenodo.17059483Paper 5: Schwarzschild curvature verification — DOI: 10.5281/zenodo.18489703Paper 6: Heisenberg uncertainty (v2, CRM over Mathlib) — DOI: 10.5281/zenodo.18519836Paper 7: Physical bidual gap, trace-class operators — DOI: 10.5281/zenodo.18527559Paper 8: 1D Ising model and LPO — DOI: 10.5281/zenodo.18516813Paper 9: Ising formulation-invariance — DOI: 10.5281/zenodo.18517570Paper 10: Logical geography of mathematical physics — DOI: 10.5281/zenodo.18527877Paper 11: Entanglement, CHSH, Tsirelson bound — DOI: 10.5281/zenodo.18527676Paper 12: Constructive history of mathematical physics — DOI: 10.5281/zenodo.18528793Paper 13: Event horizon as logical boundary — DOI: 10.5281/zenodo.18529007 Note: GitHub repository is not maintained; please see Zenodo for current files.