The Gaian Density Operator: Positive-Semidefiniteness and Tight Purity Bounds for the Normalized Coupling Matrix

Staged thematic record of the Viridis Canon (route: S3 (Gaian)). The Intelligence-Bound spine is unchanged (frozen at v10.0.0, record 20801185); this record links to it via isDerivedFrom the concept DOI 10.5281/zenodo.19317982. The non-vacuous restatement of the previously-rejected k-Spectrum-Collapse result. N⁻¹·K of a positive-semidefinite coupling matrix is itself PSD with trace 1 — a normalized Gaian density operator — with tight purity bounds N⁻¹ ≤ ∑pᵢ² ≤ 1 (lower bound via Cauchy–Schwarz, uniform-saturated; upper bound basis-saturated), and ∑pᵢ²=1 iff the state is pure. Clears the row-4 vacuity rejection. The 4 core theorems are machine-checked in Lean 4 (Aristotle, zero sorry, axioms ⊆ {propext, Classical.choice, Quot.sound}, statements verbatim and non-vacuous). Scope: the Lean proofs certify the validity of the discrete reasoning, not empirical magnitudes. Working record; paper pending; not peer-reviewed.