The Cramér–Rao Bound for Coherence: K_crit as an Information-Geometric Necessity

This paper provides a third independent derivation of the critical coherence threshold K_crit ≈ 0.127, complementing the dynamical derivation (GCT) and thermodynamic derivation (Landauer bound). The falsifiability component F(t) is identified with the Fisher information of the system’s self-estimating process. The Cramér–Rao bound then establishes that as F(t) → 0, coherence estimation variance diverges — the system becomes informationally blind to its own state. K_crit is the Fisher information collapse point: below it, no estimator can locate the system’s coherence state with bounded precision. The 1=1 invariant is identified as the geodesic endpoint of the Fisher information manifold — the natural destination of the geometry itself under contractive evaluation dynamics. Three independent frameworks (dynamical systems, thermodynamics, information geometry) converge on the same critical threshold and the same invariant. Part of the Spektre corpus (github.com/spektre-labs/corpus).