What question did this study set out to answer?

To derive the critical coherence threshold K_crit and explore its implications through information geometry.

March 8, 2026Open Access

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

Key Points

To derive the critical coherence threshold K_crit and explore its implications through information geometry.
Independent derivation of K_crit using three frameworks: dynamical systems, thermodynamics, and information geometry.
Utilization of the Cramér–Rao bound for coherence estimation variance.
Identification of the F(t) component with Fisher information in the self-estimating process.
K_crit is established at approximately 0.127.
As Fisher information F(t) approaches zero, variance in coherence estimation diverges.
The invariant 1=1 is linked to the geodesic endpoint of the Fisher information manifold.

Abstract

This paper provides a third independent derivation of the critical coherence threshold Kcrit ≈ 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. Kcrit 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).

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

Key Points

Abstract

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