Operational Characterization of the Center

Abstract

The operational center of a finite cyclic observer window consists exactly of constant observables.

Theorem 1.1 (Zero perturbation characterizes the operational center).

$$\begin{array}{c}\forall M\in {\mathbb{N}}_{>0},\\ \forall f:\mathrm{ZMod}(M)\to \mathbb{C},\\ L_{+1}(f)=0\iff \exists c\in \mathbb{C},f=(i\mapsto c).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Observer/CenterOperational.center_iff_const (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let M be positive and let f be a complex-valued observable on the cyclic window ZMod M. The seminorm L for translation by one measures the largest pointwise update defect. Its kernel is the operational center considered here.

The established seminorm-kernel theorem first identifies zero perturbation with invariance under translation by one. The existing update-defect equivalence transfers that condition to zero defect, and the cyclic-window characterization then gives a scalar c for which f is the constant function with value c. Conversely, every constant observable lies in the kernel. The result introduces no larger operator algebra or independent notion of center.

References

• Truth anchor: D5/S3/Observer/CenterOperational.center_iff_const
• Dependency: D5/S3/Observer/ObserverMetric