Ordered-Time Simplex Second-Magnus Average

Abstract

Evaluate the ordered-simplex average of the second-Magnus kernel exactly.

Theorem 1.1 (Exact ordered-simplex response).

$$\forall g\in \mathbb{R},T\in \mathbb{R},g\ne 0\Rightarrow A\left(g,T\right)=T^2-\frac{2\times (1-\cos \left(g\times T\right))}{g^2}$$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencyHolonomy/OrderedTimeSimplexSecondMagnusAverage.ordered_time_simplex_kernel_average_formula (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a nonzero frequency gap, collapsing the ordered two-time simplex to the time difference gives a triangularly weighted squared sine integral with an exact closed form.

The formula supplies a common finite horizon for each fixed gap. A uniform minimum over a finite frequency family and a Bochner-valued Magnus integral remain future transport steps.

References

• Truth anchor: D5/S3/Observer/AgencyHolonomy/OrderedTimeSimplexSecondMagnusAverage.ordered_time_simplex_kernel_average_formula
• Dependency: D5/S3/Observer/AgencyHolonomy/SecondMagnusSwapCurvature