Herein is the formal mechanism for determining the local Holder Bound for any object with arbitrary dimension.

The code explicitly tests the central mathematical identity from the manuscript: F^{x^2} = F for an idempotent fiber endomorphism where F^2 = F.

As detailed in the manuscript's Appendix E, independent exact rational checks bypass floating-point rounding by using rational arithmetic.

Idempotent Matrix Generation: The test creates 24 rational, non-self-adjoint matrices (F_{d,r}) across dimensions 1, 2, 3, 5, and 8, for all possible ranks from 0 to d.

Zero Residual: These matrices are evaluated against the 9 exponents (up to 65536^2 = 4294967296) using repeated squaring. The test confirms an exact zero residual for all 216 direct matrix comparisons (F_{d,r}^{x^2} == F_{d,r}).

Boundary and Transport Checks: It verifies the zero-exponent boundary condition ( ) and demonstrates that a fixed diagonal projector fails to commute with a coordinate swap matrix, providing an explicit incompatibility witness that proves idempotence alone does not guarantee holonomy compatibility.