A Universal Hölder Bound with runtime verification

A Universal Holder Bound (Abstract)

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.