Step-by-Step Verification of the Chronos Constant

This paper provides a complete, transparent, and reproducible mathematical verification of the Chronos constant χ\chiχ, defined by its continued-fraction expansion in Chronos Field Theory. Using only standard continued-fraction recurrence relations, the paper demonstrates how any mathematician can reconstruct the convergents pn/qnp_n / q_npn/qn, verify numerical agreement with the published decimal value, and confirm the irrationality of χ\chiχ. Each step is written explicitly, showing the derivation of the first several convergents and quantifying their convergence to the Chronos constant. A general theoretical bound is used to guarantee uniqueness and convergence of the continued fraction. The paper concludes by proving that χ\chiχ is irrational and uniquely defined, establishing a rigorous mathematical foundation for the constant used throughout Chronos physics and Dynamical Stability Theory.