Discrepancy between Julian Year and Tropical Year

Theorem
The Julian year and the tropical year differ such that the Julian calendar becomes $1$ day further out approximately every $128$ years.

Proof
By definition, the length $Y_T$ of the tropical year is defined as $\approx 365 \cdotp 24219 \, 878$ days

By definition of the Julian year:
 * $Y_J = \begin{cases} 366 \, \text {days} & : 4 \divides y \\

365 \, \text {days} & : 4 \nmid y \end{cases}$ where:
 * $Y_J$ denotes the length of the Julian year in days
 * $y$ denotes the number of the year
 * $4 \divides y$ denotes that $y$ is divisible by $4$
 * $4 \nmid y$ denotes that $y$ is not divisible by $4$.

Thus:

Thus the mean Julian year is $\dfrac 1 {128 \cdotp 18}$ days longer than the tropical year.

This means that after approximately $128$ Julian years, the Julian calendar starts one day later relative to the tropical year.