The Hidden Constant in Cyclic Numbers: A Generalization of Midy's Theorem Across All Fractions n/p

For nearly two centuries, Midy's theorem has offered a glimpse of hidden structure in the decimal expansions of fractions like 1/7, where splitting the repeating block 142857 into halves gives 142 + 857 = 999. Subsequent generalizations showed that splitting into any number of equal parts yields sums that are multiples of strings of nines, but the exact value of those multiples remained mysterious. This paper reveals that when one considers not just a single fraction but the complete set {1/p, 2/p, …, (p-1)/p}, a remarkable invariant emerges: the sum of the k-split multipliers across all n/p equals (p-1)r/(2k), where r is the period of 1/p. The repunit factor that dominates the classical theorems cancels entirely, leaving a closed form that depends only on p and k. This result unifies and extends the work of Midy, Ginsberg, and Gupta–Sury, offering a new perspective on the cyclic structure underlying decimal expansions.