Modulo-counter elapsed-time identity ($((p+d)\mathrm{mod}M-p)\mathrm{mod}M=d(0\le d<M)$)

A free-running digital counter with modulus $M$ shows only the residue of elapsed ticks. If two observations differ by a true interval $d<M$, their modular difference is congruent to $d$ and is the unique representative in $\{0,\dots ,M-1\}$, proving the displayed identity. It handles a wrap between observations without a conditional branch. A complete elapsed modulus is indistinguishable from zero, so the interval bound is essential. For a $w$-bit timer at frequency $f$, the wrap period is $2^w/f$, and unsigned integer arithmetic in C can implement the difference directly on an appropriate type.