The Collision Invariant

Abstract

For a prime p and base b, the digit function delta(r) = floor(br/p) partitions the residues {1, ..., p-1} into b contiguous bins. The collision count C(g) records how many residues share a bin with their image under multiplication by g. We prove four results about this function. First, the gate width theorem: exactly b-1 multipliers satisfy C(g) = 0, given by the explicit family g = -u/(b-u) mod p for u = 1, ..., b-1. Second, the finite determination theorem: the collision deviation S at lag l depends only on p mod b^(l+1). Third, the reflection identity: S(a) + S(m-a) = -1 for m = b^(l+1), implying a grand mean of -1/2 and a pairing symmetry across the group of units. Fourth, the half-group theorem: for every non-trivial good slice n, the wrapping set W_n has size exactly phi(m)/2. The bilateral symmetry a -> m-a swaps wrapping with non-wrapping.