The Collision Invariant

TL;DR

This paper investigates properties of a digit-based partitioning and collision count function modulo a prime, revealing explicit formulas, symmetries, and structural theorems about the distribution of residues under multiplication.

Contribution

It establishes four main theorems detailing the behavior, symmetry, and structure of the collision count function and related residue sets in prime modular arithmetic.

Findings

Exactly b-1 multipliers have zero collision count, explicitly characterized.

Collision deviation depends only on p mod b^(l+1).

A symmetry relation S(a)+S(m−a) = -1 holds for the collision deviation.

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.