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The Collision Invariant

Alexander S. Petty

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.

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.

Paper Structure

This paper contains 8 sections, 9 theorems, 13 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Linearization
  3. The Gate Width Theorem
  4. The Finite Determination Theorem
  5. The Reflection Identity
  6. The Half-Group Structure
  7. Computational Verification
  8. Remarks

Key Result

Lemma 2

For $r \in \{1, \ldots, p{-}1\}$, let $[x]_p = x \bmod p$ and set $x = [br]_p$. Then $br = p\,\delta(r) + x$ and Under the permutation $r \mapsto [br]_p$, the bins $B_d$ are exactly the residue classes modulo $b$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Definition 1
  • Lemma 2: Conjugation
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6: Gate width
  • ...and 11 more