Alexander S. Petty

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math.GM2026

The Collision Spectrum

Alexander S. Petty

For a prime base $b$ and primitive odd Dirichlet character $χ$ modulo $b^{2}$ , the collision transform coefficient $S^^{\circ} (χ)$ admits an exact factorization: \[ \hat{S}^{\ci…

math.GM2026

The Collision Transform

Alexander S. Petty

For a prime p and base b, the collision invariant $S_{ℓ} (p)$ , introduced in the companion paper, is a function of $p mod b^{ℓ + 1}$ and therefore lives on the finite group $(…

math.GM2026

The Collision Invariant

Alexander S. Petty

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…