The Collision Transform

Abstract

For a prime p and base b, the collision invariant $S_{\ell}(p)$, introduced in the companion paper, is a function of $p \bmod b^{\ell+1}$ and therefore lives on the finite group $(\mathbb{Z}/b^{\ell+1}\mathbb{Z})^{\times}$. Its Fourier expansion over Dirichlet characters modulo $b^{\ell+1}$ is the collision transform. The reflection identity forces all even-character coefficients of the centered invariant to vanish: only odd characters contribute. The centered prime harmonic sum $F^{\circ}(s) = \sum_p S^{\circ}_p / p^s$ is therefore a finite linear combination of non-trivial odd character sums $\sum_p \chi(p)/p^s$, with no principal-character term. At $s = 1$, each sum converges by Mertens' theorem for arithmetic progressions. Convergence below $s = 1$ is conditional on the absence of $L$-function zeros above a given depth. Computation indicates convergence persists to at least $s = 0.6$ in base 10 and to $s = 0.5$ in base 3. The real parts of the products $\hat{S}^{\circ}(\chi) \cdot P(s, \chi)$ have mixed signs, so convergence is a collective constraint on the joint zero distribution, not a test of each $L$-function individually. Aggregating the collision deviation across bases with a fixed convergent weighting produces the base sum, a function on primes that reveals mod-3 structure. For bases with $3 \nmid b$, the reflection $a \mapsto m - a$ fixes a unique residue class modulo 3, and the mean of $S$ over units in that class equals the grand mean $-1/2$ (the neutrality theorem). Removing the mod-3 component introduces a principal-character term that is absent from $F^{\circ}$. The base-summed harmonic sum is negligible: the collision invariant's structural content is base-specific.