The General Neutrality Theorem

The neutrality at mod 3 is not special. It happens at every odd prime.

The pattern

At every odd prime $q$ not dividing the base: the reflection identity fixes exactly one residue class modulo $q$. That class has mean $-1/2$. It is neutral.

The remaining $q - 1$ classes split into $(q-1)/2$ pairs swapped by reflection. Each pair's means sum to $-1$.

At $q = 3$: one neutral class, one pair. At $q = 5$: one neutral class, two pairs. At $q = 7$: one neutral class, three pairs. The constraint tightens at every prime scale.

The sieve

The fraction of units NOT constrained by neutrality at any prime $q \le Q$ shrinks as $Q$ grows. Each prime $q$ fixes a class at mean $-1/2$ and pairs the rest. The unconstrained units are those not in any neutral class. As more primes are included, fewer units escape.

The collision deviations are simultaneously constrained at every prime scale. The convergence of $F^\circ$ is not a single cancellation. It is an infinite family of cancellations, one at each prime, all operating in parallel.

The sign structure

Computation reveals something beyond neutrality: the products $\hat{S}^\circ(\chi) \cdot P(s, \chi)$ have mixed signs. Large collision coefficients tend to pair with characters whose prime sums are small, and vice versa. The collision spectrum and the prime distribution avoid each other.

This anti-correlation is not proved. It is observed across every base and lag tested. It suggests that the convergence of $F^\circ$ is not just permitted by the neutrality constraint but actively enforced by a deeper structural avoidance.

Try it yourself

./nfield 7               # neutrality at q=3
./nfield 97              # neutrality at q=3, 7, ...

Code: github.com/alexspetty/nfield

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Alexander S. Petty
September 2025
.:.