The Neutrality Theorem

The centered sum converges at $s = 1$. But why? The cancellation is not accidental. It is forced by the mod-3 structure of the collision invariant.

The reflection and mod 3

The reflection identity $S(a) + S(m-a) = -1$ pairs units modulo $m$. The map $a \mapsto m - a$ swaps residue classes modulo 3. Two of the three classes are exchanged. One is fixed.

The fixed class has mean $-1/2$. Exactly the grand mean. It is neutral: it contributes nothing to the divergence.

The two swapped classes have means that sum to $-1$ (by the reflection identity applied class-by-class). They are paired but not neutral individually.

Perfect cancellation

The centered sum $F^\circ$ removes the class means. The principal-character contribution, which would produce a Mertens-rate divergence, cancels algebraically. Not approximately. Exactly.

This is not a numerical observation. It is a theorem. The reflection identity forces the cancellation. The mod-3 structure is why the centered sum at $s = 1$ converges: the symmetry of the collision invariant kills the divergent term.

Alpha equals one

The cancellation exponent, which measures how completely the Mertens term is removed, equals exactly 1. Perfect cancellation. No residual growth. The digit function's bilateral symmetry is strong enough to eliminate the principal character entirely.

Try it yourself

./nfield 7               # class structure visible
./nfield 97              # same structure, larger prime

Code: github.com/alexspetty/nfield

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Alexander S. Petty
August 2025
.:.