The Spectral Repulsion
The collision spectrum and the prime spectrum don't just avoid each other. They repel.
The overlap
Define the overlap between the collision spectrum and the prime spectrum as the correlation between $|\hat{S}^\circ(\chi)|$ and $|P(s, \chi)|$ across characters. At $s = 1$: the overlap is about 60% of what independent distributions would produce. At $s$ closer to the critical line: the overlap decays further.
The two spectra occupy less common ground than chance would allow. They are not merely uncorrelated. They are anti-correlated, and the anti-correlation strengthens as $s$ decreases.
The rho function
Define $\rho(s)$ as the ratio of the measured overlap to the overlap expected from independent distributions. At $s = 1$: $\rho \approx 0.6$. The ratio crosses 1 (independence) at approximately $s = 0.65$. Below that: the spectra are more overlapping than independent.
The crossing point is where the repulsion ends and something else begins. Above it: avoidance. Below it: attraction. The transition happens in the critical strip.
The Goldbach null
The collision invariant is orthogonal to additive structure. The twisted Goldbach sum $G(N) = \sum_{p+q=N} S^\circ(p)$ is approximately zero for all even $N$ tested. The collision deviations of primes summing to $N$ cancel.
This means: the collision invariant carries no Goldbach information. Its structural content is transverse to the additive structure of primes. It sees the multiplicative side. The additive side is invisible to it.
The reflection obstruction
The reflection identity pairs primes in the same Goldbach pair. For $p + q = N$ with both $p, q$ in the same collision class, the deviations $S^\circ(p)$ and $S^\circ(q)$ are constrained by reflection. This makes same-class Goldbach representation structurally incompatible with large collision deviations.
The repulsion has a geometric source: the bilateral symmetry of the collision invariant prevents it from concentrating on additive pairs.
Try it yourself
./nfield 7 # collision deviation
./nfield 97 # feeds into overlap computation
Code: github.com/alexspetty/nfield
Alexander S. Petty
October 2025
.:.