The Double Transversality
The collision spectrum and the prime spectrum avoid each other. Not once. Twice.
The first transversality
Within a single base: characters with large collision coefficients $|\hat{S}^\circ(\chi)|$ tend to have small prime character sums $|P(s, \chi)|$. The two spectra are anti-correlated. Pearson coefficient approximately $-0.25$ across all bases tested.
The collision invariant places its weight where the primes don't. The primes concentrate where the collision invariant is quiet. They occupy complementary regions of character space.
The second transversality
Across bases: the base sum $\sum_b w(b) F^\circ_b(s)$ approximately vanishes. Each base carries a collision signal, but the signals from different bases nearly cancel when aggregated. The collision invariant's structural content is base-specific. It does not survive aggregation.
Within each base: rich structure. Across bases: cancellation. The collision invariant is a coordinate-dependent observable. Change the coordinate system (the base), and the observable changes. Sum over all coordinates, and the observable disappears.
What the double transversality means
The collision spectrum encodes something that is:
- transverse to the prime spectrum (anti-correlated within each base)
- transverse to the base-sum (cancelled across bases)
It occupies a structural niche that neither the primes nor the base aggregation can reach. The collision invariant sees what the primes miss, and it sees it in a way that depends on the choice of base.
This is not a deficiency. It is a characterization of what the collision invariant IS: a base-specific, prime-transverse structural observable.
Try it yourself
./nfield 7 # collision data in base 10
./nfield 7 --base 3 # same prime, different base
Code: github.com/alexspetty/nfield
Alexander S. Petty
October 2025
.:.