The Structure That Survives
The collision invariant is the part of the arithmetic that remains when everything else has been allowed to move. This essay explains the name and the analogy to noble gases.
The Collision Spectrum
The Fourier coefficients factor through Bernoulli numbers and L-function values at s = 1.
The Collision Transform
The collision periodic table, centered and Fourier-transformed. It cancels at s = 1.
The Collision Invariant
A finite signed table for every prime, built from the digit function. The collision invariant.
The Analytic Collision Transform
The same finite diagonal geometry now carries an exact identity at every s in the critical strip. The analytic factors move. The collision content does not.
The Collision Spectrum and the L-Function Landscape
The Fourier coefficients factor through Bernoulli numbers and diagonal character sums. The collision weight encodes L-function values at s=1. The digit function meets the critical strip.
The Spectral Repulsion
Forty percent of the expected overlap between collision weights and prime character sums is missing. The ratio is stable across every prime base from 3 to 37.
The Double Transversality
The collision invariant and the prime distribution avoid each other across the strip. The avoidance persists at every lag, across every base tested.
The General Neutrality Theorem
Neutrality holds at every odd prime, not just 3. The same reflection identity, the same vanishing, at every scale. The anti-correlation between collision weights and prime sums is universal.
The Neutrality Theorem
The mod-3 component of the collision transform vanishes. Remove it and the sum explodes. Neutrality is not decorative. It is structural.
The Collision Transform and the Critical Strip
The centered sum converges at s=1. Push it below. The penetration depth measures how far into the critical strip the collision signal survives.
The Collision Periodic Table
Forty integers, determined by the last two digits of every prime past 100. Every complement pair sums to exactly -1. At base 12, the table sorts musical intervals by tension.