Digit Collisions and the Cubic Law
The collision energy is the cube of the base, split one third to two thirds between diagonal and off-diagonal poles. The cubic constant is unity.
Alex Petty
The Orbit's Edge
The orbit is multiplicative. The boundary is additive. The Jacobi sum is the bridge. Two laws from the time of Gauss show up because the boundary forced the character to look at -2.
Carry Boundaries and Bernoulli Spectra
The collision invariant turned out to be a special case. This paper finds the source underneath it. The floor provides the weight. The boundary provides the geometry. The spectrum is their product.
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.