The Centered Collision Sum
Subtract the family bias and the divergence vanishes. The centered sum converges at s = 1.
The Collision Fluctuation Sum
I added up the collision deviations. The sum drifted downward, and the drift had a shape.
Silent Primes
At seven primes in base 10, the collision count is exactly zero. The recipe finds them all.
The Character Structure of the Collision Fluctuation
The collision fluctuation drifts at the Mertens rate. Split it by character, and four currents appear.
Bin Derangements and the Gate Width Theorem
Nine deranging multipliers at every prime in base 10. The count does not depend on the prime.
Phase-Filtered Ramanujan Sums and the Spectral Gate
The last digit of a prime looks like the least interesting thing about it. It is not.
The Autocorrelation Formula
One quantity in the spectral chain resisted a closed form longer than the others.
The Spectral Power of the Digit Function
Ten bucket sizes per prime. They determine far more than you would expect.
The Spectral Structure of Fractional Fields
The eigenvalues of the cross-alignment matrix encode the internal structure of the prime.
The Cross-Alignment Matrix
Compare every fraction to every other, digit by digit. The grid has a shape.
The Coherence Decomposition
When the alignment is high, why is it high? Signal and background come apart cleanly.
The Three-Tier Theorem
Every positive integer falls into one of exactly three tiers. No borderline cases.