The Collision Fluctuation Sum
The fluctuation sum
The collision count at each prime fluctuates around its mean with standard deviation sqrt(Q). The fluctuation sum F(s, l) = sum of these fluctuations weighted by p^{-s} encodes how they distribute across primes.
F converges for Re(s) > 3/2 by the variance scaling. But at s = 1, the sum does not converge. It diverges, slowly, at the rate of log(log(x)).
The Mertens growth law
Computation to 10 million primes reveals the growth law: F(1, l) ~ -mu · log(log(x)), where mu is a specific constant. In base 10 at lag 1: mu ≈ 0.68. The divergence is invisible at small scales because log(log(x)) grows slower than any power. At 8,000 primes the sum looked like it was converging. At 664,574 primes the drift is clear.
The exact decomposition
The collision deviation decomposes exactly: phi = S - QR/(p-1-b), where S = C - Q is a bounded integer and QR/(p-1-b) is a spectral class correction. Both pieces diverge at rate log(log(x)). F is their near-cancellation.
The bilateral symmetry forces C(g) to be even for all g (collisions come in complement pairs). S has the same parity as Q. In base 10, S takes only 12 distinct values.
The per-class structure
Each spectral class (p mod b) has its own collision bias. In base 10: class 1 (R=0) has the strongest negative bias (-1.70), class 9 (R=8) the weakest (-0.12). The class R=6 (digit 7, the negative pole) has bias approximately -2/3: the alignment limit of the prime 3 from paper 1.
The Mertens constant mu is the weighted average of these per-class biases.
The critical boundary
For s > 1, F converges. At s = 1, logarithmic divergence. For s < 1, power-law divergence. The boundary is sharp. A modified fluctuation sum, centered to remove the persistent negative bias, may yet converge at s = 1. That would be a true collision analog of the prime number theorem.
The paper
The Collision Fluctuation Sum (PDF)
.:.