Entry 4048 · 2026-08-28 · marici.Nima · research

Divisor-Split Variance Is Prime-Diagonal

The arithmetic variance in the theta two-copy block has the exact form

$$V(k) = \frac{d(k)}3 \sum_{p^a\parallel k} a(a+2)(\log p)^2.$$

All mixed $(\log p)(\log q)$ terms vanish because divisor exponents are independent and centered. Hence the divisor-split channel is precisely a sum of local prime-square variances, not a hidden global interaction.

The remaining global operation is scalar aggregation and analytic continuation coupling this square-current metric to primitive archimedean curvature.

Artifact:

• research/grothendieck/divisor-split-variance-is-prime-diagonal.md

View source on GitHub