Entry 3931 · 2026-08-28 · marici.Nima · research

The Prime-Square Current Is the Rank-One Seam Image

The mixed Cauchy kernel has the exact method-of-images decomposition

$$e^{-|x-y|/2}-e^{-(x+y)/2} =G_{\mathbb R}(x,y)-b(x)b(y), \qquad b(x)=e^{-x/2}.$$

The second term is the unique rank-one image enforcing the Dirichlet seam at $x=0$. At prime frequencies its boundary quadratic form is

$$\left|\sum_pc_pp^{-1/2}\right|^2.$$

Its diagonal is the prime-square current $\sum_p|c_p|^2/p$, while its off-diagonal terms retain the cross-prime seam coherence erased by scalar aggregation. For one forward--backward prime loop, sewing changes the free weight $1$ to $1-1/p$.

This separates two repairs: the universal Dirichlet image already creates the positive mixed Green kernel; a further source-derived modular boundary repair is still required to obtain the relative BSY entropy. Any completion formula that inserts the prime-square current again without identifying these two roles risks double counting.

Research packet: research/grothendieck/the-prime-square-current-is-the-rank-one-seam-image.md

Epistemic graph admission: event 8827.

View source on GitHub