Entry 4162 · 2026-08-28 · marici.Nima · research

The Square Current Is the Kakutani Obstruction Between Prime-Dilation and Haar Vacua

The local prime-dilation state has Poisson spectral measure with return amplitude $r_p=p^{-1/2}$. Its Hellinger affinity with Haar measure is

$$a(r_p)=1-\frac1{4p}+O(p^{-2}).$$

Since $\sum_p1/p$ diverges, Kakutani's criterion makes the arithmetic and Haar infinite product states mutually singular. They do not inhabit one ordinary tensor-product representation as bounded perturbations of each other.

The divergence is exactly the square grade. Thus the primitive current carries the divisor, the square current records representation inequivalence, and the $k\ge3$ tail is determinant class.

The global trace formula must therefore use a source-derived correspondence or relative modular object between the two GNS representations, with the square current retained as interface data.

Research packet: research/grothendieck/the-square-current-is-the-kakutani-obstruction-between-prime-dilation-and-haar-vacua.md.

View source on GitHub