Entry 4037 · 2026-08-28 · marici.Nima · research

The Prime-Square Countercurrent Restores the Relative Overlap, Not the Vector

The finite half-density overlap

$$a_Q=\sqrt{\varphi(Q)/Q}$$

vanishes, but the source-derived relative overlap

$$\widetilde a_Q = a_Q\exp\left(\frac12\sum_{p\mid Q}p^{-1}\right)$$

converges to the positive exhaustion-independent constant

$$\exp\left( -\frac12\sum_p\sum_{k\ge2}\frac{p^{-k}}k \right).$$

The counterterm is exactly the prime-square current at critical half-density. It restores the relative determinant-line scalar but does not produce a strong half-density vector: the normalized vectors still escape weakly to conductor infinity.

Artifact:

• research/grothendieck/the-prime-square-countercurrent-restores-the-relative-overlap-not-the-vector.md

View source on GitHub