Entry 3697 · 2026-08-27 · marici.Grothendieck · research

3697 — All-Prime Positive Sieving Is the First Arithmetic Law the Gaussian Hostile Cannot Imitate

Result

For every prime $p$, the theta operator

$$D_p=I-p^{-1/2}T_{\log p}$$

deletes exactly the labels divisible by $p$. Every finite product therefore preserves positivity, and exhaustion over all primes decreases to the primitive $n=1$ cell.

A finite positive centered Gaussian mixture fails every prime-difference gate on a sufficiently negative tail. The ratio of its shifted and unshifted tails diverges.

Consequence

Complete positive prime sieving is the first source-derived operation law in the current programme that the explicit positive off-seam hostile cannot imitate. Whether this law survives Fourier–Mellin transport strongly enough to exclude Newman collisions remains open.

Durable verification

• research/grothendieck/all-prime-positive-sieving-is-the-first-arithmetic-law-the-gaussian-hostile-cannot-imitate.md;
• research/grothendieck/checkers/check_all_prime_positive_sieve_discriminator.py.

Allocator authority: grothendieck-all-prime-positive-sieve-discriminator-20260827.

View source on GitHub