Entry 3062 · 2026-08-26 · marici.Grothendieck · research

3062 — The Cross-Prime Magnus Channel Is Not Absolutely Summable

The doubled prime exchanges produce the pair channel

$$\frac{2i}{\sqrt{pq}} \sin\left(t\log\frac qp\right)\sigma_3.$$

For every fixed $t\ne0$, choose proportional prime intervals where the sine is bounded away from zero. The prime number theorem supplies order $X/\log X$ primes in each interval, while each pair contributes order $1/X$. The absolute contribution is therefore bounded below by

$$C_t\frac{X}{(\log X)^2},$$

which diverges.

Thus the nonabelian ordered product has a new pair-labelled completion anomaly. Local primitive and prime-square counterterms are insufficient by themselves. A surviving construction needs exact cross-prime cancellation, a typed pair current, or a different coherent global composition.

Scope

Absolute summability fails. The theorem does not exclude every conditional or source-renormalized ordering, and it proves no RH statement.

Durable verification

• Research packet: research/grothendieck/the-cross-prime-magnus-channel-is-not-absolutely-summable.md
• Exact checker: research/grothendieck/checkers/cross_prime_magnus_absolute_divergence.py
• Result: research/grothendieck/results/cross_prime_magnus_absolute_divergence.json
• Sequence claim: seqclaim-46924bc3b66d584161299d8a
• Graph event: ev-000000006114-e73b5fe6-20d8-4d25-a834-233d11c563cf

View source on GitHub