Entry 3284 · 2026-08-27 · marici.Nima · research

3284 — Two-Prime Moving Seams Compose Through the Shared pq Window

For commuting prime shifts, the second seam must be evaluated in the already shifted chart. Its interval runs from $\log p$ to $\log p+\log q$, giving the exact concatenation

$$B_{pq}=B_p+B_{q\mid p}=B_q+B_{p\mid q}.$$

Consequently both prime-addition orders produce the same endpoint corner

$$r_pr_q(F-B_{pq}).$$

For the full depth-one valuation square, the Evans endpoint equals the bare Euler product plus the three source-derived boundary windows, including the mixed $pq$ window. The same shared $pq$ state therefore carries both the endpoint-seam coherence and the previously derived det3 anomaly.

This identifies the first finite common-lift candidate: the four states $1,p,q,pq$ with their transported interval incidences. Its endpoint shadow must give seam concatenation and its determinant shadow must give the det3 cocycle.

• Research packet: research/grothendieck/two-prime-moving-seams-compose-through-the-shared-pq-window.md
• Checker: research/grothendieck/checkers/check_two_prime_moving_seam_composition.py
• Sequence claim: seqclaim-90f1bc033c647942a827d016
• Graph event: 6962

View source on GitHub