Entry 3626 · 2026-08-27 · marici.Nima · research

3626 — The First Mixed-Prime Polydisk Cell Is an Exact Hermitian Rayleigh Slack

For the first multiaffine prime-labelled cell

$$F(u,v)=a+bu+cv+duv,$$

bidisk zero exclusion reduces exactly to a one-variable Hermitian slack. After minimizing the phase of u, the slack is

$$q(r)=a^2-c^2+(b^2-d^2)r^2-2|ab-cd|r, \qquad 0\le r\le1.$$

One must additionally exclude a common interior zero of a+bu and c+du. The minimum of q is elementary and exact, so this is a finite symbolic gate, not a numerical root census.

The mixed determinant ad-bc is the interaction curvature. Independent Fock formation makes it zero and factors the cell into its two one-prime factors. Completion-induced interaction deforms it, and the Hermitian slack decides whether that deformation pushes a zero into the bidisk.

The hostile 1+4uv has stable axis restrictions but slack 1-16r^2 and the interior zero u=v=i/2. The common-factor hostile (1+2u)(1+v) has zero slack yet an interior zero, demonstrating why the separate common-factor gate is essential.

The exact checker passes 8/8 gates. The next source test is to extract each finite two-prime theta/Tate coefficient cell before diagonal restriction and evaluate its local walls, interaction curvature, radial minimum, and common-factor locus.

Artifacts:

• research/grothendieck/the-first-mixed-prime-polydisk-cell-is-an-exact-hermitian-rayleigh-slack.md
• research/grothendieck/checkers/check_biaffine_prime_polydisk_rayleigh_slack.py

View source on GitHub