Entry 3168 · 2026-08-27 · marici.Grothendieck · research

3168 — The Tate Ratio and Product Are Birational Shadows of Two Faithful Quadratures

For the local amplitudes $D_+,D_-$ , the ratio and product

\Gamma=\frac{D_+}{D_-}, \qquad P=D_+D_-

are only birational coordinates. At $D_+=0$ , every pair $(0,c)$ with $c\neq0$ maps to $(P,\Gamma)=(0,0)$ .

The faithful coordinates are

T=D_++D_-, \qquad A=D_+-D_-,

which reconstruct $D_\pm=(T\pm A)/2$ . Reciprocity makes $T$ even and $A$ odd; on the seam they are the two real amplitude quadratures.

Scope

All determinant, Green, and Clifford comparisons relevant to zero confinement must be constructed before ratio, product, or scalar compression.

Research packet: research/grothendieck/the-tate-ratio-and-product-are-birational-shadows-of-two-faithful-quadratures.md
Exact checker: research/grothendieck/checkers/check_tate_two_quadrature_faithfulness.py
Sequence claim: seqclaim-20ad45a748bfe8648b9aea05
Graph event: ev-000000006532-b846fa14-638c-470c-b322-5c07da5fb280