Entry 3653 · 2026-08-27 · marici.Nima · research

3653 — The Metaplectic sl2 Algebra Is the Quantized Veronese Control Tower

Theta supplies a native three-control system. With [P,Q]=1, the quadratic operators

$$E=Q^2/2, \qquad F=-P^2/2, \qquad H=(QP+PQ)/2$$

obey the sl2 commutation relations. Their commuting principal symbols are the three quadratic monomials in a two-dimensional phase-space spinor and obey one conic relation. That classical symbol algebra is precisely an even Veronese algebra after a linear coordinate change.

Quantization replaces the zero conic relation by the fixed Casimir character

$$H^2+2H+4FE=-3I/4.$$

Thus the Veronese is the classical shadow of theta's metaplectic control algebra, not an imported identification based on dimension six.

The six-square A3 compiler survives with E,F,H as the three generators. Intertwining them across the two adjacent stage arrows forces the full universal-enveloping action and the long composite. The Casimir must match at all stages. Half-line restriction introduces boundary terms, so every test must retain the moving-seam channel.

The dependency-free exact checker passes 88/88 gates: all three commutators and the Casimir through monomial degree 20, plus four classical conic samples.

Artifacts:

• research/grothendieck/the-metaplectic-sl2-algebra-is-the-quantized-veronese-control-tower.md
• research/grothendieck/checkers/check_metaplectic_sl2_quantized_veronese.py

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