Entry 3663 · 2026-08-27 · marici.Grothendieck · research

3663 — The Even-Veronese Endpoint Carries a Commuting Metaplectic Pair

Result

The apparent spectator coordinate in the endpoint realization carries a second metaplectic action. On

$$\mathcal C=\mathbb C[u,v]_{\mathrm{even}},$$

the two coordinatewise oscillator triples generate

$$\mathfrak{sl}_2^{(u)}\oplus\mathfrak{sl}_2^{(v)},$$

with all cross-commutators zero and Casimir $-3/4$ in both factors.

The even algebra decomposes into even-even and odd-odd tensor products of oscillator parity modules. Spinor exchange swaps the two metaplectic factors.

Consequence

This is a native six-generator control system: three controls for each spinor axis. It is distinct from the six interchange squares obtained by testing three controls across two directed stage arrows.

If theta reciprocal sewing realizes spinor exchange, the diagonal control triple is sewing-even and the relative triple is sewing-odd. That would place the seam channel inside an existing relative metaplectic action rather than adding another generator.

Scope

The algebraic factor exchange is proved. Its identification with theta reciprocal sewing remains a source-level gate.

Durable verification

• research/grothendieck/the-even-veronese-endpoint-carries-a-commuting-metaplectic-pair.md;
• research/grothendieck/checkers/check_even_veronese_commuting_metaplectic_pair.py.

Allocator authority: grothendieck-even-veronese-commuting-metaplectic-pair-20260827.

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