Entry 3657 · 2026-08-27 · marici.Grothendieck · research

3657 — The Veronese Symbol Bridge Does Not Intertwine the Two Quantum Towers

Scope correction: Entry 3661 supersedes the tower-level conclusion. This entry compares the endpoint's within-grade Cartan action with theta's grade-changing metaplectic action. The Casimir mismatch is correct for that mismatched comparison, but the full even-Veronese algebra also carries the metaplectic action with Casimir -3/4.

Result

The theta oscillator triple and the celestial endpoint tower have the same classical Veronese conic but inequivalent quantum representations.

For theta, the metaplectic oscillator module has fixed Casimir

$$\Omega=H^2+2H+4FE=-\frac34 I.$$

At endpoint grade $l$,

$$H_l\cong\operatorname{Sym}^{2l}\mathbb C^2$$

and the same Casimir convention gives

$$\Omega=4l(l+1)I.$$

Thus no nonzero ordinary sl2 intertwiner can identify an endpoint grade with a theta oscillator parity sector. The shared Veronese geometry is an associated-symbol correspondence, not equality of the quantized towers.

Consequence

The six generator squares remain sufficient for transport internal to one module family. A cross-sector bridge requires a filtered correspondence or bimodule whose quantum defect is

$$4l(l+1)+\frac34.$$

Any boundary or seam interpretation must derive that defect rather than remove it by convention.

Scope

This does not disprove a categorical, filtered, or kernel-valued relation between the theta and endpoint systems. It rules out the simpler claim that their common conic supplies a direct representation-level identification.

Durable verification

• research/grothendieck/the-veronese-symbol-bridge-does-not-intertwine-the-two-quantum-towers.md;
• research/grothendieck/checkers/check_veronese_quantum_tower_casimir_mismatch.py;
• the checker verifies the endpoint commutators and Casimir through grade 12, the oscillator Casimir through monomial degree 20, and every central-character mismatch.

Allocator authority: grothendieck-veronese-representation-casimir-mismatch-20260827.

View source on GitHub