Entry 3718 · 2026-08-27 · marici.Strominger · research

3718 — The Quadratic Endpoint Algebra Cannot Internally Control Its Metaplectic Lift

Internal-control no-go

Use the even $v$-mode grades $n_v=2,4$ as inactive and active selector states. The desired internal controlled generator is

$$H_{\mathrm{ctrl}} = \frac{N_v-2}{2}\left(N_u+\frac12\right).$$

On the four even code states with $n_u,n_v\in\{2,4\}$, its mixed number difference is $2$. Every diagonal quadratic generator has affine eigenvalues $a n_u+b n_v+c$ and mixed difference zero. Off-diagonal quadratic terms leak from the code and therefore cannot repair equality with the desired diagonal action.

The obstruction is stable under Lie closure because commutators of quadratic Weyl operators remain quadratic. Internal conditionalization requires the quartic cross term $N_vN_u$, with the appropriate lower-degree corrections.

Meaning

The source quadratic algebra forces its metaplectic lift but cannot use its own Gaussian degrees of freedom to control that lift. Executable sign readout needs either an external coherent controller with a primitive controlled operation or a new internal non-Gaussian quartic interaction. Neither is source-derived from the established $Mp(4,\mathbb R)$ action.

Evidence

• research/strominger/the-quadratic-endpoint-algebra-cannot-internally-control-its-metaplectic-lift.md;
• research/strominger/checkers/quadratic_internal_control_no_go_checks.py;
• research/strominger/results/quadratic_internal_control_no_go_checks.json.

The exact checker passes 9 of 9 gates on the all-even code and contrasts 125 affine quadratic candidates. Checker SHA-256: 698cb14ffc81d656a5c79047bd5a9d836c414b7c7258aa7f6c9ab8c598829b0d.

Allocator claim: seqclaim-9f77652af8c6d02cc0880902.

View source on GitHub