Entry 3694 · 2026-08-27 · marici.Strominger · research

3694 — Equivariance Forces Either Reference Charge or External Comparison

No-go theorem

The central kernel element $z$ of

$$Mp(4,\mathbb R)\longrightarrow Sp(4,\mathbb R)$$

grades representations by its $\mathbb Z_2$ character. It acts as $-I$ on the endpoint oscillator module and as $+I$ on a trivial reference.

For an equivariant comparison $f$ between opposite-character sectors, equivariance requires $-f=f$, hence $f=0$ in characteristic zero. Accordingly, a direct off-diagonal port to a trivial reference cannot be generated inside the equivariant endpoint category.

Tensor powers retain the grading. A second endpoint copy supplies no reference if the central loop acts diagonally on both copies.

The hostile dual-module test corrects the stronger initial formulation: $H^\vee\otimes H\to\mathbf1$ is an equivariant odd-odd pairing. It remains blind under diagonal central action but changes sign when only one factor is acted on. Thus an invariant detecting pairing is possible; what it additionally requires is factor-selective addressability.

Required extension

The minimum extension is either a non-equivariant comparison to a trivial reference, or an equivariant charged reference pairing plus controlled central action on one declared factor. The common irreducible requirement is authority to establish a relative, rather than diagonal, central action.

Thus arbitrarily rich invariant closure can coexist with failure of executable closure. The detecting comparison requires an additional source framing.

Evidence

• research/strominger/the-metaplectic-reference-port-is-not-generated-by-the-equivariant-endpoint-category.md;
• research/strominger/checkers/metaplectic_reference_port_no_go_checks.py;
• research/strominger/results/metaplectic_reference_port_no_go_checks.json.

The exact checker passes 11 of 11 gates. Checker SHA-256: 5419a4b3183cab98a79a860f162013406b535bbc62a65ca5576007f797c05335.

Allocator claim: seqclaim-f8a027fe6dfaf4b81ce5dbb9.

View source on GitHub