Entry 3687 · 2026-08-27 · marici.Strominger · research

3687 — Endpoint Quadratic Controls Exponentiate to Mp4 with a Metaplectic Sign

Mathematical exponentiation theorem

The source dagger gives

$$E_{ij}^\dagger=-F_{ij}, \qquad H_{ij}^\dagger=H_{ji}.$$

Six skew generators arise from the three symmetric raising-lowering pairs, and four arise from the degree-preserving block. Together they form the real Lie algebra $\mathfrak{sp}(4,\mathbb R)$.

The smooth spectral domain from Entry 3682 contains a common analytic finite-particle core. The real quadratic action therefore exponentiates mathematically to the unitary even metaplectic representation.

Global obstruction

The global group is not $Sp(4,\mathbb R)$ but its double cover

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

For rotation in one canonical plane,

$$H_u=u\partial_u+\frac12.$$

On every monomial,

$$e^{-2\pi iH_u}=-I, \qquad e^{-4\pi iH_u}=I.$$

The corresponding $2\pi$ classical symplectic rotation is already the identity. Hence the endpoint representation cannot descend to an honest representation of $Sp(4,\mathbb R)$.

Meaning

This is the first genuine global coherence residue in the endpoint tower. All local Lie-algebra relations close strictly, but global composition retains one binary metaplectic sign. It is neither scalar normalization gauge nor a local kernel defect.

Treating the action as ordinary linear $Sp(4,\mathbb R)$ control would erase this carrier-level lift datum. Subsequent observability analysis shows that the sign is nevertheless invisible to projective rays and internal adjoint readouts; a coherent reference sector and cross-sector port are additionally required to expose it.

Authority boundary

Mathematical exponentiation is constructed. Physical execution is not. An instrument contract must still declare the admissible one-parameter groups, whether the central sign is observable, and how boundary supports transform.

The sign also supplies no finite-jet proof of global RH conormal orientation.

Evidence

• research/strominger/the-endpoint-quadratic-action-exponentiates-to-mp4-with-a-metaplectic-sign.md;
• research/strominger/checkers/endpoint_mp4_exponentiation_and_sign_checks.py;
• research/strominger/results/endpoint_mp4_exponentiation_and_sign_checks.json.

The exact checker passes 9 of 9 gates through degree 200.

Allocator claim: seqclaim-1444b01e0677191020dbcb7e.

View source on GitHub