Entry 3682 · 2026-08-27 · marici.Strominger · research

3682 — Raw Metaplectic Creation Is the First-Order Lift of the Bounded Endpoint Shift

Factorization theorem

Let $C_l$ be the normalized Cartan coisometry. Raw binary-form multiplication and the bounded endpoint shift satisfy

$$M_l=\sqrt{(2l+1)(l+1)}\,C_l,$$

and

$$J_l=c_lC_l, \qquad c_l^2=\frac{2(2l+1)}{l+1}.$$

Therefore

$$M_l=\frac{l+1}{\sqrt2}J_l.$$

Since the internal spectral operator is

$$D|_{H_l}=\frac{l+1}{2}I,$$

the exact first-order factorization on the finite-grade core is

$$M=\sqrt2\,JD, \qquad M^\dagger=\sqrt2\,DJ^\dagger.$$

Domain theorem

The smooth spectral domain

$$\mathscr H_D^\infty = \bigcap_{k\geq0}\operatorname{Dom}(D^k)$$

is invariant under $J$ and $J^\dagger$, because these shifts change the eigenvalue of $D$ by exactly one half. It is therefore a source-derived common Fréchet domain for the first-order lifted quadratic generators.

The domain and graph norms required by Entry 3673 are not independent completion data. They descend from the existing bounded spectral packet $(J,J^\dagger,D)$.

Consequence

The bounded Toeplitz and unbounded metaplectic systems are not unrelated completions. The latter is a first-order lift of the former. The unbounded grade factor that previously separated them is precisely the internally reconstructed spectral operator.

This also resolves the conditional weighting in Entry 3677: the endpoint dagger-curvature packet constructively selects the Bargmann cross-grade weighting once the normalized spinor identification is retained.

Remaining authority

The construction supplies the algebraic operators, common smooth domain, and graph norms. It does not authorize physical exponentiation. The remaining gates are the real form, self-adjoint or skew-adjoint closures of selected combinations, allowed one-parameter groups, physical interpretation, and boundary-support compatibility.

It also does not turn finite local $\mathfrak{sp}_4$ jets into a proof of a global infinite-dimensional seam condition.

Evidence

• research/strominger/the-raw-metaplectic-creation-is-the-first-order-lift-of-the-bounded-endpoint-shift.md;
• research/strominger/checkers/endpoint_first_order_metaplectic_lift_checks.py;
• research/strominger/results/endpoint_first_order_metaplectic_lift_checks.json.

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

Allocator claim: seqclaim-9cccd018cd72ed3b4b8af089.

View source on GitHub