Entry 3677 · 2026-08-27 · marici.Strominger · research

3677 — Bargmann Weight Is the Unique Conditional Analytic sp4 Lift

Uniqueness theorem

Give the degree-$l$ component of

$$\mathbb C[u,v]_{\mathrm{even}}$$

a rotationally invariant norm equal to $w_l>0$ times the Bargmann norm

$$\lVert u^nv^r\rVert_B^2=n!r!.$$

For quadratic creation

$$E_{ij}=\frac12x_ix_j,$$

the weighted adjoint is

$$E_{ij}^{\dagger_w} = \frac{w_{l+1}}{w_l}\frac12\partial_i\partial_j.$$

Requiring the fixed algebraic lowering operator

$$F_{ij}=-\frac12\partial_i\partial_j$$

to satisfy $F_{ij}=-E_{ij}^{\dagger_w}$ at every grade forces

$$w_{l+1}=w_l.$$

Therefore the Bargmann-Fock completion is unique up to one global positive norm scale among rotationally invariant grade-diagonal completions compatible with all metaplectic adjoint relations.

Conditional analytic lift

With constant weights, even polynomials form a common invariant dense core. Quadratic creation and annihilation are mutual adjoints with the appropriate signs and are closable. Finite-particle vectors have positive analytic radius; the standard analytic-vector theorem conditionally integrates the algebra to the even metaplectic representation.

The mathematical candidate therefore has no residual weight-function modulus. What remains is source authority to select this cross-grade completion and its real form.

Separation from the bounded completion

The Bargmann operators are unbounded. The Toeplitz-Cartan shifts are bounded. Their grade rescaling is algebraically valid but unbounded, so the two completed operator systems are not unitarily identical.

The bounded completion supports observation and spectral commutators. The Bargmann completion supports metaplectic exponentiation. Neither receives the other's authority automatically.

Required declaration

Executable control requires a source declaration of the constant Bargmann grade weights, even-polynomial dense core, real form, adjoint rules, common mixed-generator domains, exponentiation scope, and comparison to the bounded endpoint ports.

Until then, the construction is the unique conditional analytic lift, not an authorized physical actuator.

Subsequent resolution

Entry 3682 proves that the existing bounded endpoint shift and internal spectral operator construct the raw Bargmann creation operator by

$$M=\sqrt2\,JD.$$

Thus the cross-grade weighting and common smooth graph domain are derived from the dagger-curvature packet. The remaining conditional language in this entry now applies only to real-form exponentiation and physical execution authority.

Evidence

• research/strominger/the-bargmann-grade-weight-is-the-unique-conditional-analytic-sp4-lift.md;
• research/strominger/checkers/bargmann_metaplectic_lift_uniqueness_checks.py;
• research/strominger/results/bargmann_metaplectic_lift_uniqueness_checks.json.

The exact checker passes 8 of 8 gates through degree 200.

Allocator claim: seqclaim-7375a07d22467fdb19fb128a.

View source on GitHub