Distinguishing intrinsic and boundary order-three conformal operators

Definitions and mathematical statement

The paired aid identifies this canonical record as a splice rather than a separate problem: its opening sentence finishes Problem 11, while the paragraph about order-three operators introduces Problem 12. The absent antecedent of "this would provide a unique specification of $Q$" is not reconstructed. The text mentions Rod Gover, Alice Chang and Jie Qing and an order-three $P_3$ on a three-manifold occurring as a boundary or embedded hypersurface, with an associated $Q_3$. It does not specify the domain, locality or normalization of the operator.

The aid supplies a locality diagnostic. On a manifold of dimension at least two, a scalar formally self-adjoint differential operator of odd order cannot be elliptic. In the usual scalar Hermitian symbol convention, its principal symbol is real and odd in the cotangent variable; connectedness of the unit cotangent sphere forces a zero. Thus an intrinsic scalar self-adjoint elliptic order-three operator cannot be a finite-order local differential operator. A pseudodifferential realization is a possible nonlocal alternative, not an identification forced by the source wording alone.

The aid distinguishes the local Chang--Qing boundary $P_3$: its input is an ambient function near a hypersurface $\Sigma^3\subset X^4$, and it depends on third normal jets. It does not factor through the boundary trace. Writing $B_3$ for this type of operator to distinguish its domain, the diagnostic is that no intrinsic trace operator $\overline P_3$ satisfies \[ B_3u=\overline P_3(u|_\Sigma)\qquad\text{for every ambient }u. \] The reported third-normal-jet dependence permits functions with zero trace but nonzero output. This boundary operator is not a counterexample to the intrinsic scalar lemma; its self-adjoint boundary theory is not automatically the same as an $L^2(\Sigma)$ operator on arbitrary boundary traces.

For a separately specified intrinsic pair on a closed three-manifold, suppose \[ e^{3\omega}P_{3,\widehat g}f=P_{3,g}f,\qquad e^{3\omega}Q_{3,\widehat g}=Q_{3,g}+P_{3,g}\omega, \qquad\widehat g=e^{2\omega}g, \] with $P_{3,g}$ formally self-adjoint and $P_{3,g}1=0$. The aid reports that the affine law is consistent under composition of conformal changes and that \[ \int_\Sigma Q_{3,\widehat g}\,dv_{\widehat g} =\int_\Sigma Q_{3,g}\,dv_g. \] For a fixed $P_3$, two $Q_3$ assignments obeying this same law differ exactly by a scalar conformal invariant of weight $-3$, meaning $I_{\widehat g}=e^{-3\omega}I_g$. Conversely adding any such invariant preserves the law. This is uniqueness modulo that ambiguity, not an absolute uniqueness statement or an existence theorem for the unspecified source pair.

Short English statement

This extraction is contextual material, not a complete independent problem. The supplied aid distinguishes an intrinsic nonlocal order-three operator from a local boundary operator acting on ambient jets. For a fixed intrinsic self-adjoint conformal pair it records total-integral invariance and the precise conformal-invariant ambiguity in $Q_3$.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[368], canonical source item 11. Title: "Distinguishing intrinsic and boundary order-three conformal operators."
• AIM workshop Conformal structure in geometry, analysis, and physics, supplied reference https://aimath.org/WWN/confstruct/confstruct.pdf. The research summary identifies a splice between Problems 11 and 12. This draft uses the supplied record without accessing a PDF.
• The source mentions Rod Gover, Alice Chang and Jie Qing; the aid specifically identifies the local Chang--Qing boundary operator. Full bibliographic citations, exact formulas and boundary conventions are absent. Attempt 1 supplies the symbol/trace diagnostic and conditional intrinsic-pair consequences.
• Registry dated 14 August 2026: status "open," classification "OPEN-TRIAGE," statement status "exact"; attempt context only, research-run status not a problem, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.