Literal quotient is two-dimensional in the four-dimensional parity-even model

Definitions and mathematical statement

Problem 6 asks whether $L/(L'+L'')$ is one-dimensional and generated by the class of $Q$. The extracted question does not define its three spaces or dimension. The paired aid reads $L$ literally as the full space of critical-weight local invariants and gives a counterexample in a specified four-dimensional category; that interpretation must be distinguished from a more restrictive numerator.

In the aid's fixed-dimension parity-even polynomial model, consider universal natural scalar densities in dimension four, modulo identities valid for all metrics, rather than values on one special metric. Set \[ J=\tfrac16R,\qquad P=\tfrac12(\operatorname{Ric}-Jg), \] where $W$ is Weyl curvature, and choose a consistent Laplacian convention. The reported complete critical-weight space is \[ L=\operatorname{span}\{J^2\,dv_g,\ |P|^2\,dv_g, |W|^2\,dv_g,\ (\Delta J)\,dv_g\}. \] In this category the aid identifies the local conformal-gradient subspace and pointwise conformal-invariant subspace as \[ L'=\operatorname{span}\{(\Delta J)\,dv_g\},\qquad L''=\operatorname{span}\{|W|^2\,dv_g\}. \] Here local conformal gradients refer to the variational category used by the aid, whose full original definition is not supplied.

With the convention \[ Q_4=-\Delta J-2|P|^2+2J^2, \] the reported conclusion is \[ \dim L/(L'+L'')=2, \qquad [J^2\,dv_g]\text{ and }[Q_4\,dv_g]\text{ are independent}. \] The sign of the displayed Laplacian term is convention dependent and lies in the denominator subspace, so it does not affect this independence. Thus the one-dimensional claim is false under this full parity-even polynomial interpretation.

The aid contrasts a repaired numerator: in its conformal-index/formally-self-adjoint-density interpretation the allowable space is \[ L_{\mathrm{restricted}}= \operatorname{span}\{Q_4\,dv_g,\ (\Delta J)\,dv_g,\ |W|^2\,dv_g\}. \] Formal self-adjointness here concerns the conformal linearization of a density as an operator on conformal factors, not a scalar function being an operator by itself. The reported repaired quotient is one-dimensional and generated by $[Q_4\,dv_g]$. The term conformal-index and its equivalence to that linearization condition need the missing definitions. This restricted result is not the same assertion as the literal full-space quotient, and the four-dimensional calculation is not a classification of every possible invariant category or dimension.

Short English statement

The question depends on the omitted definitions of its spaces. The aid reports that the full four-dimensional parity-even polynomial quotient is two-dimensional, with an extra $J^2$ class. A narrower conformal-index/self-adjoint numerator instead leaves the expected single $Q_4$ class. These two versions must not be identified.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[362], Problem 6. Title: "Literal quotient is two-dimensional in the four-dimensional parity-even model."
• AIM workshop Conformal structure in geometry, analysis, and physics, supplied reference https://aimath.org/WWN/confstruct/confstruct.pdf. The extracted question supplies no definitions, remarks or literature entries; this draft uses the supplied text without accessing a PDF.
• Supplied attempt 1: complete weight-four basis, denominator identifications, extra quotient class and restricted-numerator comparison. No authoritative external bibliographic citation is supplied.
• Registry dated 14 August 2026: "solved," "SOLVED-BY-YOU," statement status "exact"; research-run status partially solved, result "counterexample," rigor "proved," novelty confidence low. These are source metadata, not authorship or review claims of this draft. The machine-generated, non-peer-reviewed aid requires independent verification before citation.