A source and tractor-slot audit of Other routes to Q

Definitions and mathematical statement

Context-only page split. The first line, "Problem 30, in the form $\mathrm{const}\cdot\operatorname{Pff}+L+V$?", is identified by the supplied verification/aid as the end of Problem 32, not a complete newly numbered problem. The remainder is the exposition "Other routes to $Q$", leading toward Problem 33 outside this record. Footnotes say this section recompiles a document of T. Branson and R. Gover and that Problems 29--32 are conjectures addressed there. The missing antecedents are not supplied by inventing a decomposition target.

For a smooth Riemannian conformal manifold of dimension $n\ge4$, write $J=R/(2(n-1))$ and $P=(\operatorname{Ric}-Jg)/(n-2)$. The standard tractor bundle splits in a metric as \[ \mathcal T=\mathcal E[1]\oplus(T^*M\otimes\mathcal E[1]) \oplus\mathcal E[-1], \] with normal tractor connection \[ \nabla_a^{\mathcal T}\begin{pmatrix}\sigma\\\mu_b\\\tau\end{pmatrix} =\begin{pmatrix} \nabla_a\sigma-\mu_a\\ \nabla_a\mu_b+g_{ab}\tau+P_{ab}\sigma\\ \nabla_a\tau-P_a{}^b\mu_b \end{pmatrix}. \] This is an explicit interpretation of the extracted middle-slot $V\sigma$ and contracted bottom-slot notation. Let $I_g=(2-n,0,J)\in\Gamma(\mathcal T[-1])$. Put $\Delta_0=\nabla^a\nabla_a$ and define the coupled operator \[ \Box=-\nabla^a\nabla_a+\frac{(n-2)R}{4(n-1)}. \] The scalar $K$ in the extracted coefficient is interpreted as scalar curvature $R$; a full convention check remains necessary.

Supplied recovery certificate. The aid gives the unspecialized metric-splitting calculation \[ \Box I_g=\begin{pmatrix} -\tfrac12n(n-4)J\\ (n-4)\,{\,\mathrm{d}} J\\ -\Delta_0J+\tfrac n2J^2-(n-2)|P|^2 \end{pmatrix}. \] In dimension four the first two slots vanish and the third is \[ Q_4=-\Delta_0J+2J^2-2|P|^2. \] The unspecialized identity is a coupled-connection calculation; it does not assert that this same second-order operator on $\mathcal T[-1]$ is conformally invariant in every dimension. The conformally invariant coupled Laplacian at these weights is the dimension-four case.

Higher even-dimensional source construction. For even $n\ge4$, the source asserts an operator $\Box_{n-2}$ with leading part $(-\Delta_0)^{n/2-1}$ plus lower-order terms and the appropriately weighted identities \[ \Box_{n-2}I_g=(0,0,Q_n),\qquad I_{e^{2\omega}g}=I_g+D\omega,\qquad \Box_{n-2}D\omega=(0,0,P_n\omega). \] Here $D$ is the second-order tractor $D$ operator and $P_n$ the critical GJMS operator; the composition has order $n$. The pure bottom slot is interpreted through the canonical bottom filtration, with conformal densities, rather than as an unweighted triple invariant under every change of splitting. The resulting scalar law is \[ e^{n\omega}Q_n(e^{2\omega}g)=Q_n(g)+P_n(g)\omega, \] or, with $q_g=Q_n(g)\,{\,\mathrm{d}} V_g$, $q_{e^{2\omega}g}=q_g+(P_n(g)\omega)\,{\,\mathrm{d}} V_g$. Conformal covariance of $P_n$ makes successive scale changes agree with one change by their sum. The extracted unweighted-looking $\widehat Q_n=Q_n+P_n\omega$ suppresses these identifications.

The exposition says an ambient-to-tractor algorithm expands the result into curvature and derivative formulas in small dimensions while avoiding dimensional continuation. It does not provide all lower-order terms of $\Box_{n-2}$ or a new complete question here. The $n-4$ slot factors, order count and density cocycle are the aid's proposed consistency checks for recovering the text, not a claimed new solution of a separate problem.

Short English statement

This mixed extract is contextual tractor exposition, beginning with a cut-off cross-reference. The supplied aid checks the three slots of the coupled operator and recovers $Q_4$ when the upper slots vanish. Its higher-dimensional statements require density weights and the conformal transformation factor; no independent missing problem is reconstructed.

Sources

• Local UnsolvedMath registry, permanent ID 20002056, source code catalogue entry; canonical aim-geometry-notes.json, notes[393].
• AIM Conformal structure in geometry, analysis, and physics, page-split continuation and "Other routes to $Q$", equations (41)--(42): https://aimath.org/WWN/confstruct/confstruct.pdf.
• Extracted footnotes attribute the underlying document and Problems 29--32 to T. Branson and R. Gover; full document details and a separate proving citation for the slot calculation are absent here.
• Registry research attempt 1, dated 2026-08-14: context only, status at run not a problem, rigor "proved", novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.