Scale invariance implies local conformal invariance for a class of four-dimensional unitary QFTs

Theorem 1.1: under the paper's framework (unitary, positive energy, Poincare- and scale-invariant vacuum, discrete bounded-below dimension spectrum with finite multiplicities, finite scaling support, bounded local net, physical stress tensor and virial current), there is a Hermitian dimension-two physical scalar $L$ with $\partial^\mu V^{(3)}_\mu=\Box L$, so $T^{(4)}_{\mu\nu}+\tfrac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L$ is symmetric, conserved and traceless with unchanged translation and Lorentz charges, and conformal currents have unbroken local Ward identities. It does not treat theories outside these hypotheses (continuous spectra, no bounded net, no local virial current), and integration to a global conformal action on the net is left open.

Theorem 1.1: under the paper's framework (unitary, positive energy, Poincare- and scale-invariant vacuum, discrete bounded-below dimension spectrum with finite multiplicities, finite scaling support, bounded local net, physical stress tensor and virial current), there is a Hermitian dimension-two physical scalar $L$ with $\partial^\mu V^{(3)}_\mu=\Box L$, so $T^{(4)}_{\mu\nu}+\tfrac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L$ is symmetric, conserved and traceless with unchanged translation and Lorentz charges, and conformal currents have unbroken local Ward identities. It does not treat theories outside these hypotheses (continuous spectra, no bounded net, no local virial current), and integration to a global conformal action on the net is left open.