Conformal stress improvement in the finite-scaling-support operational framework

Conformal stress improvement in the finite-scaling-support operational framework

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The designated four-dimensional scale-invariant operational QFT class is claimed to admit a traceless stress-tensor improvement and unbroken local conformal Ward identities.

Mathematical statement of the claimed result

Result 1.

The framework assumes vacuum-generated four-dimensional Wightman fields and their adjoints on a common invariant polynomial domain, positive translation spectrum, Poincaré covariance, graded locality, and a vacuum invariant under continuous dilations. Original field labels decompose algebraically into finite scaling support, \[ \mathcal F=\bigoplus_{\Delta\in\Sigma}\mathcal F_\Delta, \] where real $\Sigma$ is bounded below and locally finite and each Lorentz-stable grade is finite-dimensional. Adjoints preserve grades and differentiation raises dimension by one. A fixed covariant isotone local net contains closed affiliated realizations; new physical fields require arbitrarily small localization, all joint finite-length distributions, compatible domains/adjoints, and affiliation in that original net. The original stress tensor is symmetric and conserved, with translation/Lorentz Ward identities, and a virial field generates dilations through $D_\mu=x^\nu T_{\mu\nu}-V_\mu$, $T^\mu{}_{\mu}=\partial^\mu V_\mu$. In this specified framework the claim supplies a Hermitian physical dimension-two scalar $L$ with \[ \partial^\mu V^{(3)}_\mu=\Box L,\qquad t_{\mu\nu}=T^{(4)}_{\mu\nu} +\tfrac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L, \] symmetric, conserved, traceless, and carrying the same affine Ward charges. Conformal-Killing currents $X^\nu t_{\mu\nu}$ extend the unbroken local Ward action. This theorem retains finite scaling support and the operational field-membership conditions. The common invariant domain must contain \[ D_{\mathrm{old}}=\operatorname{span}\{\Omega,A_1\cdots A_n(h)\Omega: n<\infty,\ h\in\mathcal S((\mathbb R^4)^n)\}, \] with the original fields acting by their original insertion rule. Here $h$ is an arbitrary joint test kernel, without a finite-rank restriction. Homogeneous scaling has convention \[ U_D(s)A_\Delta(x)U_D(-s)=e^{s\Delta}A_\Delta(e^s x). \]

Outline of the argument

Bounded null-cut identities and two-horizon spectral comparisons isolate a localizable scalar wave. Quadratic-charge and time-cone estimates establish compatible local products, enabling stress improvement without changing Poincare charges; the conclusion concerns local currents rather than a global action on the entire net.

Formal verification

Family 282 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, Scale and conformal symmetry in four-dimensional operational quantum field theory, 2026-09-26; citation key OAI:Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI catalogue entry.