Scale and conformal invariance in four-dimensional quantum field theory

Scale and conformal invariance in four-dimensional quantum field theory concerns whether a unitary quantum field theory that is invariant under uniform changes of length is also invariant under local conformal transformations. A manuscript in OpenAI's public mathematics collection, dated 26 September 2026, claims this implication in four spacetime dimensions for an operational class of theories defined by stated hypotheses on local algebras and fields. The collection has no Lean formalization of this result.

Background

As recounted in the manuscript's introduction:
- The improvement problem. In four dimensions a conserved dilatation current has the form x^νT_μν − V_μ, where the trace of the stress tensor is the divergence of the virial current V. Conformal invariance asks for an improvement of the stress tensor to a traceless one with the same charges; a local scalar L whose wave operator gives the trace suffices. Improvements of this kind go back to Callan, Coleman and Jackiw.
- Two dimensions and perturbation theory. In two dimensions the implication follows from Zamolodchikov's c-theorem and Polchinski's argument under suitable hypotheses. In four dimensions, local renormalization-group consistency conditions give a perturbative result, in the work of Fortin, Grinstein and Stergiou and of Jack and Osborn; Nakayama reviews the distinctions involved.
- Nonperturbative arguments. The a-theorem of Komargodski and Schwimmer led Luty, Polchinski and Rattazzi, and then Dymarsky, Komargodski, Schwimmer and Theisen, to arguments based on dilaton scattering; the manuscript describes these as leaving a passage from on-shell dilaton constraints to an off-shell local trace identity and as retaining infrared-regulation hypotheses. Bzowski and Skenderis, and Dymarsky, Farnsworth, Komargodski, Luty and Prilepina, analysed contact terms and a generalized-free trace. Yonekura and Sachs gave arguments aimed directly at the local scalar under further assumptions.

The original result

Claim, 26 September 2026. The main theorem of Scale and conformal symmetry in four-dimensional operational quantum field theory concerns a unitary, positive-energy theory on four-dimensional Minkowski space with a Poincaré- and dilatation-invariant vacuum, whose fields have a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities and finite scaling support, have compatible realizations in a causally commuting net of bounded local algebras, and include a physical stress tensor and virial current with local Ward identities. Under these hypotheses it states that there is a physical Hermitian scalar field L of dimension two with which the stress tensor admits a symmetric, conserved, traceless improvement with the same translation and Lorentz charges, so that the currents of conformal Killing vector fields have unbroken local Ward identities.
Scope, as stated: the conclusion is a local conformal Ward action; integration to a global conformal action on the whole net is a separate question. The hypotheses delimit the class, and the manuscript discusses the free massless scalar as an example in which the scalar is already present.
Method, as described: the proof uses modular theory of the vacuum on wedges, cones and diamonds and half-sided modular inclusions instead of a Lagrangian or scattering states. Bounded null cuts give positive generators; a dispersion comparison between two null horizons removes a critical scalar tail; actual modular symmetries extract a dimension-two scalar wave from the virial field; and a reconstruction of all mixed products, a common domain and affiliation with the original net make this wave a physical field.
Sources. Commit adc7f12 of the openai/math repository: Scale and conformal symmetry in four-dimensional operational quantum field theory.
Formalization. The collection lists no Lean formalization or Comparator challenge for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-07	Scale and conformal symmetry: reconstructing the improvement field, an article of the Latent Seminar blog at lowrank.github.io	Reading, exposition	Explains family 282: fields, local operations and currents; the theorem and its hypotheses; a free-field example; the trace wave and modular comparisons; eliminating the critical spectral tail; from physical modular motion to an accessible scalar; mixed products and the operator domain; local conformal Ward identities.	Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. Read on 31 October 2026.
2026-10-16	J. Jardine, Lattice24, referee review rank_122 in JJardine919/openai-math-audit	Review, AI-assisted audit	Reports no gap in the parts it read: the introduction, the main theorem, the framework section and parts of the dispersion and short-product sections, with symbolic checks of the free-scalar example and several identities. It states that the bounded-profile Ward formula, the cone modularity and time-Möbius action, the extraction of the scalar and the mixed-product reconstruction were not verified in detail, so that the verdict covers only the parts listed.	Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.
2026-10-18	AI Proved What?, episode 282 Scale implies conformal, proved at last?, remade version of 23 October 2026	Reading, exposition	Video explainer of the 112-page preprint and its hypotheses. It recalls that Polchinski wrote in 1988 that neither a proof nor a counterexample was known in three or more dimensions, that four-dimensional perturbative proofs date from 2012 and that a nonperturbative proof claimed in 2013 was later withdrawn. It states that the preprint does not claim a global conformal action, theories outside its hypotheses or curved spacetime, and that it knows of no interacting theory shown to satisfy the hypotheses.	Episode status, last checked 30 October 2026: posted as a preprint, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, no. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.

Notes

The background summarizes the manuscript's own account of earlier work.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.