Existence, Continuum Limits, and Classification Frontiers

An existence or classification result applies to a specified model, dimension, observable class, and equivalence notion. This chapter compares major mathematical-QFT problems by the exact conclusion already proved and the next inference still requiring proof.

Enter this chapter

Use the first topic to formulate one concrete claim, then follow its model-specific branch. The comparison table retains its stated September and August 2026 source cutoffs. Helpful background. The non-Abelian spectral formulation distinguishes a vacuum gap from confinement observables. Chiral gauge theories on the lattice supplies the regulator problem, and the rigorous-status guide separates theorem, construction, and evidence.

Topics

1. Existence, Uniqueness, and Equivalence Claims — Separates constructing one theory from proving uniqueness or equivalence.
2. Two- and Three-Dimensional Constructive Model Atlas — Compares successful constructions in their exact low-dimensional regimes.
3. Four-Dimensional Scalar QFT: Existence and Triviality — States the model classes covered by scalar triviality results.
4. Yang–Mills Existence and the Mass Gap — Defines the joint continuum-existence and positive-gap problem.
5. Rigorous Mass-Gap, Scattering, and Confinement Obligations — Distinguishes a gap from particle completeness and confinement.
6. Chiral Gauge Theories and Standard Model Construction — Identifies what anomaly-free regulator proposals still need to establish.
7. Interacting Curved-Spacetime and Gauge Existence Problems — Separates formal local constructions from global physical representations.
8. Continuum Limits and Universality Problems — Specifies the topology and observables in a universality claim.
9. Lattice-to-Continuum Constructive Proof Obligations — Lists the uniform estimates needed to remove lattice regulators.
10. Classification and Comparison Problems across QFT Frameworks — Keeps framework comparisons directional and hypothesis-dependent.
11. Conformal-Net, VOA, TQFT, and Categorical Classification Frontiers — Fixes the class and realization assumptions behind classification statements.
12. Reviewing Progress on Open Mathematical-QFT Problems — Shows how a new result changes a precisely bounded open problem.

How the ideas fit together

Nearby results can solve different problems. Scalar Gaussianity concerns the stated critical scaling class Aizenman and Duminil-Copin 2021, Theorem 1.2 (PDF). The gauge–Higgs theorem gives a massive Gaussian limit under coupled parameter scaling Chatterjee 2026, Theorem 3.2 (PDF); it does not solve the Clay Mathematics Institute Yang–Mills problem. Likewise, the ordinary-categorical AQFT comparison does not settle its infinity-localization problem Benini et al. 2026, abstract and § 5 (PDF). The chiral-regulator row refers to the specific constructions of Thorngren, Preskill, and Fidkowski 2026, abstract (PDF).

Hypotheses and licensed conclusions

Representative existence and classification frontiers: scalar and Yang–Mills sources checked 2026-09-07; other dated comparisons retain the 2026-08-10 cutoff

Claim target	Data that must remain fixed	Primary theorem or construction class	Licensed conclusion	Excluded upgrade and first open obligation
Low-dimensional constructive models	Exact interaction, two or three dimensions, ultraviolet and volume regulators, coupling regime, correlation hierarchy, and reconstruction hypotheses	Model-specific constructive estimates for P(φ)2, weakly coupled φ43, or massive Gross–Neveu2	Continuum correlations and further properties only for the stated model and regime	No transfer to four dimensions and no automatic scattering completeness; prove each added spectral statement separately
Four-dimensional ferromagnetic scalar scaling	Nearest-neighbor Ising-type or lattice-cutoff λφ4 class, critical or near-critical scaling, normalized correlations, and corrected hypotheses	Aizenman–Duminil-Copin marginal-triviality theorem, 2021 with 2024 corrigendum	Gaussianity of subsequential scaling limits in the theorem’s classes	Not a theorem for every scalar action or continuum trajectory; extend the argument or construct a different non-Gaussian target
Four-dimensional pure Yang–Mills	Compact simple gauge group, pure gauge fields on Euclidean four-space, axiomatic continuum object, positivity, infinite volume, and vacuum spectrum	Jaffe–Witten problem statement and dated institutional status	A precise open construction-plus-gap problem; nearby lattice and Yang–Mills–Higgs theorems remain partial evidence	No constructed pure continuum theory with the required positive gap; prove regulator removal and the separate spectral lower bound
Anomaly-free chiral gauge theory	Gauge group and representations, local regulator, exact gauge symmetry, anomaly cancellation, target chirality, positivity, and continuum limit	Exact lower-dimensional Hamiltonians and partial four-dimensional symmetry-disentangler constructions	A rigorous route for stated lower-dimensional models and specified four-dimensional symmetry data	Anomaly cancellation alone does not construct the Standard Model; supply the missing local Hamiltonian, interface proof, gauging, and continuum recovery
Interacting theory on curved spacetime	Globally hyperbolic background, compact interaction support, field content, gauge complex, renormalization prescription, and coefficient ring	Locally covariant perturbative algebras or abstract local C-star relations	Formal local observables, or an abstract local algebra, at the strength of the selected construction	No generic convergent finite-coupling vacuum or physical representation; construct positive states and control global or adiabatic limits
Lattice continuum and universality	Bare-parameter trajectory, order of ultraviolet and volume limits, observable family, topology, boundary conditions, and symmetry class	Uniform moment, correlation, renormalization-group, tightness, or comparison estimates	Subsequential or full convergence and universality only in the proved topology and observable sector	Finite-size collapse or matching critical exponents is not full QFT convergence; control complete observables and reconstruction hypotheses along a declared limit. Prove uniqueness or universality separately, and nontriviality if an interacting limit is claimed
AQFT–prefactorization comparison	Lorentzian site, additivity, time-slice conditions, target category, weak-equivalence class, and localization	Benini–Carmona–Grant-Stuart–Schenkel one-categorical equivalence	Equivalence under the theorem’s additive and target-category hypotheses	No unrestricted cochain-valued infinity-categorical equivalence; solve the stated infinity-localization problem
Conformal-net, VOA, and extended-TQFT classification	Central charge or finiteness class, unitarity and analytic bounds, bordism dimension, tangential structure, extension depth, and target category	Complete subcritical net classifications, conditional VOA–net bridges, and cobordism-hypothesis classifications in fixed targets	Classification or realization for the explicitly named class	No classification of all QFTs and no automatic converse; characterize the essential image and prove realization and completeness

Structured table data (JSON).

Review the chapter

A sequence of lattice theories has matching critical exponents and stable low-order correlators. Has universality of its continuum QFT been proved?

Solution

No. The result must control the declared full observable family in a specified limiting topology, with the necessary positivity and reconstruction conditions. Matching selected quantities can motivate a universality conjecture without proving uniqueness or equivalence of the limiting theories.

Where to go next

Return to theorem-first claim grammar to formulate a proposed advance precisely, or select the relevant constructive or categorical branch for its proof tools.

References

• Aizenman, Michael, and Hugo Duminil-Copin. “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\phi^4_4$ Models.” Annals of Mathematics 194 (2021): 163–235; corrigendum 199 (2024): 479. DOI; Corrigendum.
• Benini, Marco, Victor Carmona, Alastair Grant-Stuart, and Alexander Schenkel. “On the Equivalence of AQFTs and Prefactorization Algebras.” Letters in Mathematical Physics 116 (2026): 13. DOI; Open PDF.
• Chatterjee, Sourav. “A Scaling Limit of $SU(2)$ Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI; Open PDF, arXiv v4.
• Clay Mathematics Institute. “Yang–Mills and the Mass Gap.” Checked 7 September 2026. Problem page.
• Thorngren, Ryan, John Preskill, and Łukasz Fidkowski. “Chiral Lattice Gauge Theories from Symmetry Disentanglers.” arXiv:2601.04304 (2026). arXiv.