Existence, Continuum Limits, and Classification Frontiers

The frontier of mathematical quantum field theory is not a single list of famous conjectures. An existence statement is meaningful only after the model, spacetime dimension, regulator, limiting procedure, observable class, topology of convergence, positivity condition, and uniqueness relation have been fixed. A mass gap, confinement, scattering theory, comparison functor, or classification then requires a further theorem. This chapter develops that separation for constructive models, four-dimensional scalar and Yang–Mills theories, chiral gauge theories, curved-spacetime interactions, continuum limits, and categorical frameworks. All statements of current status use an evidence cutoff of 2026-08-10.

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.

Enter this chapter

A useful frontier claim has the form

$$\mathfrak C=(\mathcal M,d,R,\Lambda,\mathcal O,\tau,P,F,t_*).$$

Here $\mathcal M$ is the model, $d$ the spacetime dimension, $R$ the regulator, $\Lambda$ the ordered collection of limits, $\mathcal O$ the observables being controlled, $\tau$ the convergence topology, $P$ the positivity or Hilbert-space condition, $F$ the proposed reconstruction or comparison map, and $t_*$ the source cutoff. Omitting one entry can change a theorem into an analogy. For example, convergence of a projected lattice gauge field in the topology of distributions does not imply convergence of every gauge-invariant correlation function, while construction of a formal power series does not imply convergence at fixed coupling.

The first diagram shows the dependencies that must be proved in order. Its arrows are implications, not a timetable: different models stop at different nodes, and a physical property such as a mass gap never follows merely from the word “continuum.”

Existence, reconstruction, physical properties, and classification are separate stages. Low-dimensional constructive models traverse substantial parts of the chain; four-dimensional scalar, pure Yang–Mills, and chiral-gauge programs have model-specific partial results and open steps; categorical theorems classify objects only in their declared targets. The diagram is schematic and not to scale. Structured description and source data (JSON)

Read the chain from left to right. A regulated family must first satisfy estimates uniform in the regulator. Compactness or convergence must then be proved in a named topology for a sufficiently complete set of observables. Euclidean data require full reflection positivity and the other reconstruction hypotheses before they define a relativistic Hilbert-space theory. A mass gap, scattering theory, confinement criterion, comparison equivalence, or classification completeness is an additional conclusion with its own hypotheses. A partial result remains mathematically valuable precisely when its stopping point is stated accurately.

The chapter sequence

The twelve pages move from claim grammar through model-specific frontiers to a reproducible treatment of open status. Each page can be read independently after its background note, but the order makes the logical dependencies visible.

1. Existence, Uniqueness, and Equivalence Claims types the objects, quantifiers, topologies, maps, and uniqueness relations that an existence or equivalence assertion must contain.
2. Two- and Three-Dimensional Constructive Model Atlas compares genuinely constructed scalar, fermionic, gauge, and scaling models without transferring their results across dimensions.
3. Four-Dimensional Scalar QFT: Existence and Triviality separates Gaussian scaling theorems for specified ferromagnetic classes from broader claims about every scalar interaction.
4. Yang–Mills Existence and the Mass Gap states the four-dimensional pure-gauge construction and spectral targets and explains what finite lattices and nearby models do not prove.
5. Rigorous Mass-Gap, Scattering, and Confinement Obligations distinguishes a vacuum spectral gap, isolated particles, scattering-state existence, asymptotic completeness, and confinement observables.
6. Chiral Gauge Theories and Standard Model Construction follows anomaly cancellation, regulator locality, gauge invariance, positivity, continuum removal, and the recovery of the intended chiral spectrum.
7. Interacting Curved-Spacetime and Gauge Existence Problems distinguishes local formal constructions and abstract local algebras from finite-coupling states and global nonperturbative theories.
8. Continuum Limits and Universality Problems identifies the uniform estimates, tuned parameters, convergence modes, and observable comparisons required by a universality theorem.
9. Lattice-to-Continuum Constructive Proof Obligations turns lattice existence or data into a sequence of tightness, positivity, reconstruction, and nontriviality obligations.
10. Classification and Comparison Problems across QFT Frameworks tests directional maps among Euclidean, Wightman, algebraic, factorization-algebraic, VOA, conformal-net, and functorial descriptions.
11. Conformal-Net, VOA, TQFT, and Categorical Classification Frontiers records exactly which objects are classified in fixed analytic or higher-categorical settings and which realization problems remain.
12. Dated Open-Problem Evidence Search and Research Handoff gives a source-first method for updating a frontier without promoting evidence, expectation, or a nearby theorem into a proof.

Hypotheses and licensed conclusions

The comparison below is deliberately asymmetric. Its fourth column records the strongest conclusion supported by the stated source class; the fifth records the tempting upgrade that has not been supplied.

Representative existence and classification frontiers at the 2026-08-10 evidence 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; establish uniqueness, positivity, complete correlations, and nontriviality
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) preserves the caption, headers, rows, and reading order.

The scalar row is bounded by the random-current and ferromagnetic hypotheses of Aizenman and Duminil-Copin 2021, Theorem 1.2 and §§ 1.4–1.6, pp. 163–185, corrected 2024. The Yang–Mills row uses the exact continuum-and-gap target in Jaffe and Witten 2000, §§ 3–4, PDF pp. 5–7, which the Clay Mathematics Institute 2026 problem page continued to list as unsolved at the cutoff. Chatterjee’s nearby theorem instead gives a projected massive Gaussian field for a jointly scaled $SU(2)$ Yang–Mills–Higgs model 2026, Theorem 3.2 and § 3.3, pp. 12–17; its Higgs matter and Gaussian limit cannot be deleted from the statement.

The chiral-gauge row is equally specific. Thorngren, Preskill, and Fidkowski construct exact anomaly-free $1+1$-dimensional Hamiltonians and develop relevant $3+1$-dimensional symmetry data, but explicitly leave the full four-dimensional Hamiltonian and interface construction unfinished 2026, § 5, pp. 27–28. For framework comparison, Benini, Carmona, Grant-Stuart, and Schenkel prove the one-categorical result under stated hypotheses and retain the infinity-categorical localization step as an open problem 2024, Theorems 3.3–3.4 and Open Problem 5.6. These are advances of different mathematical types, not points on one numerical scale.

Failure boundaries

Most false frontier claims arise by preserving the scientific nouns while changing a quantifier, topology, implication direction, or model label. The second diagram performs five such perturbations and records the strongest conclusion that survives.

Each dashed branch removes one indispensable element. A low-dimensional theorem cannot be relabeled as four-dimensional; stable finite-cutoff data do not supply tightness; partial correlations do not supply reflection positivity of a full hierarchy; a one-way construction does not supply a converse or essential surjectivity; and a strong source consensus does not close an open lemma. The diagram is schematic and not to scale. Structured description and source data (JSON)

Five checks catch these failures early. First, keep subscripts and matter content: $\phi^4_3$, pure Yang–Mills, and Yang–Mills–Higgs are different models. Second, write limits with their order and topology; $a\downarrow0$ followed by $L\uparrow\infty$ is not automatically interchangeable with the reverse order. Third, ask whether all correlations, a generating functional, a net, or only one projected observable is controlled. Fourth, distinguish a construction functor from an equivalence by checking full faithfulness, essential image, and any inverse. Fifth, attach an open-status sentence to a date and primary source, because a later theorem may close only a narrower or neighboring obligation.

A reproducible frontier decision

For any proposed result, begin with a one-sentence target: “Construct this object for this model and dimension, in this topology, with these positivity and covariance properties.” Then separate four questions.

1. Existence: Is there a genuine continuum object, only a regulated object, a formal series, an abstract algebra, or a subsequential scaling field?
2. Recovery: Are the intended local observables, states, gauge-invariant sector, and Lorentzian theory recovered, with a stated uniqueness relation?
3. Additional physics: Which gap, particle, scattering, confinement, or phase statement has been proved independently?
4. Comparison or classification: What are the source and target categories, what functor is constructed, and which of faithfulness, fullness, essential surjectivity, realization, and converse are established?

The answer should end at the first missing proof obligation. Numerical data, perturbative expansions, exact solvable limits, anomaly checks, and categorical analogies can motivate that obligation or test a consequence. They do not alter its logical status. Conversely, saying “open” should not erase partial theorems: a Gaussian scaling theorem, a lower-dimensional Hamiltonian, a formal local construction, or a classification within a fixed target is a precise result that deserves its full hypotheses and conclusion.

Check your understanding

A paper proves that one smeared lattice gauge potential converges in distribution to a massive Gaussian one-form as the lattice spacing tends to zero in a joint weak-coupling, large-Higgs-length limit. Classify the conclusion and list three additional results needed before it could contribute to a proof of the four-dimensional pure Yang–Mills mass-gap problem.

Solution

The conclusion is a continuum scaling theorem for a projected observable in a gauge–Higgs model and in a specified parameter regime. It is not a construction of pure Yang–Mills, because Higgs matter remains part of the regulated model and the limiting field is Gaussian. At minimum one would need: (1) removal or decoupling of the Higgs sector while controlling a non-Abelian pure-gauge continuum limit; (2) construction of a sufficiently complete gauge-invariant observable algebra or correlation hierarchy with positivity, covariance, locality, and a physical Hilbert-space representation; and (3) a separate infinite-volume spectral theorem proving a positive vacuum gap uniformly after continuum removal. Identifying confinement or asymptotic completeness would require still further observable-specific theorems.

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.” arXiv:2412.07318 (2024). arXiv.
• Chatterjee, Sourav. “A Scaling Limit of $SU(2)$ Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI; Open PDF.
• Clay Mathematics Institute. “Yang–Mills and the Mass Gap.” Current through 2026-08-10. Problem page.
• Jaffe, Arthur, and Edward Witten. “Quantum Yang–Mills Theory.” In The Millennium Prize Problems, 129–152. Clay Mathematics Institute and American Mathematical Society, 2006; problem description released 2000. Official PDF.
• Thorngren, Ryan, John Preskill, and Łukasz Fidkowski. “Chiral Lattice Gauge Theories from Symmetry Disentanglers.” arXiv:2601.04304 (2026). arXiv.