Constructive Existence Ledger by Model, Dimension, and Observable

Constructive existence is always indexed by a model, dimension, regulator family, observable class, convergence topology, and proved properties. The useful comparison is therefore not “rigorous or not,” but exactly which mathematical object has been constructed and which later conclusions have separately been established.

Required background. The constructive program and cutoff removal supplies the construction stages; from Euclidean measures to relativistic models supplies the reconstruction boundary.

Helpful background. Source authority, status, and specialist review helps assess theorem versions; counterexamples, nonconverses, and hypothesis stress tests supplies the claim test.

Four representative constructions

The following cases illustrate why “the model exists” is too coarse.

Model	Constructed object and convergence	Further proved structure	Boundary
Massive $P(\phi)_2$	Infinite-volume Euclidean measure and Schwinger functions after volume removal; broad stable polynomials, with regime depending on method	OS/Wightman axioms; weak-coupling clustering, mass gap, isolated particle, scattering	Does not prove four-dimensional scalar existence or general asymptotic completeness
$\phi^4_3$	Renormalized ultraviolet and volume limits of Schwinger functions with mass and vacuum counterterms	OS/Wightman theory; mass-gap and particle results in established regimes	Three-dimensional counterterms and estimates do not transfer to $d=4$
Massive Gross–Neveu$_2$	UV and volume limits of fermionic Schwinger functions at small renormalized coupling	Fermionic OS axioms; Borel summability; some two-particle results	Not the full massless dynamical-mass claim at arbitrary coupling
Weak-coupling lattice Yang–Mills–Higgs	Projected gauge field converges as a random distributional one-form under joint $\varepsilon,g,\alpha$ scaling	Massive Gaussian Proca limit; Abelian smeared curvature follows	No non-Gaussian limit, Wilson-loop area law, pure Yang–Mills, or relativistic reconstruction

For $P(\phi)_2$, the finite-volume density and weak infinite-volume limit are described in Summers 2016, §3.1, pp. 10–14, with primary particle analysis in Glimm, Jaffe, and Spencer 1974, pp. 585–632. For $\phi^4_3$, the simultaneous regulated construction and OS result are summarized in Summers 2016, §3.2, pp. 16–17, with the weak-coupling Wightman and mass-gap theorem in Feldman and Osterwalder 1976, pp. 80–135.

For the massive Gross–Neveu model, Summers 2016, §3.5, pp. 23–24 records cutoff removal and OS properties, while Gawędzki and Kupiainen 1985, pp. 1–30 gives a convergent constructive expansion. For Yang–Mills–Higgs, Chatterjee 2026, Theorems 3.1–3.2 and §3.5, pp. 10–17 proves the Gaussian scaling limit and states the non-Gaussian open problem.

A theorem-first decision procedure

To classify a new claim, first name the approximants: finite-volume Hamiltonians, lattice measures, smoothed random distributions, or Schwinger functions. Next specify every limit parameter and whether convergence is weak convergence of laws, convergence of moments, convergence in a distribution norm, or strong resolvent convergence. Then list the determining observables. Finally, attach later properties only with their own hypotheses: reflection positivity, clustering, a mass gap, a particle shell, scattering, or completeness.

This procedure gives the first application as a comparison among $P(\phi)_2$, $\phi^4_3$, Gross–Neveu$_2$, and a three-dimensional gauge–Higgs scaling limit. The comparison returns to rigorous status, construction, and open problems without enlarging any theorem.

An independent check uses implication monotonicity. If a row claims Wightman reconstruction, it must also identify OS or another reconstruction input. If it claims scattering, it must identify a reconstructed Hilbert theory and isolated particle spectrum. If one prerequisite cell is empty, the downstream claim is unsupported rather than automatically inherited.

Existence-status boundaries

“Finite cutoff” means only that the regulated object exists. “Continuum Schwinger functions” means selected Euclidean correlations converge. “Euclidean measure” additionally identifies a positive law. “OS reconstruction” supplies a relativistic Hilbert theory. “Mass gap,” “particle,” and “scattering” are increasingly specific spectral results. None is a synonym for the previous one.

The converse arrows generally fail. A collection of convergent moments need not determine a measure; a measure need not be reflection positive; a reconstructed theory need not have an isolated particle; a scattering theory need not be asymptotically complete.

Adversarial evidence test

Insert a proposed four-dimensional Yang–Mills or scalar continuum theory based only on perturbation theory or Monte Carlo scaling. Those are important evidence, but they do not supply a proved tight family of continuum measures, regulator-independent Schwinger functions, or OS reconstruction. The proper classification preserves the evidence and rejects the constructive-existence label.

A second test takes a proved $d=3$ theorem and erases the dimension. Power counting, counterterms, and uniform norms change with $d$; the erased statement is strictly stronger and unsupported.

Exercises

1. Classify a result. A paper proves tightness of lattice fields and convergence of their two-point functions, but not uniqueness of subsequential laws. What is established?

Solution

There are subsequential continuum laws and a common two-point limit along the controlled subsequences. A unique measure, full Schwinger hierarchy, OS reconstruction, and interaction are not yet established.

2. Read a scattering claim. Which minimum chain of objects must appear before a Haag–Ruelle conclusion is credible?

Solution

One needs a relativistic Hilbert-space theory with locality and positive energy, an isolated mass shell with a nonzero local interpolating operator, and wave packets with separated velocities. A Euclidean measure alone is insufficient.

References

• Chatterjee, Sourav. “A Scaling Limit of SU(2) Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI. Open PDF.
• Feldman, Joel, and Konrad Osterwalder. “The Wightman Axioms and the Mass Gap for Weakly Coupled $\phi^4_3$ Quantum Field Theories.” Annals of Physics 97 (1976): 80–135. DOI.
• Gawędzki, Krzysztof, and Antti Kupiainen. “Gross–Neveu Model Through Convergent Perturbation Expansions.” Communications in Mathematical Physics 102 (1985): 1–30. DOI.
• Glimm, James, Arthur Jaffe, and Thomas Spencer. “The Wightman Axioms and Particle Structure in the $P(\phi)_2$ Quantum Field Model.” Annals of Mathematics 100 (1974): 585–632. DOI.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.