Dated Open-Problem Evidence Search and Research Handoff

An open-problem record must preserve the exact theorem target while allowing evidence to evolve. It dates the literature search, distinguishes proofs from partial constructions and physical evidence, records contrary results and failed implications, and lists the remaining lemmas. Citation count, prize status, expert expectation, simulations, and successful phenomenology can motivate a problem; none is a proof field.

Required background. Existence, uniqueness, and equivalence supplies the claim grammar. Yang–Mills existence and the mass gap, chiral gauge construction, and continuum limits and universality supply the principal frontier cases. Helpful background. Framework-comparison problems, the quantum-gravity open-problem route, rigorous nonperturbative status, higher-dimensional supersymmetric fixed points, quantum extremal surfaces, semiclassical information limits, the quantum focusing conjecture, evaporation endpoints, phenomenological evidence ceilings, and the quantum-gravity handoff show how other mutable claims are bounded.

The dated record

Every entry should contain the following fields in ordinary scientific language:

1. Exact claim. Mathematical object, dimension, signature, group or interaction, axioms, observable class, limit topology, and conclusion.
2. Evidence cutoff. Calendar date, search venues, source versions, corrections, and retractions checked.
3. Established results. Theorems with hypotheses and exact locators, separated from conditional results.
4. Partial constructions. Which regulator, model, dimension, or observable has been controlled.
5. Physical evidence. Lattice, perturbative, experimental, duality, or semiclassical evidence, with its inference ceiling.
6. Contrary constraints. No-go results, countermodels, nonconverses, and regimes where the claimed implication fails.
7. Open obligations. A dependency-ordered list of lemmas or constructions still missing.
8. Update triggers. A new proof, counterexample, correction, theorem-version change, or failure of a cited assumption.

The status vocabulary is small: proved under stated hypotheses, conditional, partially constructed, evidence, conjecture, obstruction, or open. A probability or consensus score is not a mathematical status.

First application: four-dimensional Yang–Mills

The theorem target is the one stated by Jaffe and Witten: for every compact simple $G$, construct a nontrivial quantum Yang–Mills theory on $\mathbb R^4$ with axiomatic properties at least as strong as the cited frameworks and prove a gap $\Delta>0$ 2000, §§3–4, pp. 5–7. At the cutoff 2026-08-10, the Clay problem page still states that no proof is known.

The evidence streams must remain separate.

Stream	Strongest bounded content	What it does not prove
Constructive	Exact two-dimensional Yang–Mills; selected lower-dimensional and gauge–Higgs scaling results	Four-dimensional pure-gauge continuum existence
Lattice	Positive finite-cutoff measures, transfer matrices, numerical continuum extrapolations, glueball spectra	Tightness and convergence of a full continuum hierarchy
Perturbative	Renormalizability and asymptotic freedom near weak coupling	Finite-coupling nonperturbative existence or an infrared gap
Spectral	Quantitative evidence for massive gauge-invariant channels	A uniform lower bound for the complete continuum Hilbert space
Confinement	Area-law, string-tension, and flux-tube evidence in specified regimes	Equivalence of confinement definitions or axiomatic construction
Quantum gravity and holography	Structural models and consistency constraints	A proof of the fixed four-dimensional nongravitational theory

Chatterjee’s $SU(2)$ Yang–Mills–Higgs theorem is a precise partial construction: a projected gauge field converges to a massive Gaussian field under a joint weak-coupling and large-Higgs scaling 2026, Theorem 3.2 and §3.3, pp. 12–17. Its Higgs matter, Gaussian limit, and tuning prevent it from closing the pure-gauge obligation. The record should preserve this advance rather than either ignoring it or inflating it.

The resulting living dossier belongs with open problems, research handoffs, and update records. The durable mathematical statement remains the fixed theorem target and dependency graph; the dated record can change when new primary sources appear.

Search procedure

Search the official problem statement and current institutional status first, then primary theorem and construction papers, their corrections, and sources that explicitly limit or contradict the proposed inference. Search by the complete tuple—model, dimension, regulator, topology, observable, and conclusion—rather than by a slogan such as “mass gap solved.” For a claimed new proof, check the exact theorem statement, dependencies, publication or preprint version, subsequent corrections, and whether an independent specialist analysis has identified a gap.

This procedure matters beyond Yang–Mills. Aizenman and Duminil-Copin prove Gaussian scaling for stated four-dimensional Ising-type and $\lambda\phi^4$ classes, not every scalar theory 2021, Theorem 1.2, pp. 163–177, corrected 2024. Thorngren, Preskill, and Fidkowski give exact $1+1$-dimensional chiral Hamiltonians and a partial $3+1$-dimensional hypercharge route, while leaving the full Standard Model Hamiltonian unfinished 2026, §5, pp. 27–28. Benini and collaborators prove a one-categorical AQFT–prefactorization equivalence but retain an infinity-categorical localization problem 2024, Open Problem 5.6. A good search detects the noun immediately following “under.”

Open obligations for the worked case

For four-dimensional pure Yang–Mills, the dependency order is:

1. construct a continuum, infinite-volume, gauge-invariant object with a separating local observable class;
2. prove Euclidean or Lorentzian axioms, positivity, locality, covariance, and nontriviality;
3. reconstruct or directly construct the physical Hilbert space and translation representation;
4. identify the vacuum sector and prove the uniform positive spectral gap;
5. only then attach separately proved scattering or confinement conclusions.

A new result can close one node without closing later nodes. The record should state exactly which node changed and which remain open.

Failure test: evidence entered as proof

Enter “large citation count,” “million-dollar problem,” or “accurate lattice prediction” in the theorem field. The entry is rejected because none supplies a derivation from the stated hypotheses. The information can remain under institutional context or physical evidence. The strongest mathematical status stays open until a valid proof of the exact target is available.

Exercises

A preprint proves a uniform mass gap for a four-dimensional lattice Hamiltonian at every fixed spacing and volume. Which nodes remain open?

Solution

One still needs the tuned infinite-volume and continuum object, uniformity strong enough to pass the spectral bound to it, positivity and reconstruction, vacuum control, nontrivial local observables, and the axioms in the official target. The regulated theorem is meaningful partial progress, but it is not yet the continuum result.

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.
• 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.