Exact source · explicit scope
The records below distinguish what the authors state, what this site examined, and what remains unestablished. Reading a source does not certify its proof.
Unbroken slab center symmetry implies a divergent potential
conditional theorem · conditional bridge
Theorem 2.2 obtains a rectangular-loop bound with a static potential diverging at large separation under the defined slab-center condition.
- Model
- Lattice d ≥ 2; Closed connected subgroup G of U(n), with a center-charged irreducible finite-dimensional unitary representation; Pure gauge
- Geometry and regulator
- Lattice slabs and boxes; Lattice. Boundary details: see the exact source conditions.
- Assumptions
- Definition 2.1 unbroken slab center symmetry; irreducible unitary representation nontrivial on the center.
- Publication
published; source date 2021-03-26 (day). First submitted: 2020-06-29.
- What remains
- The conclusion is a divergent static potential, not the stronger rectangular area law. Definition 2.3 uniform exponential decay is an additional premise of the separate Theorem 2.4 record, not a premise asserted here.
- The slab-center hypothesis must itself be established in the intended regime. This result does not remove the lattice cutoff or provide a physical continuum mass-gap theorem.
- Proof-obligation relation
- volume — conditional: A conditional lattice confinement implication under the stated slab-center condition; it neither establishes the stronger area-law hypothesis nor completes the continuum target.
- Author claim
- A scoped conditional theorem in the model above; no full four-dimensional solution is asserted by this record.
- Technical assessment
- selected sections; latest recorded action scope checked on 2026-10-24. No independent proof verification is recorded.
- Community acceptance
- No consensus assessment is inferred from publication or this site’s reading. External assessments, if documented, are separate records.
- Clay recognition
- These are partial-result records, not prize-recognition claims. See the separately sourced problem status.
Assessment history and unresolved checks
2026-10-24: scope checked
selected sections · Codex editorial source inspection; inherited and current actions remain distinct.
Examined: Section 2.1 model; Definition 2.1 and Theorem 2.2; Definition 2.3 and Theorem 2.4.
For pure lattice theory with closed connected G subgroup of U(n), d>=2, the slab-center condition implies divergent static potential for center-charged irreducible finite-dimensional unitary representations. The stronger arbitrary-boundary exponential-correlation hypothesis yields rectangular area law for every Gibbs measure.
- Definition 2.3 constants are uniform over cubes and every boundary condition and bounded edge-local observables. Decay in one vacuum is insufficient; no weak-coupling continuum bound is supplied.
- No independent proof certification or numerical simulation reproduction.
2026-10-23: scope checked
selected sections · Inherited webmaster-kit planning inspection; not a new reading on import.
Examined: Section 2.1; Definition 2.1; Theorem 2.2.
Theorem 2.2 obtains confinement with a divergent static potential from the defined slab-center condition for center-charged representations. Theorem 2.4 adds exponential correlation decay uniform under arbitrary boundary conditions to establish an area law.
- These hypotheses must not be identified with decay of one vacuum correlator. This does not establish the required weak-coupling continuum control for the Millennium problem.
- No independent expert proof verification or computational reproduction is recorded.
Uniform boundary-condition decay implies an area law
conditional theorem · conditional bridge
Theorem 2.4 obtains an area law using the exponential-correlation hypothesis of Definition 2.3, uniform under arbitrary boundary conditions.
- Model
- Lattice d ≥ 2; Closed connected subgroup G of U(n), with a center-charged irreducible finite-dimensional unitary representation; Pure gauge
- Geometry and regulator
- Lattice slabs and boxes; Lattice. Every boundary condition admitted in Definition 2.3, not only a selected vacuum.
- Assumptions
- Definition 2.3, including its quantifiers over boxes, boundary conditions and local observables; center-charged representation.
- Publication
published; source date 2021-03-26 (day). First submitted: 2020-06-29.
- What remains
- The boundary-condition-uniform hypothesis is stronger than decay in one vacuum correlator. Its required weak-coupling continuum control is not established here.
- Definition 2.3 constants are uniform over cubes and every boundary condition and bounded edge-local observables. Decay in one vacuum is insufficient; no weak-coupling continuum bound is supplied.
- Proof-obligation relation
- volume — conditional: Uniform boundary-condition decay implies an area law: The boundary-condition-uniform hypothesis is stronger than decay in one vacuum correlator. Its required weak-coupling continuum control is not established here.
- Author claim
- A scoped conditional theorem in the model above; no full four-dimensional solution is asserted by this record.
- Technical assessment
- selected sections; latest recorded action scope checked on 2026-10-24. No independent proof verification is recorded.
- Community acceptance
- No consensus assessment is inferred from publication or this site’s reading. External assessments, if documented, are separate records.
- Clay recognition
- These are partial-result records, not prize-recognition claims. See the separately sourced problem status.
Assessment history and unresolved checks
2026-10-24: scope checked
selected sections · Codex editorial source inspection; inherited and current actions remain distinct.
Examined: Section 2.1 model; Definition 2.1 and Theorem 2.2; Definition 2.3 and Theorem 2.4.
For pure lattice theory with closed connected G subgroup of U(n), d>=2, the slab-center condition implies divergent static potential for center-charged irreducible finite-dimensional unitary representations. The stronger arbitrary-boundary exponential-correlation hypothesis yields rectangular area law for every Gibbs measure.
- Definition 2.3 constants are uniform over cubes and every boundary condition and bounded edge-local observables. Decay in one vacuum is insufficient; no weak-coupling continuum bound is supplied.
- No independent proof certification or numerical simulation reproduction.
2026-10-23: scope checked
selected sections · Inherited webmaster-kit planning inspection; not a new reading on import.
Examined: Definition 2.3; Theorem 2.4.
Theorem 2.2 obtains confinement with a divergent static potential from the defined slab-center condition for center-charged representations. Theorem 2.4 adds exponential correlation decay uniform under arbitrary boundary conditions to establish an area law.
- These hypotheses must not be identified with decay of one vacuum correlator. This does not establish the required weak-coupling continuum control for the Millennium problem.
- No independent expert proof verification or computational reproduction is recorded.
Editorial history
2026-10-24 · scope correction
Separated provenance and claim type for Unbroken slab center symmetry implies a divergent potential.
Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.
Previous statement: “Source scope checked, 22 October 2026 (inherited undifferentiated label).”
2026-10-24 · scope correction
Separated provenance and claim type for Uniform boundary-condition decay implies an area law.
Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.
Previous statement: “Source scope checked, 22 October 2026 (inherited undifferentiated label).”
2026-10-22 · new result
Inherited dossier: A precise criterion connecting center symmetry and confinement.
Original editorial snapshot, retained during migration. This is not the date the source result first appeared.
Download public records (JSON) · Source versions (BibTeX) · Compare in the atlas
In A probabilistic mechanism for quark confinement, Theorem 2.2 uses the paper’s definition of unbroken center symmetry on slabs. For a representation that acts nontrivially on the center, it obtains a rectangular-loop bound with a potential diverging at large separation.
Theorem 2.4 starts from a stronger premise: exponential decay of appropriate local correlations uniformly under arbitrary boundary conditions, as specified in Definition 2.3. It obtains unbroken center symmetry and a rectangular Wilson-loop area law.
The first conclusion should not be silently replaced by linear confinement, and the second premise should not be shortened to “one correlator has a mass.” Sections 2.3–2.4 are the precise starting point.
A Wilson loop transports a test color charge around a closed contour and takes a trace. The trace removes the arbitrary choice of gauge frame at the starting point. Rectangular loops are useful because one side can be interpreted as a Euclidean propagation time for static sources.
In a situation where the appropriate large-time limit exists, the schematic relation
V(R)=−T→∞limT1log∣⟨W(R,T)⟩∣
connects the loop with a static-source energy, after handling normalization and source self-energies. A behavior proportional to the rectangle’s area suggests an energy growing proportionally to separation.
Notice that this observable contains external probes. The vacuum excitation spectrum is a different object. Relating these questions requires hypotheses; neither a shared physical picture nor a common use of the word “confinement” can replace them.
Consider an ordinary statistical system in a large box. A boundary can favor one phase even when correlations measured in that chosen phase are short-ranged. Consequently, short-range correlations under one boundary condition do not imply that every possible boundary has negligible influence.
A theorem that permits arbitrary boundaries is testing a stronger kind of independence. Its constants must keep working as the boundary configuration changes. This makes the estimate suitable for gluing or conditioning arguments, where intermediate boundaries are generated by the rest of the system rather than chosen for convenience.
A helpful schematic inequality is
∣CovB,δ(f,g)∣≤K1e−K2dist(f,g).
The logical force comes from the quantifiers: which boxes B, boundary data δ, and bounded observables f,g are allowed, and what may influence K1,K2? Removing those quantifiers changes the statement.
The criterion tells us how one strong property implies another. To use it for the continuum problem, one would need to establish suitable hypotheses in a regime connected to that limit, retain the necessary uniformity, and identify the physical limiting theory.
The representation assumption also matters. A center-sensitive loop cannot be used unchanged as a universal diagnostic for every gauge group and every probe representation. Group and global-form information therefore belongs in the research record.
| Question | Assessment |
|---|
| Is there a rigorous implication? | Yes, with the definitions in the cited theorems |
| Are the assumptions just one measured decay rate? | No |
| Is the lattice cutoff removed by this implication? | No |
| Does this establish the Clay mass gap for all groups? | No |
Read Definition 2.3 and write its quantifiers in words. Now imagine that the decay constant depends on the box size and approaches zero. Which step of an infinite-volume argument has become unavailable?
Next compare two claims: “correlations decay exponentially in a selected state” and “the same estimate holds for all admitted boundary data.” Give a statistical-mechanical example explaining why the second demands more information.
Start with physical states and finite volume if these distinctions are new. Then compare this criterion with the different conditional-slab argument in the dynamical area-law dossier.
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