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.
Elimination beyond a loop cutoff supplies indirect relations
method · related method
The framework generates direct loop equations and relations obtained by eliminating longer intermediate loops, with SU(2) applications in two, three and four lattice dimensions.
- Model
- 2D, 3D, 4D lattice; SU(2) applications; Pure gauge
- Geometry and regulator
- Euclidean lattice; Lattice. Boundary details: see the exact source conditions.
- Assumptions
- Finite retained loop basis, exact input identities and elimination rules; no convergence of the full hierarchy is claimed.
- Publication
preprint; source date 2026-01-21 (day). First submitted: 2026-01-07.
- What remains
- An equation-generation method is not a proof that finite truncations converge, nor a spectral or continuum construction result.
- Completeness in reported finite comparisons is not convergence of the infinite hierarchy. Auxiliary loop-space enlargement, truncation and projected relations remain explicit; no continuum or spectral result.
- Proof-obligation relation
- This method or constraint does not establish a target proof obligation; its value is in organizing restricted lattice information.
- Author claim
- A scoped method 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 1; Section 5.1, Table 6; Section 5.2 improved filtering and Section 5.3 application examples.
Direct loop equations and indirect relations obtained by elimination of longer intermediate loops, with finite SU(2) loop sets in 2D/3D/4D. Vertex-support filtering is a computational heuristic tested against the stated finite sets.
- Completeness in reported finite comparisons is not convergence of the infinite hierarchy. Auxiliary loop-space enlargement, truncation and projected relations remain explicit; no continuum or spectral result.
- No independent proof certification or numerical simulation reproduction.
2026-10-23: scope checked
abstract · Inherited webmaster-kit planning inspection; not a new reading on import.
Examined: Abstract; Version history.
The abstract describes systematic construction of direct loop equations and indirect relations obtained through elimination of longer intermediate loops, with SU(2) lattice examples in 2D, 3D, and 4D.
- This record does not assert convergence of truncated hierarchies, a continuum construction, or a spectral-gap result. Detailed completeness and independence claims require checking the text.
- No independent expert proof verification or computational reproduction is recorded.
Editorial history
2026-10-24 · scope correction
Separated provenance and claim type for Elimination beyond a loop cutoff supplies indirect relations.
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: Direct and indirect loop equations.
Original editorial snapshot, retained during migration. This is not the date the source result first appeared.
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Liu and Yang’s Direct and Indirect Loop Equations in Lattice Yang–Mills Theory organizes equations for Wilson-loop expectations. It constructs local direct equations geometrically and investigates additional relations obtained by eliminating longer intermediate loops. Applications concern SU(2) in two, three, and four lattice dimensions.
The tracked publication state is the preprint version reviewed here. Its relevance is better control of the equations entering a calculation, rather than a claimed continuum construction.
A Wilson loop traces the ordered product of link variables around a closed lattice path. Gauge transformations at neighboring endpoints cancel; taking the trace removes the remaining conjugation. This makes loops natural gauge-invariant observables.
The path integral also supplies relations among observables. On a compact gauge group, invariance of Haar measure lets one change an integration variable without changing the integral. Infinitesimal variations give Schwinger–Dyson identities, the path-integral analogue of integration by parts. Matrix identities provide further trace relations.
The complication is closure. An equation involving a short loop can also contain a longer one. Keeping only loops up to a chosen length does not guarantee that every equation stays inside that list. The omitted variables still matter even when the target observable is short.
For background, review the lattice definition and Haar measure. The finite-dimensional analogy below explains the algebraic issue without requiring the paper’s loop notation.
Consider a toy system with retained variables u,v and an intermediate variable w:
u+2w=1,v−w=0.
Neither equation involves only retained variables. Deleting both because they contain w would leave u and v unconstrained. But eliminating w gives
u+2v=1.
The intermediate variable disappears while its consequence remains. This example is an illustration of elimination, not an equation from Yang–Mills theory.
The lesson becomes more significant when thousands of variables are involved. A relation may require a sequence of substitutions through variables beyond the working cutoff. A list that is complete under one local construction rule need not contain every consequence of a larger system.
There are therefore two distinct bookkeeping questions: have we generated the intended direct equations, and have we retained all relevant consequences of equations involving excluded variables? Answering the first does not automatically answer the second.
Read §3 for plaquette-cut and subloop-cut constructions, then §5.1 for the origin of indirect relations. Section 5.2 describes filtering strategies; §5.3 gives applications. Appendix B supplies concrete indirect-equation examples. This route puts the conceptual difficulty before the combinatorial growth discussed in §4.
For each example, make a short ledger: variables retained, variables eliminated, exact input identities, and resulting relation. That ledger makes it possible to reproduce a claimed improvement without confusing an extra equation with an extra approximation.
A positivity bootstrap searches over candidate expectation values consistent with its imposed equations and inequalities. A missing exact relation leaves candidates that should have been excluded. Adding a valid relation can shrink the feasible set, potentially tightening the range of a target observable.
This is a mathematical implication of constraint inclusion, not a promise that every added equation improves a given numerical bound. A new equation may be redundant, nearly dependent numerically, or irrelevant to the chosen objective.
Checking rank and checking numerical conditioning are also different tasks. Exact algebra can establish dependence; floating-point rank can depend on tolerance. A reproducible calculation should record the conventions and tolerances used for both equation reduction and optimization.
A finite equation system does not by itself construct a positive continuum expectation functional. Nor does satisfying many identities prove uniqueness of the underlying state. Distinct states or candidate functionals can share a finite collection of moments.
To connect this method to the mass-gap problem, one needs additional control over growing truncations, long-distance observables, regulator removal, and the reconstructed physical spectrum. The useful near-term question is concrete: Does a newly generated relation produce a certified improvement for an observable with a clear physical interpretation?
Read the companion positivity-bootstrap dossier next, then locate the open continuum and spectral steps on the proof-obligation map. The assessment history above records what text was examined; it does not certify a complete hierarchy or a proof of the Clay problem.