View source for Conformal invariance of square-lattice random-cluster interfaces

'''Conformal invariance of square-lattice random-cluster interfaces''' is the subject of six manuscripts dated September and October 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). For the critical random-cluster (Fortuin–Kasteleyn) model on the square lattice with cluster weight ''q'', the manuscripts claim that Dobrushin interfaces converge to chordal SLE<sub>κ</sub> with κ = 4π/arccos(−√''q''/2), and that nested loop collections converge to CLE<sub>κ</sub>; at ''q'' = 1 this would give Cardy's formula for square-lattice bond percolation. A public audit claims a critical gap in the holomorphicity step of the main proof. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a critique is not presented here as independent verification.

== The original results ==
* '''Claim (1 ≤ ''q'' ≤ 4).''' For every fixed 1 ≤ ''q'' ≤ 4, critical square-lattice random-cluster Dobrushin interfaces converge to chordal SLE<sub>κ(''q'')</sub> and the complete nested plane loop collections converge to whole-plane CLE<sub>κ(''q'')</sub>; at ''q'' = 1 the manuscript states Cardy's formula for square-lattice bond percolation with free boundary edges. Sources: the manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026 Square-lattice FK interfaces and nested loops for 1 ≤ q < 4], dated 23 September 2026, and the manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026 Conformal limits of critical square-lattice random-cluster interfaces], dated 5 October 2026, which states the range 1 ≤ ''q'' ≤ 4. The proof passes through a discrete observable whose limit is shown to be holomorphic (Theorem "Holomorphic disk limit"), using reflection inequalities, angular extraction and a "strict power saving" in the cost of the reflected norm diagrams.
* '''Claim (0 < ''q'' < 1).''' The manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026 Self-dual random-cluster interfaces below one] (23 September 2026) claims convergence to SLE<sub>κ</sub> for smooth Jordan domains by a different boundary observable.
* '''Related manuscripts.''' The family also contains ''Natural occupation measures for critical square-lattice FK interfaces'' (5 October 2026), which takes the trace convergence of the first manuscript as an input, and two FK–Ising (''q'' = 2) manuscripts on quenched disorder and on thermal perturbations with massive SLE.
* '''Formalization.''' The collection's Lean documentation has no scope note for this family.

== Earlier published results ==
* As recounted in the first manuscript: Smirnov proved Cardy's formula for critical site percolation on the triangular lattice, and Camia and Newman constructed its full continuum loop limit. For the FK–Ising model (''q'' = 2), Smirnov's fermionic observable and the isoradial analysis of Chelkak and Smirnov led to convergence of the Dobrushin interface to SLE<sub>16/3</sub> (Chelkak, Duminil-Copin, Hongler, Kemppainen and Smirnov), followed by loop-ensemble results of Kemppainen and Smirnov. For 1 ≤ ''q'' ≤ 4, Duminil-Copin, Kozlowski, Krachun, Manolescu and Oulamara proved asymptotic rotational invariance without identifying the limits. For square-lattice bond percolation the manuscript lists earlier public claims by Tsai, Yam and Zhou and by Zhou (the latter under additional boundary conditions).

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-15 || [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (GitHub; J. Jardine, Lattice24) || Critique (claimed gap) || Claims a critical gap in the holomorphicity proof of ''Square-lattice FK interfaces and nested loops for 1 ≤ q < 4''. According to the README, the strict power saving of the section "The complete cost of the norm diagrams" never uses the mixed difference stencil, so the same argument would also bound the undifferenced observable; combined with the manuscript's gap-closure and complete-wall factorization statements, this would force the limiting observable to vanish, contradicting the holomorphicity proof. The README concludes that at least one of three stated ingredients (the leaf bound with the reflected weights, the extra costs of reflected copies, or the identification of the baseline with the normalization) must be false as stated, and that ''Natural occupation measures'' inherits the gap. It describes the audit as performed by AI review agents with three independent refutation passes, reports that 192 of 547 unformalized manuscripts in the collection were reviewed with this the only confirmed critical gap, and invites a pointer to any misread step. || Public repository with the review and refutation-pass records.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]