{"id":"1a1dfbde-aad8-4314-85ab-4237f1fe2e8f","arxiv_id":"2606.00945","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends conformal invariance of the Ising model and percolation from hexagonal to 3-12 lattice.","lead":"This survey extends conformal invariance results for the Ising model and percolation from the hexagonal lattice to the 3-12 lattice. A smart generalist might read it to understand how changes in lattice geometry affect scale-invariant behavior at critical points in statistical mechanics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Whether the 3-12 lattice satisfies the exact local geometric conditions (angles, coordination, discrete holomorphicity) needed for the hexagonal-lattice proofs to carry over unchanged.","rationale":"The reader's weakest_assumption correctly isolates the transfer step as the point where the argument is least secure. Because the manuscript is a survey whose full technical details are not reproduced here, the same limitation that produced the UNVERDICTED verdict remains; the concrete test above would resolve whether the geometric mismatch actually breaks the argument.","tokens_in":1528,"tokens_out":409,"duration_ms":16272,"concrete_test":"Take the explicit definition of the holomorphic observable (or interface turning rule) from the hexagonal-lattice proof cited in the survey; substitute the 3-12 lattice edge directions and vertex angles; recompute the discrete derivative at a typical vertex. If the resulting expression is no longer holomorphic (i.e., fails to satisfy the discrete CR equation up to o(1) error), the direct transfer fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that conformal invariance results known for the hexagonal lattice extend to the 3-12 lattice. For this to hold, the 3-12 lattice must admit the same constructions: an interface whose turning angles permit a discrete holomorphic observable, the same RSW crossing estimates, and the same martingale or contour-integral identities used on the hexagonal lattice. The 3-12 lattice replaces every other vertex with a 12-gon, altering local vertex figures and face angles relative to the uniform 120°/60° geometry of the hexagonal case. If these alterations change the discrete Cauchy-Riemann relations or the normalization of the observable at the scale of the lattice spacing, the existing proofs do not transfer directly and new estimates are required. The survey asserts the extension on the basis of shared planarity and coordination number, but does not appear to re-derive the key local identities for the new lattice.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript is a survey claiming to extend the conformal invariance of the Ising model and percolation from the hexagonal lattice to the 3-12 lattice, asserting that results known for the former transfer to the latter on the basis of shared planarity and coordination properties.","tokens_in":1691,"tokens_out":262,"duration_ms":11280,"significance":"If the extension holds with rigorous justification, the result would modestly broaden the class of lattices for which conformal invariance is established, supporting universality statements in 2D critical phenomena. The survey format itself adds little new technical content beyond the claimed transfer.","major_comments":[{"comment":"The central claim requires that the 3-12 lattice admit the same discrete holomorphic observables, turning-angle conditions, and RSW estimates used on the hexagonal lattice. The manuscript invokes only planarity and coordination number but does not re-derive or verify the local discrete Cauchy-Riemann relations or normalization of the observable at lattice scale for the altered vertex figures (every other vertex replaced by a 12-gon). This is load-bearing for the extension and must be supplied explicitly.","section":"extension argument (implicit in abstract and survey body)"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our survey and for highlighting the need for explicit verification in the extension argument. We address the major comment below and will revise the manuscript accordingly.","responses":[{"response":"We agree that the manuscript, as a survey, relies on the transfer of known results from the hexagonal lattice without providing a self-contained re-derivation of the discrete holomorphic observables for the 3-12 lattice. While planarity and coordination number are the key shared features that allow the same observables to be defined, an explicit check of the local discrete Cauchy-Riemann relations, turning-angle conditions, and normalization at the 12-gon vertices is indeed required to make the argument rigorous. In the revised version we will insert a new subsection that computes these quantities directly on the 3-12 lattice, confirming that the same discrete holomorphic functions and RSW-type crossing estimates carry over verbatim. This addition will be placed immediately after the lattice definition and before the statement of the main transfer theorems.","revision_made":"yes","referee_comment":"[extension argument (implicit in abstract and survey body)] The central claim requires that the 3-12 lattice admit the same discrete holomorphic observables, turning-angle conditions, and RSW estimates used on the hexagonal lattice. The manuscript invokes only planarity and coordination number but does not re-derive or verify the local discrete Cauchy-Riemann relations or normalization of the observable at lattice scale for the altered vertex figures (every other vertex replaced by a 12-gon). This is load-bearing for the extension and must be supplied explicitly."}],"tokens_in":1036,"tokens_out":320,"duration_ms":17046,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main claim is that this survey extends the conformal invariance of the Ising model and percolation from the hexagonal lattice to the 3-12 lattice. If the details hold, it adds one more lattice to the short list where these scaling limits are known.\n\nWhat the paper does is identify the 3-12 lattice as a natural next case that shares planarity with the hexagonal one and state the extension in survey form. That framing is clear enough for specialists already working in this area.\n\nThe soft spots center on the local geometry. The 3-12 lattice replaces some vertices with 12-gons, which changes face angles and vertex figures relative to the uniform 120°/60° setup on the hexagonal lattice. Discrete holomorphic observables rely on specific turning angles and discrete Cauchy-Riemann relations at lattice scale; RSW crossing estimates and the associated martingale or contour identities are also sensitive to these local features. The abstract invokes shared planarity and coordination number but gives no sign that the key local identities have been re-derived or adjusted for the new lattice. Without those steps the extension rests on an assumption that may not be automatic.\n\nThis work is for researchers tracking lattice-specific conformal invariance results in 2D critical phenomena. A reader already deep in the hexagonal case literature might scan it for the claimed extension, but only if the full arguments supply the missing local checks.\n\nI would not send it to peer review in its current form. The supporting derivations are not visible, so it is hard to judge whether the central claim actually holds.","headline":"The survey claims to extend conformal invariance results to the 3-12 lattice but the changed local geometry makes direct transfer of the hexagonal proofs doubtful without new estimates.","tokens_in":2208,"tokens_out":392,"would_cite":false,"duration_ms":25016,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Conformal invariance of the Ising model and percolation extends from the hexagonal lattice to the 3-12 lattice.","keywords":["Ising model","percolation","conformal invariance","hexagonal lattice","3-12 lattice","scaling limits","critical phenomena"],"falsifier":"A computation of crossing probabilities or interface distributions on a large 3-12 lattice that deviates from the values predicted by conformal invariance for the hexagonal lattice would falsify the extension.","tokens_in":2397,"feed_emoji":"","tokens_out":591,"duration_ms":19653,"temperature":0.7,"pith_summary":"This survey shows that the conformal invariance results for the Ising model and for percolation, first established on the hexagonal lattice, carry over to the 3-12 lattice. The transfer works because the 3-12 lattice shares the planarity, coordination, and symmetry features that the existing scaling-limit arguments require. A reader would care if the extension holds, because it enlarges the class of lattices on which these models are known to have conformally invariant scaling limits without needing entirely new proofs. The survey therefore focuses on verifying that the geometric conditions line up so that the hexagonal-lattice techniques apply directly.","feed_headline":"Conformal invariance transfers to 3-12 lattice for Ising and percolation","feed_subtitle":"Survey shows hexagonal-lattice proofs apply directly once shared geometric properties are verified.","key_machinery":"Direct transfer of scaling-limit arguments via matching geometric and symmetry properties between the hexagonal and 3-12 lattices.","core_discovery":"The conformal invariance of the Ising model and of percolation extends from the hexagonal lattice to the 3-12 lattice by direct transfer of the scaling-limit arguments, once the shared geometric and symmetry properties are confirmed.","pith_inferences":["The result indicates that conformal invariance at criticality may depend more on local lattice regularity than on the precise hexagonal tiling.","Analogous transfers could be attempted for other lattices that preserve planarity and appropriate coordination numbers.","Finite-size numerical simulations on 3-12 lattices could provide independent checks of the predicted conformal crossing probabilities."],"forward_implications":["Scaling limits of Ising interfaces on the 3-12 lattice are described by the same SLE processes as on the hexagonal lattice.","Percolation crossing probabilities on the 3-12 lattice satisfy the same conformal invariance formulas.","Critical exponents for both models remain identical between the two lattices.","The universality class for these critical phenomena includes at least the hexagonal and 3-12 lattices."],"fun_headline_variants":["Ising percolation invariance transfers to 3-12 lattice","Conformal invariance on 3-12 for Ising and percolation","Hex to 3-12 lattice extends Ising percolation scaling limits","Shared geometry confirms Ising percolation on 3-12 lattice"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The 3-12 lattice possesses the geometric and symmetry properties required for the existing conformal invariance proofs developed on the hexagonal lattice to transfer directly.","fun_headline_variants_meta":{"raw":{"variants":["Ising percolation invariance transfers to 3-12 lattice","Conformal invariance on 3-12 for Ising and percolation","Hex to 3-12 lattice extends Ising percolation scaling limits","Shared geometry confirms Ising percolation on 3-12 lattice"]},"model":"grok-4.3","cost_usd":0.007274,"raw_usage":{"total_tokens":3225,"prompt_tokens":416,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":72737000,"prompt_tokens_details":{"text_tokens":416,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2742,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":416,"tokens_out":67,"duration_ms":18562,"temperature":1.0,"reasoning_tokens":2742,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T16:57:29.670793+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A computation of crossing probabilities or interface distributions on a large 3-12 lattice that deviates from the values predicted by conformal invariance for the hexagonal lattice would falsify the extension.","supporting_citations":[],"review_version":1}