{"id":"8ab39991-4a95-4cb0-b94a-6e5ae8a2734e","arxiv_id":"2508.06451","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Aztec rectangles with odd rectangular holes have product formulas for domino tilings, yielding a dual of the Aztec diamond theorem and a symmetry in correlations of diagonal slits.","lead":"This paper proves simple product formulas for the number of domino tilings of Aztec rectangles with odd rectangular holes, and derives a dual of the Aztec diamond theorem. It matters because exact tiling counts with holes were long sought, and the results reveal a symmetry in slit correlations on the square grid.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Product formulas are proved for odd-rectangle holes, not the original Aztec-diamond windows; the title's motivating geometry may remain open.","rationale":"The reader's weakest assumption identifies exactly the same scope gap: the abstract's product formulas are for odd Aztec rectangle holes, not the original Aztec-diamond windows. My read agrees and sharpens it: the title's 'Aztec windows' may refer to the new odd-rectangle holes rather than the historical diamond holes, but the abstract does not clarify the relationship. Without the full text, I cannot determine whether the proofs extend to the original geometry, so the appropriate verdict remains UNVERDICTED. No internal inconsistency is apparent from the abstract; the concern is about the scope of the central claim and the unavailability of the proof details.","tokens_in":781,"tokens_out":2822,"duration_ms":35238,"concrete_test":"Check whether the original Aztec-window region—an Aztec diamond of order n with a central Aztec diamond hole of order m—belongs to any of the paper's proved families. If it does not, compute its tiling count for the smallest case (e.g., n=3, m=1) and test it against the claimed product formula. A failure would show the central claim does not cover the motivating geometry; a success or an explicit inclusion proof would resolve the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract explicitly says the original Aztec-window problem—an Aztec-diamond-shaped hole—gave non-round counts, and then pivots to 'a very closely related shape of holes (namely, odd Aztec rectangles).' The central claim is that these odd-rectangle-hole regions have product formulas, plus a dual Aztec diamond theorem for toroidal Aztec diamonds with such holes. The load-bearing assumption is that this new family is what the title calls 'round Aztec windows.' If the proof does not also cover the original Aztec-diamond-hole regions, the paper leaves the motivating 'Aztec window' problem unresolved; it proves a product formula for a different, albeit related, family. The abstract gives no bridge (e.g., a limiting or inclusion argument) from odd Aztec rectangles to Aztec diamonds, and no theorem statement is visible to check. The 'curious symmetry' and 'dual' results may likewise be specific to rectangular-hole geometry, so the advertised relevance to the original problem is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript (arXiv:2508.06451) studies domino tilings of Aztec-rectangle-like regions with holes of a shape the abstract calls 'odd Aztec rectangles'. It announces product formulas for the number of domino tilings of such regions and of certain symmetric cruciform regions; a formula for the number of perfect matchings of toroidal Aztec diamonds with such holes under a 'natural evolution' of the holes, leading to a claimed dual of the Aztec diamond theorem; and a symmetry in the correlation of diagonal slits on the square grid. The abstract explicitly notes that the original Aztec-window problem—a smaller Aztec-diamond-shaped hole—produced non-round counts, and states that the paper instead considers a 'very closely related' family of holes.","tokens_in":1012,"tokens_out":3211,"duration_ms":38304,"significance":"If the proofs are correct, the results would be a meaningful addition to enumerative combinatorics, extending the product-form family beyond the classical Aztec diamond theorem and connecting to correlation functions of slits. The purported dual Aztec diamond theorem and the unexpected slit-correlation symmetry are potentially significant. That said, this review is based on the abstract only; no derivations, theorem statements, or proof sketches are available for verification. The claims are plausible but unverified at this stage.","major_comments":[{"comment":"The abstract states that the original Aztec-window problem (a smaller Aztec diamond as a hole) gave non-round counts, and then pivots to 'a very closely related shape of holes (namely, odd Aztec rectangles).' The title 'Round Aztec windows' and the opening frame the paper as resolving the motivating question, but the announced product formulas are for a different family. The abstract gives no bridge — neither a limiting argument, an inclusion, nor a separate theorem — from odd Aztec rectangles to Aztec diamonds or vice versa. If the proofs do not also cover the original Aztec-diamond-hole regions, the paper leaves the motivating problem unresolved and the advertised relevance is only indirect. This is load-bearing for the central claim, because the novelty depends on whether the 'related' family is the correct generalization and whether the original non-roundness is explained or bypassed","section":"Abstract"},{"comment":"The claimed 'dual of the Aztec diamond theorem' is not stated in any checkable form. The abstract says only that a 'simple formula governs the way the number of their perfect matchings changes under a natural evolution of the holes' and that this 'yields in particular a natural dual.' Without the precise statement (e.g., the exact evolution rule, the class of toroidal regions, and the formula) it is impossible to assess whether this is a genuine dual or a related identity. The same applies to the 'correlation of diagonal slits' symmetry: the object and the symmetry are not defined in the available text. A theorem statement or precise definitions are necessary for verification.","section":"Abstract"}],"minor_comments":[{"comment":"'Round Aztec windows' is not defined in the abstract. The relationship between the odd-Aztec-rectangle holes and the original 'Aztec window' (Aztec-diamond hole) should be clarified early, since the phrase appears to be the authors' term for the new family.","section":"Title/Abstract"},{"comment":"The phrase 'a large variety of regions' is vague; a few examples or a precise class description would help the reader understand the scope.","section":"Abstract"},{"comment":"Terms such as 'symmetric cruciform regions,' 'correlation of diagonal slits,' and 'natural evolution of the holes' are used without definitions or references. Adding definitions or pointers to the full text would improve accessibility.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This review is based solely on the abstract; the full text was not available. I could not verify any of the mathematical claims. The editor may wish to obtain the complete manuscript before making a decision. In particular, the scope mismatch between the motivating Aztec-window problem and the announced odd-Aztec-rectangle results should be explicitly addressed in the full paper, and the abstract should avoid overstating the connection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious abstract, but only an abstract. The main claim—product formulas for Aztec rectangles and cruciform regions with odd-rectangle holes—looks like a genuine extension of known work, and the dual of the Aztec diamond theorem for toroidal regions is striking if it checks out. However, the title says 'round Aztec windows' and the abstract deliberately shifts the hole shape from an Aztec diamond to odd Aztec rectangles, so the original window problem is not resolved. The stress-test flags this, and it holds up from the abstract alone. Ciucu is a respected figure, so the results likely have real content, but I can't verify anything without the full text.\n\nWhat's new: the abstract contrasts with earlier pattern-only results and claims actual product formulas for a family that hasn't had them; the toroidal dual is a distinct structural statement, not just a tweaked count. The correlation symmetry, while only mentioned in passing, could be useful. These are valuable if they are true.\n\nSoft spots: the title is a bit generous—'round Aztec windows' is introduced for the new family, not the original motivating shape. The paper should be explicit that the classical Aztec-diamond-hole problem remains open, and readers should know that before going in. Also, the abstract gives no theorem statements, so the exact scope of 'large variety' and the form of the dual/symmetry are unknown. There is no way to audit the proofs.\n\nI'd say: send it to refereeing. A competent enumerative combinatorialist should look at the derivations. If the proofs are correct, this is a publishable contribution, even if not the full answer to the original window question. The referee should push for a clear distinction between the new family and the original problem.\n\nWould I cite it? If the proofs hold, yes, but I'd want to read the full version first. I'd bring it to a reading group once the full text is available, not just the abstract.","headline":"New product formulas for odd-rectangle-hole regions, but the original Aztec-window problem remains open; the title overpromises.","tokens_in":1421,"tokens_out":2602,"would_cite":true,"duration_ms":27483,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05B45","05C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves product formulas for domino tilings of Aztec rectangles with odd Aztec-rectangle holes and a dual of the Aztec diamond theorem for toroidal versions.","keywords":["round Aztec windows","domino tilings","Aztec diamond theorem","product formulas","toroidal Aztec diamond","perfect matchings","odd Aztec rectangles","slit correlations"],"falsifier":"Directly enumerate domino tilings of an Aztec rectangle with an odd Aztec-rectangular hole (for example, a small parameter choice within the stated family) and divide by the paper's product formula: any ratio different from 1 refutes the central claim. For the toroidal dual, compute perfect matchings before and after one hole-evolution step and compare the ratio with the formula.","tokens_in":704,"feed_emoji":"🧩","tokens_out":5892,"duration_ms":55092,"temperature":0.7,"pith_summary":"Domino tilings of an Aztec rectangle normally lose simple product formulas when a central 'window' hole is cut out; the paper shows that if the hole is chosen to be an odd Aztec rectangle rather than an Aztec diamond, a large family of such holed regions does have tiling counts given by simple closed-form products. The same happens for symmetric cruciform regions. On toroidal Aztec diamonds, the paper proves that the number of perfect matchings changes under hole evolution by a formula that is a natural dual of the Aztec diamond theorem. It also identifies an unexpected symmetry in the correlation of diagonal slits on the square grid. A sympathetic reader should take away that the earlier non-round counts for Aztec-window holes are not an obstacle to product formulas when the window is replaced by this closely related odd-rectangle shape.","feed_headline":"Product formulas count tilings with odd-rectangle windows","feed_subtitle":"A dual of the Aztec diamond theorem controls how hole evolution changes matchings on the torus.","key_machinery":"The central object is the odd Aztec rectangle—an Aztec rectangle with odd dimensions—used as the shape of the hole. It supplies the admissible window whose boundary makes the tiling counts factor. The dual-of-the-Aztec-diamond evolution formula is the identity that carries the toroidal part of the argument: it expresses the change in the perfect matching count of a toroidal Aztec diamond under hole evolution in simple multiplicative terms. Correlation of the holes, measured for diagonal slits on the square grid, is the quantitative object exhibiting the paper's symmetry.","core_discovery":"The central claim is that 'round' Aztec windows belong to a family with product-formula tiling counts. Concretely, the paper establishes that regions obtained from Aztec rectangles by removing holes shaped like odd Aztec rectangles have domino-tiling numbers equal to simple product formulas, and that symmetric cruciform regions share this property. On graphs obtained from a toroidal Aztec diamond by making the same kind of holes, the paper proves a formula describing how the number of perfect matchings changes under a natural evolution of the holes; specializing this gives a dual of the Aztec diamond theorem. The paper further derives consequences for correlations of holes, including an unex","pith_inferences":["If the odd-rectangle window family is the right setting, the original Aztec-diamond-window counts may be recoverable as finite sums of these product formulas by inclusion-exclusion over rectangles; a direct test would be to write such a decomposition.","The toroidal evolution formula may extend beyond Aztec diamonds to other toroidal boards whose perfect matchings are already product-counted, providing a general 'dual' operation for adding holes.","The diagonal-slit correlation symmetry suggests a determinant or Pfaffian expression symmetric under exchanging slit endpoints; one testable extension is to checkerboard or weighted edge versions of the square grid."],"forward_implications":["Domino tilings of Aztec rectangles with odd Aztec-rectangular holes are counted by explicit product formulas, not just recurrences or determinants.","Symmetric cruciform regions formed from Aztec rectangles share the same product-formula behavior.","For toroidal Aztec diamonds, hole evolution changes perfect matching counts by a simple multiplicative rule, yielding a dual of the Aztec diamond theorem.","The correlation of diagonal slits on the square grid satisfies a symmetry that is not forced by obvious grid symmetries.","Together these give a broader family of 'round' window problems all solvable by closed forms, even though the original Aztec-diamond-window counts were not round."],"supporting_citations":[],"fun_headline_variants":["Odd-rectangle windows yield product formulas for Aztec tilings","Dual of Aztec diamond theorem from hole evolution on torus","Unexpected symmetry in diagonal slit correlations from Aztec","Product-form tiling counts for Aztec rectangles with odd holes"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The load-bearing premise is that the well-behaved holes are odd Aztec rectangles; if the hole is instead an Aztec diamond, the paper does not claim the same product formulas, and the resolution of the historical 'Aztec window' problem depends on accepting this replacement of the hole shape.","fun_headline_variants_meta":{"raw":{"variants":["Odd-rectangle windows yield product formulas for Aztec tilings","Dual of Aztec diamond theorem from hole evolution on torus","Unexpected symmetry in diagonal slit correlations from Aztec","Product-form tiling counts for Aztec rectangles with odd holes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000269,"raw_usage":{"total_tokens":1468,"prompt_tokens":766,"completion_tokens":702,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":643}},"tokens_in":510,"tokens_out":702,"duration_ms":7261,"temperature":1.0,"reasoning_tokens":643,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:41:35.936183+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly enumerate domino tilings of an Aztec rectangle with an odd Aztec-rectangular hole (for example, a small parameter choice within the stated family) and divide by the paper's product formula: any ratio different from 1 refutes the central claim. For the toroidal dual, compute perfect matchings before and after one hole-evolution step and compare the ratio with the formula.","supporting_citations":[],"review_version":1}