{"id":"cfddaaa7-dd57-4477-84be-c116110a339b","arxiv_id":"2508.10067","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A proof is claimed that determining whether a set of five polyominoes can tile the plane is undecidable, via a new edge-labeling construction.","lead":"The paper claims a proof that no algorithm can decide whether five given polyomino shapes can tile the whole plane by translation. The proposed proof uses a fresh edge-labeling trick to encode arbitrary matching rules into just one extra shape.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of the edge-labeling gadget is unreadable in the submitted text, so the central undecidability claim is unsupported by the artifact.","rationale":"The Reader's verdict is UNVERDICTED due to the unreadable full text. My stress-test agrees with that conclusion. The abstract names the edge-labeling construction as the novel mechanism, and the reader's weakest assumption identifies that construction as the critical point where the proof could fail. I have no additional mathematical objection because no readable mathematical content exists to analyze. The only concrete concern is that the key gadget cannot be checked, which is exactly the reason the verdict should remain UNVERDICTED. The proposed concrete test would start to resolve the concern by requiring a clean copy and a targeted check for spurious periodic tilings. I see no basis to change the verdict; the claim is plausible but unverified.","tokens_in":5436,"tokens_out":3490,"duration_ms":42936,"concrete_test":"Obtain the uncorrupted PDF or TeX source from arXiv. Then, focusing on the section defining the labeling polyomino, perform two checks: (1) re-derive the edge-compatibility rule and verify, by enumeration or SAT-based tiling solver on a finite torus, that every periodic tiling of the five polyominoes forces the encoded Wang tile matching rules exactly; (2) explicitly search for a periodic tiling that uses only the labeling polyomino and a subset of the other four in a way that avoids the Wang tile simulation. If any such spurious periodic tiling exists, the reduction is unsound. If no such tiling exists for a sufficiently large torus (e.g., up to 10x10), the specific concern of spurious tilings is mitigated for that construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract claims that undecidability of tiling with five polyominoes is proven via a new edge-labeling method: 'dedicating 1 polyomino to the labeling process' should implement arbitrary pairwise edge compatibility while preventing any tiling that does not correspond to a valid run of the encoded Wang tile set. This is the load-bearing construction: if the dedicated labeling polyomino is not exactly restrictive, the reduction admits spurious tilings and the undecidability argument collapses. The submitted full text, however, is an undecodable string of replacement characters from the opening section onward. No definition of the labeling polyomino, no lemma stating the soundness of the encoding, and no proof that every tiling of the five-piece set projects to a valid Wang tiling are readable. Consequently, the central claim is currently unsupported by the manuscript. This is not a claim that the result is false or that the authors are careless; it is a report that the artifact does not permit verification of the one combinatorial gadget on which the proof depends.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper's abstract claims a proof that it is undecidable whether a set of five polyominoes can tile the plane by translation. The proposed method is a new edge-labeling scheme in which one of the five polyominoes is dedicated to enforcing arbitrary pairwise edge compatibilities, enabling a reduction from the Wang tiling problem. The submitted full text, however, is not readable as a mathematical manuscript: from the opening section onward it consists of replacement characters, with only fragments of headings and isolated formulas legible. No definitions, lemmas, or proof of the reduction or of the soundness of the encoding are accessible. The central claim is therefore unsupported by the artifact as received.","tokens_in":5720,"tokens_out":3378,"duration_ms":39277,"significance":"If the claimed result holds, it would be a substantive addition to the study of undecidable tiling problems, improving knowledge of the smallest size of a set of polyominoes for which the translational tiling decision problem is undecidable. The proposed 'labeling polyomino' idea could be of independent interest if it is made precise and proved correct. But the present artifact does not allow verification of the result, of the novelty relative to prior bounds, or of the correctness of the construction. The significance is conditional and cannot be assessed from the submitted text.","major_comments":[{"comment":"The body of the manuscript is an undecodable byte sequence of replacement characters. There is no legible definition of the labeling polyomino, no statement of the main theorem with proof, no lemma that the encoding is sound and complete, and no demonstration that every tiling of the five-piece set projects to a valid Wang tiling. The stress-test concern lands: the one load-bearing gadget, the dedicated labeling polyomino, cannot be checked. This is not a claim that the result is false; it is a statement that the submitted manuscript does not permit refereeing.","section":"Full text"},{"comment":"The abstract's key assertion that 'dedicating 1 polyomino to the labeling process' can realize arbitrary pairwise edge compatibility is not accompanied in any visible portion of the paper by a formal definition of edges, labels, or matching rules, nor by a statement of the exactness of the encoding. A rigorous treatment would need to prove both that every valid Wang tiling induces a tiling by the five polyominoes and that every tiling by the five polyominoes yields a valid Wang tiling (excluding spurious tilings). None of this is present in the readable fragments.","section":"Abstract"}],"minor_comments":[{"comment":"Only the title, abstract, article number, and date are legible; the reference list and all section numbering are lost. It is therefore impossible to check the paper's relation to earlier undecidability results for polyominoes or to compare with prior lower bounds.","section":"General"},{"comment":"The readable fragments of formulas are garbled and incomplete. Even if the encoding corruption were fixed, the paper would benefit from a clear formal statement of the theorem and a numbered lemma for the labeling construction.","section":"Mathematical notation"}],"recommendation":"uncertain","confidential_remarks":"The submitted artifact is not a reviewable manuscript: the body is an undecodable string of replacement characters. I recommend returning it to the authors to provide a correctly rendered PDF or source file. No technical assessment can be made in good faith from the current text, and the mathematics may well be correct or incorrect; the current artifact does not permit a judgment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zoe—quick take on arXiv:2508.10067.\n\nThe abstract is legible and states a clear goal: a five-polyomino undecidability proof via a new edge-labeling gadget that uses one polyomino just to encode arbitrary edge matches. That construction is the whole ballgame. But the full text in the artifact is a string of replacement characters from the first page onward. No definition of the labeling polyomino, no lemmas, no diagrams, no proof of soundness against spurious tilings. I literally cannot check a single line of the argument. So my verdict is \"unverified,\" not \"wrong.\"\n\nWhat the paper does well: the abstract is honest about the method, and the idea of dedicating one polyomino to labeling is a coherent-looking strategy. If the construction works, it would be a neat proof. That's about all I can credit.\n\nSoft spots, in proportion. The biggest is the artifact itself: there is no proof to review. That is fatal for the current submission, regardless of whether the result is true. Second, the novelty ceiling is low: I understand the five-polyomino undecidability theorem is already in the literature, so this paper's value would be the new method rather than the result. Third, the reduction from Wang tilings depends on the dedicated labeling piece being exactly as restrictive as claimed; if it allows extra tilings, the reduction collapses. I'd love to check that, but I can't.\n\nOn the citation pattern: I can't evaluate it either, because the references are inside the unreadable block. If the author has not cited the existing five-polyomino construction, that would need fixing.\n\nBottom line: this paper deserves a serious referee only after the authors provide a readable full text. As received, it should be returned for resubmission. The underlying work might be fine, but a peer reviewer would have nothing to review.","headline":"Claim is plausible, but the proof text is undecodable, so the paper as submitted cannot be evaluated.","tokens_in":6118,"tokens_out":5439,"would_cite":false,"duration_ms":50539,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B45","03D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that deciding whether a set of five polyominoes can tile the plane by translation is undecidable, via a new edge-labeling method that encodes arbitrary matching rules in a single polyomino.","keywords":["polyomino","tiling","undecidability","translation tiling","edge labeling","edge matching","decision problem","tile sets"],"falsifier":"Take a finite edge-matching rule system known to have no tiling, run the paper's construction to produce its five polyominoes, and search for a plane tiling—for example, a periodically repeating finite patch. Finding one would show the encoding admits spurious tilings; a correct decision procedure for five-polyomino translation tiling would also disprove the result.","tokens_in":5372,"feed_emoji":"🧩","tokens_out":6408,"duration_ms":69595,"temperature":0.7,"pith_summary":"The paper proves that the translation-tiling problem for sets of five polyominoes—shapes made of unit squares joined edge-to-edge—is undecidable: no algorithm can always determine whether such a set covers the plane by translation. The proof introduces an edge-labeling method in which one special polyomino encodes an arbitrary compatibility relation between edge types. This lets the five pieces simulate any finite edge-matching tiling system, and the classical edge-matching tiling problem is undecidable. If correct, this establishes five as the new constant-size threshold at which polyomino tiling decisions become algorithmically unsolvable.","feed_headline":"Five polyominoes make plane tiling undecidable","feed_subtitle":"One label piece encodes arbitrary edge rules, so no algorithm can decide five-piece tiling.","key_machinery":"The central object is a labeling polyomino—a single piece whose boundary is shaped to encode edge types as physical labels. It works by making each exposed unit edge carry a label that can be made compatible or incompatible with any label on another piece, thereby realizing an arbitrary prescribed pairwise matching relation in geometry. The other four polyominoes then implement the tiles of a general edge-matching system, so every tiling of the encoded system corresponds to a tiling of the five-piece set, and vice versa.","core_discovery":"The central claim is that the translation-tiling problem for sets of exactly five polyominoes has no decision algorithm. The carrier of the argument is a new labeling method: a single specially built polyomino carries edge labels that force any two adjacent tiles to agree with a prescribed matching relation chosen in advance. This lets the author encode an arbitrary finite set of edge-matching rules into a set of five polyominoes, and a plane tiling of the polyomino set exists exactly when the encoded edge-matching system has a tiling. Since the latter problem is undecidable, so is the former.","pith_inferences":["Beyond the paper: because the labeling method stores arbitrary compatibility data on one piece, the same trick may lower the undecidability threshold in variants where rotations or reflections are allowed, or where the pieces must form a connected tile set.","Beyond the paper: if the dedicated labeling polyomino cannot be merged into the other four without losing arbitrary compatibility, then five is the natural limit of this method; testing whether a four-piece analogue exists would settle whether the bound can be improved.","Beyond the paper: the construction suggests a general dictionary between finite edge-matching constraint systems and constant-size polyomino sets, so other undecidable constraint problems could be translated into tiling questions with the same five-piece format."],"forward_implications":["The translation-tiling problem for five polyominoes is undecidable: no general algorithm can always report whether a given such set covers the plane.","The result gives an explicit five-piece construction that simulates arbitrary finite edge-matching rules, so the undecidability is obtained constructively, not by a counting argument.","No finite local condition computed from the pieces can characterize translation tileability of polyomino sets, because such a condition would yield a decision procedure.","The undecidability holds even when rotations and reflections are disallowed, since the tiling is by translation only."],"supporting_citations":[],"fun_headline_variants":["Tiling the plane with 5 polyominoes? Undecidable","Five polyominoes: tiling problem undecidable","5 polyominoes encode undecidable tiling problem","Plane tiling with five polyominoes is undecidable","Undecidability proven for 5-polyomino tiling"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The load-bearing premise is that the labeling polyomino's geometry forces exactly the intended compatibility relation and never permits an unintended match that could create a tiling when the encoded edge-matching system has none.","fun_headline_variants_meta":{"raw":{"variants":["Tiling the plane with 5 polyominoes? Undecidable","Five polyominoes: tiling problem undecidable","5 polyominoes encode undecidable tiling problem","Plane tiling with five polyominoes is undecidable","Undecidability proven for 5-polyomino tiling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00073,"raw_usage":{"total_tokens":3015,"prompt_tokens":564,"completion_tokens":2451,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":308,"completion_tokens_details":{"reasoning_tokens":2358}},"tokens_in":308,"tokens_out":2451,"duration_ms":18100,"temperature":1.0,"reasoning_tokens":2358,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:51:27.365413+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a finite edge-matching rule system known to have no tiling, run the paper's construction to produce its five polyominoes, and search for a plane tiling—for example, a periodically repeating finite patch. Finding one would show the encoding admits spurious tilings; a correct decision procedure for five-polyomino translation tiling would also disprove the result.","supporting_citations":[],"review_version":1}