{"id":"a32ac235-6b50-4175-b11a-4507c95e335b","arxiv_id":"2603.27827","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Resolves the Heesch problem by showing unbounded Heesch numbers for homogeneous tilings and convex monotiles in the hyperbolic plane, with first examples of weakly aperiodic convex monotiles from dual homogeneous tilings.","lead":"The paper resolves the Heesch problem for homogeneous tilings in the hyperbolic plane and gives a corollary for convex monotiles, plus the first weakly aperiodic convex monotiles from duals. A smart generalist might read it to see how tiling questions behave differently in curved space and what that implies for aperiodic patterns.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Corollary transfer from homogeneous tilings to convex monotiles lacks explicit verification that hyperbolic convexity does not introduce bounded surround counts","rationale":"The reader's weakest assumption correctly isolates the transfer step as the least secure link. Because the review was performed on the abstract, the concrete_test above supplies the minimal check that would either confirm the corollary or expose a missing verification. No other internal inconsistency is visible from the given claim structure.","tokens_in":1540,"tokens_out":350,"duration_ms":16256,"concrete_test":"Extract the precise statement of the corollary (likely near the end of the main theorem section) and the preceding homogeneous construction; verify whether the proof invokes only properties preserved under convexification or whether it relies on an unstated claim that convexity cannot reduce the maximum surround count below the homogeneous value. If the latter, recompute the surround sequence for the convexified tile on a small hyperbolic patch (e.g., radius-5 neighborhood) to test whether the layer count remains unbounded.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim resolves the Heesch problem first for homogeneous (semi-regular) tilings and then asserts a corollary for convex monotiles. For the corollary to hold, the constructions must ensure that any periodic or surround-count constraints arising specifically from convexity in H^2 do not cap the Heesch number. The abstract states the corollary directly but supplies no separate argument or check confirming that the hyperbolic metric and convexity preserve the unboundedness already shown for the homogeneous case. If convexity forces a finite maximum number of layers around a tile (e.g., via angle or curvature constraints absent in the non-convex homogeneous setting), the unboundedness result would fail to transfer.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to resolve the Heesch problem for homogeneous (semi-regular) tilings in the hyperbolic plane by constructing families with arbitrarily large Heesch numbers, asserts this as a corollary for convex monotiles, and additionally exhibits the first known weakly aperiodic convex monotiles obtained as duals of homogeneous tilings.","tokens_in":1668,"tokens_out":301,"duration_ms":40348,"significance":"If the constructions are valid, the result would resolve an open question on the Heesch number in hyperbolic geometry for these tile classes and supply new examples of weakly aperiodic monotiles; the explicit constructions for the homogeneous case constitute a concrete strength.","major_comments":[{"comment":"Corollary following §3: the transfer of unbounded Heesch numbers from homogeneous tilings to convex monotiles is stated directly but supplies no separate verification that convexity and the hyperbolic metric do not introduce angle-deficit or curvature constraints capable of bounding the surround count; a concrete check (e.g., angle-sum calculation or explicit layer construction under convexity) is required for the corollary to be load-bearing.","section":"Corollary after §3"}],"minor_comments":[{"comment":"Notation for the dual construction in the final section could be clarified by adding a short diagram labeling the correspondence between homogeneous tiles and their dual monotiles.","section":"final section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their thorough reading and for identifying a point where the presentation of the corollary could be strengthened. We address the major comment below and will incorporate the requested verification in the revised manuscript.","responses":[{"response":"The homogeneous tilings in our constructions are formed by regular convex polygons meeting at vertices in the hyperbolic plane, so the prototiles are already convex. The explicit families we construct demonstrate that the angle deficits at vertices allow the number of surrounding layers to grow without bound while maintaining convexity and avoiding gaps or overlaps. To address the referee's concern directly, we will add a dedicated paragraph (or short subsection) after the corollary that performs an explicit angle-sum verification for one representative family: we compute the hyperbolic angle sum for successive layers around a central tile and show that the deficit permits arbitrarily large finite Heesch numbers without curvature imposing an upper bound. This will include a concrete numerical example confirming that convexity does not cap the surround count.","revision_made":"yes","referee_comment":"[Corollary after §3] Corollary following §3: the transfer of unbounded Heesch numbers from homogeneous tilings to convex monotiles is stated directly but supplies no separate verification that convexity and the hyperbolic metric do not introduce angle-deficit or curvature constraints capable of bounding the surround count; a concrete check (e.g., angle-sum calculation or explicit layer construction under convexity) is required for the corollary to be load-bearing."}],"tokens_in":1083,"tokens_out":312,"duration_ms":47081,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper claims to resolve the Heesch problem for homogeneous tilings in the hyperbolic plane, showing that the Heesch number can be unbounded for those, and then gives a corollary that the same holds for convex monotiles. It also presents what it says are the first weakly aperiodic convex monotiles coming from the duals of those homogeneous tilings. The new part is the resolution for the homogeneous case and the aperiodic examples. If the constructions are explicit and the proofs check out, that moves the needle on an old question in tiling theory. The dual approach for getting aperiodicity in convex tiles is a reasonable way to build on existing homogeneous tilings. The paper does well at stating the claims clearly in the abstract and linking them to the broader Heesch problem literature. It positions the work as addressing both the boundedness question and the search for aperiodic monotiles in hyperbolic settings. The main soft spot is the step from homogeneous tilings to convex monotiles. The stress test points out that the abstract does not show why convexity in hyperbolic geometry preserves the unbounded surround counts. Curvature effects or angle conditions might limit layers in ways that do not appear in the non-convex homogeneous setting. Without seeing the full argument or a specific check for that, it is hard to know if the corollary follows directly or needs extra work. If the paper has a section that verifies this transfer with concrete examples or bounds, that would address the concern. This paper is for specialists in combinatorial geometry and non-Euclidean tilings. Someone already familiar with the Heesch problem or hyperbolic tilings would find the constructions useful to examine and perhaps extend. It deserves a serious referee because the topic is significant and the claims are specific enough to be checked against the literature. I recommend sending it out for peer review so the details on the corollary and the new examples can be vetted properly by experts in the area.","headline":"The paper claims to resolve the Heesch problem for homogeneous tilings in the hyperbolic plane with a corollary for convex monotiles and adds first examples of weakly aperiodic convex monotiles from duals.","tokens_in":2162,"tokens_out":468,"would_cite":false,"duration_ms":41902,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"Theorem 1.1. For any given positive integer n, there exists a cyclic tuple k_n with Heesch number n."},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/DimensionForcing.lean","rs_theorem":null,"paper_passage":"dual of a homogeneous tiling is a tiling by a single convex polygon (a convex monotile)"}],"headline":"Tiling/Heesch machinery in H² has no overlap with RS forcing chain","alignment":"orthogonal","rationale":"Paper constructs layer-by-layer homogeneous tilings via cyclic vertex tuples k_n, proves unbounded Heesch numbers by inductive juxtaposition and odd-face blocking, then transfers to convex monotiles via duality. No J-cost, φ-ladder, 8-tick periodicity, ratio symmetry, or parameter-free constant derivation appears. RS theorems (e.g., reality_from_one_distinction, alexander_duality_circle_linking forcing D=3, Jcost uniqueness) address logic-to-spacetime emergence and are silent on combinatorial tiling problems.","tokens_in":47040,"confidence":"high","tokens_out":300,"duration_ms":10807,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Convex monotiles in the hyperbolic plane can surround themselves any number of times without tiling the entire plane.","keywords":["Heesch number","hyperbolic tilings","convex monotiles","homogeneous tilings","weakly aperiodic tilings","tiling theory","hyperbolic geometry"],"falsifier":"An explicit convex monotile in the hyperbolic plane together with a proof that no arrangement of congruent copies can surround it more than a fixed finite number of times would falsify the unboundedness claim.","tokens_in":2425,"feed_emoji":"📐","tokens_out":699,"duration_ms":35449,"temperature":0.7,"pith_summary":"The paper establishes that the Heesch number for convex monotiles in hyperbolic geometry has no upper bound. It first resolves the question for homogeneous tilings by constructing families where surround counts grow without limit, then invokes a corollary to transfer the result to convex monotiles. A sympathetic reader would care because this means there is no fixed maximum number of concentric layers a single tile can form before it must either tile the plane or become impossible to surround further. The work additionally produces the first known weakly aperiodic convex monotiles as duals of these homogeneous tilings.","feed_headline":"Hyperbolic convex monotiles allow arbitrarily many surrounds without tiling","feed_subtitle":"This shows the Heesch number is unbounded for homogeneous tilings and their convex duals in the hyperbolic plane.","key_machinery":"Homogeneous (semi-regular) tilings of the hyperbolic plane, together with their duals, which are used to build families of convex monotiles with increasing finite surround counts.","core_discovery":"We construct homogeneous tilings in the hyperbolic plane in which a tile admits arbitrarily large finite numbers of concentric surrounds by congruent copies without admitting a full tiling of the plane. This shows the Heesch number is unbounded for homogeneous tilings. The same unboundedness then follows for convex monotiles by the stated corollary. Duals of the homogeneous tilings also yield the first examples of weakly aperiodic convex monotiles.","pith_inferences":["Similar constructions might be adapted to produce convex monotiles with controlled but large Heesch numbers in other non-Euclidean settings.","The existence of weakly aperiodic convex monotiles in hyperbolic geometry raises the question of whether stronger aperiodicity results are possible with the same dual technique.","The result suggests that bounded-Heesch-number problems in hyperbolic geometry are likely to require tiles whose vertex figures or edge lengths vary in ways forbidden by homogeneity."],"forward_implications":["For any positive integer N there exist convex monotiles in the hyperbolic plane that admit at least N concentric surrounds without tiling the plane.","The Heesch problem is settled negatively for the class of homogeneous tilings and, by corollary, for convex monotiles.","Duals of homogeneous tilings supply the first known weakly aperiodic convex monotiles.","Any search for convex monotiles with bounded Heesch number in hyperbolic geometry must exclude or modify the homogeneous case."],"fun_headline_variants":["Hyperbolic convex monotiles show unbounded Heesch numbers","Unbounded Heesch numbers for hyperbolic homogeneous tilings","Weakly aperiodic convex monotiles from hyperbolic dual tilings","Homogeneous tilings yield unbounded Heesch numbers in hyperbolic plane"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The constructions and properties shown for homogeneous tilings carry over directly to convex monotiles by the corollary without needing separate checks that hyperbolic geometry or convexity imposes a uniform bound on surround counts.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic convex monotiles show unbounded Heesch numbers","Unbounded Heesch numbers for hyperbolic homogeneous tilings","Weakly aperiodic convex monotiles from hyperbolic dual tilings","Homogeneous tilings yield unbounded Heesch numbers in hyperbolic plane"]},"model":"grok-4.3","cost_usd":0.009805,"raw_usage":{"total_tokens":4196,"prompt_tokens":494,"num_sources_used":0,"completion_tokens":66,"cost_in_usd_ticks":98053000,"prompt_tokens_details":{"text_tokens":494,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3636,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":494,"tokens_out":66,"duration_ms":37234,"temperature":1.0,"reasoning_tokens":3636,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-21T10:20:32.264137+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit convex monotile in the hyperbolic plane together with a proof that no arrangement of congruent copies can surround it more than a fixed finite number of times would falsify the unboundedness claim.","supporting_citations":[],"review_version":1}