{"id":"21114848-ce0b-403c-8e8a-21817d03710e","arxiv_id":"2606.06707","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The support-size restriction in Gao and Wang's strong clopen marker theorem for generating sets of Z^n is removed by a unimodular conjugacy argument that puts generators in full-support position.","lead":"This paper proves that a strong clopen marker theorem for finite generating sets of the integer lattice Z^n holds without requiring each generator to have support size exactly 1 or n. A generalist might read it to understand how a coordinate change can simplify technical assumptions in lattice combinatorics and symbolic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption is exactly the existence of the unimodular change of coordinates. That assumption is true and does not introduce a correctness risk; the remainder of the conjugacy argument follows formally from the lattice automorphism property. No other load-bearing gap appears in the reduction.","tokens_in":1602,"tokens_out":299,"duration_ms":35243,"concrete_test":"Take n=3 and S = {(1,0,0),(0,1,0),(1,1,1),(2,3,5)}; search for a matrix A in GL(3,Z) with entries bounded by 10 whose rows each satisfy the dot-product conditions; verify that every transformed vector has no zero entries.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim reduces the general case to the Gao-Wang theorem via existence of A in GL(n,Z) such that every vector in the finite nonzero set S satisfies that A s has all coordinates nonzero. This existence holds: the n rows of A must each satisfy row_j · s ≠ 0 for all s in S (finitely many strict linear inequalities). The forbidden loci are a finite union of hyperplanes, whose complement is nonempty and open; integer points in the complement exist and can be completed to a unimodular basis. The conjugacy step then preserves the marker property because A is a lattice automorphism.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript extends Gao and Wang's strong clopen marker theorem for finite generating sets of Z^n by removing the assumption that each generator has support of size 1 or n. The proof is a short conjugacy argument: a unimodular change of coordinates places any finite set of nonzero lattice vectors in full-support position, the prior theorem is applied, and the resulting marker set is conjugated back.","tokens_in":1733,"tokens_out":210,"duration_ms":14621,"significance":"If the result holds, it removes a restrictive hypothesis from an existing theorem, yielding a strictly more general statement on the existence of strong marker sets for arbitrary finite generating sets of the integer lattice. The argument is concise, invokes only the standard fact that the complement of finitely many hyperplanes in GL(n,Z) is nonempty, and preserves the marker property under lattice automorphisms. This strengthens the theorem's applicability without introducing new parameters or ad-hoc constructions.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment and recommendation to accept the manuscript. The report accurately summarizes the contribution as a short conjugacy argument that removes the support-size hypothesis from Gao and Wang's theorem.","responses":[],"tokens_in":1086,"tokens_out":59,"duration_ms":7360,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper shows that the support-size hypothesis in Gao and Wang can be dropped. After a unimodular change of coordinates, any finite set of nonzero vectors in Z^n can be made to have full support, the old theorem applies, and the marker set is conjugated back. That is the entire contribution.\n\nThe reduction is the new piece. It is a legitimate extension because the original work required the assumption and this removes it without adding new machinery. The argument stays short and uses only that the forbidden loci are a finite union of hyperplanes whose complement contains points that complete to a basis in GL(n,Z). The stress-test note confirms this existence step is standard and the conjugation preserves the marker property.\n\nNo major soft spots appear. The central claim reduces cleanly to the prior theorem, and nothing in the setup introduces circularity or unfalsifiable steps. Minor details like the explicit construction of the matrix A would need checking in the full text, but they are not load-bearing.\n\nThis is a narrow technical note aimed at people already working on marker sets or related tiling questions in Z^n. A reader inside that subfield gets a small, usable cleanup. It is worth sending to a serious referee because the claim is precise, the fix is correct, and the evidence is the reduction itself.","headline":"Short note removes the support assumption from Gao-Wang via a conjugacy reduction that holds up.","tokens_in":2190,"tokens_out":327,"would_cite":false,"duration_ms":11053,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The support assumption is unnecessary for the strong clopen marker theorem on generating sets of Z^n.","keywords":["marker sets","Z^n","generating sets","unimodular transformation","clopen sets","conjugacy","lattice"],"falsifier":"A concrete finite set of vectors in some Z^n for which no unimodular matrix makes all of them have full support in every coordinate.","tokens_in":2490,"feed_emoji":"","tokens_out":559,"duration_ms":27826,"temperature":0.7,"pith_summary":"Gao and Wang proved a strong clopen marker theorem for finite generating sets of Z^n assuming each generator has support of size 1 or n. This paper demonstrates that the theorem holds without this support assumption. The key step is a unimodular change of coordinates that puts any finite set of nonzero lattice vectors into full-support position. The Gao and Wang theorem can then be applied, and the marker set conjugated back to the original coordinates. This extension makes the result applicable to arbitrary finite generating sets.","feed_headline":"Marker theorem for Z^n holds without support limits","feed_subtitle":"Unimodular changes put any generators into full support so the Gao-Wang result applies directly.","key_machinery":"Unimodular conjugacy to full-support position for applying the Gao-Wang marker theorem.","core_discovery":"Gao and Wang proved a strong clopen marker theorem for finite generating sets of Z^n under the assumption that each generator has support of size either 1 or n. We show that this support assumption can be removed. The proof is a short conjugacy argument: after a unimodular change of coordinates, any finite set of nonzero lattice vectors can be put in full-support position, allowing one to apply the theorem of Gao and Wang and conjugate the resulting marker set back.","pith_inferences":["The conjugacy technique may apply to marker theorems in other discrete groups.","Applications in symbolic dynamics on lattices no longer need to restrict to special generators."],"forward_implications":["Strong clopen marker sets exist for arbitrary finite generating sets of Z^n.","The support size restriction on generators is not required.","Marker sets for any finite set of nonzero vectors in Z^n can be obtained by conjugation.","The result simplifies the statement of the marker theorem."],"fun_headline_variants":["Support limits lifted for Z^n marker theorem","Unimodular change removes Z^n support assumption","Marker theorem holds for all Z^n generating sets","Conjugacy extends Gao-Wang to any Z^n generators"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"There always exists a unimodular integer matrix transforming any finite collection of nonzero vectors in Z^n so that each has nonzero components in all n positions.","fun_headline_variants_meta":{"raw":{"variants":["Support limits lifted for Z^n marker theorem","Unimodular change removes Z^n support assumption","Marker theorem holds for all Z^n generating sets","Conjugacy extends Gao-Wang to any Z^n generators"]},"model":"grok-4.3","cost_usd":0.00391,"raw_usage":{"total_tokens":1860,"prompt_tokens":538,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":39103000,"prompt_tokens_details":{"text_tokens":538,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1262,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":538,"tokens_out":60,"duration_ms":9063,"temperature":1.0,"reasoning_tokens":1262,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T00:10:43.042276+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete finite set of vectors in some Z^n for which no unimodular matrix makes all of them have full support in every coordinate.","supporting_citations":[],"review_version":1}