{"id":"fda5d8cc-a094-4f0b-b6ec-a8ab1ffb5dff","arxiv_id":"2606.27063","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Brick-infinite algebras admit infinitely many non-τ-rigid bricks.","lead":"The paper proves that for any finite-dimensional algebra over an algebraically closed field, having infinitely many bricks implies infinitely many of those bricks are not τ-rigid. This settles in full generality a question previously known only under tameness assumptions and advances related conjectures on brick-finiteness.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the algebraically closed finite-dimensional setting as the weakest assumption is accurate and matches the paper's explicit scope. No further load-bearing gap is visible from the claim structure itself.","tokens_in":1676,"tokens_out":248,"duration_ms":22737,"concrete_test":"Locate the statement of the main theorem (likely Theorem A or 1.1) and confirm it matches the abstract claim exactly; then check that the proof invokes only standard facts about bricks and τ-rigidity that hold over algebraically closed fields without further hidden hypotheses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that for any finite-dimensional algebra A over an algebraically closed field, brick-infiniteness implies the existence of infinitely many non-τ-rigid bricks. This is presented as a direct proof in full generality within the stated hypotheses, with no internal steps that visibly presuppose tameness, specific representation type, or additional restrictions beyond those declared in the abstract. The reduction to a weaker form of the 2nd brick-Brauer-Thrall conjecture is logically consistent with the stated implication.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for any finite-dimensional algebra A over an algebraically closed field, if all but finitely many bricks are τ-rigid then A is brick-finite. Equivalently, every brick-infinite algebra admits infinitely many non-τ-rigid bricks. The result is obtained in full generality without tameness hypotheses and is presented as verifying a weaker form of the open 2nd brick-Brauer-Thrall conjecture on rigid bricks; it settles a question previously known only for E-tame algebras.","tokens_in":1738,"tokens_out":343,"duration_ms":41157,"significance":"If the derivation is correct, the result is significant: it removes the E-tameness restriction from a previously known case and supplies a general statement linking brick-infiniteness directly to the existence of infinitely many non-τ-rigid bricks. The work strengthens earlier results on bricks and τ-rigid modules and advances the study of the 2nd brick-Brauer-Thrall conjecture in representation theory of algebras.","major_comments":[],"minor_comments":[{"comment":"Abstract, final paragraph: the phrase 'settle a question that was previously known only for E-tame algebras' would benefit from an explicit citation to the E-tame result being extended.","section":"Abstract"},{"comment":"The notation for τ-rigidity and brick-finiteness is used throughout without a dedicated preliminary subsection; a short paragraph collecting the relevant definitions and standard facts would improve readability for readers outside the immediate subfield.","section":"Introduction"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and positive evaluation of the manuscript. We are grateful for the recommendation to accept and for recognizing the result as a general strengthening of prior work on the 2nd brick-Brauer-Thrall conjecture without tameness assumptions.","responses":[],"tokens_in":1184,"tokens_out":73,"duration_ms":12896,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that any brick-infinite algebra has infinitely many bricks that are not τ-rigid. The authors prove the contrapositive in full generality: if only finitely many bricks fail to be τ-rigid, then the algebra has only finitely many bricks total. This settles the question without the E-tame assumption that limited earlier work and gives a weaker form of the open conjecture on rigid bricks.\n\nThe result is new in its scope. The abstract makes clear that previous statements required extra conditions on the algebra, and this version drops them while staying inside the standard setup of finite-dimensional algebras over algebraically closed fields. The logical reduction to the second brick-Brauer-Thrall conjecture is consistent and direct, with no visible circularity or invented entities.\n\nThe soft spot is that only the abstract is available here, so the actual derivation steps cannot be checked for gaps in handling infinite families or edge cases in the module category. If the proof turns out to lean on some unstated property of the Auslander-Reiten translate or the endomorphism rings, that would need fixing, but nothing in the stated claim flags an obvious problem.\n\nThis is for readers already working on τ-tilting theory and the brick conjectures. A specialist following these questions will find the statement useful even before the full proof is digested. It is not broad enough for a general audience.\n\nThe paper deserves a serious referee. The claim is precise, the context is active, and the removal of the tameness hypothesis is substantive enough to warrant external review rather than desk rejection.","headline":"The paper proves brick-infiniteness implies infinitely many non-τ-rigid bricks for any finite-dimensional algebra over an algebraically closed field, removing the E-tame restriction from prior results.","tokens_in":2196,"tokens_out":403,"would_cite":false,"duration_ms":26705,"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":"Any brick-infinite algebra admits infinitely many bricks that are not τ-rigid.","keywords":["bricks","τ-rigid modules","brick-finite algebras","representation theory of algebras","brick-Brauer-Thrall conjectures"],"falsifier":"An explicit brick-infinite algebra in which all but finitely many bricks are τ-rigid would falsify the claim.","tokens_in":2566,"feed_emoji":"","tokens_out":515,"duration_ms":33514,"temperature":0.7,"pith_summary":"The paper proves that for a finite-dimensional algebra over an algebraically closed field, the condition that all but finitely many bricks are τ-rigid forces the algebra to have only finitely many bricks altogether. The argument proceeds by establishing the contrapositive statement that brick-infinite algebras must contain infinitely many non-τ-rigid bricks. This settles, without any tameness hypothesis, a question previously resolved only for E-tame algebras and supplies a weaker form of an open conjecture that replaces τ-rigidity by rigidity.","feed_headline":"Brick-infinite algebras admit infinitely many non-τ-rigid bricks","feed_subtitle":"Proves that only finitely many non-τ-rigid bricks forces the algebra to have only finitely many bricks total.","key_machinery":"The direct implication, proved in full generality, that an algebra with only finitely many non-τ-rigid bricks must be brick-finite.","core_discovery":"If all but finitely many bricks of the algebra A are τ-rigid, then A is brick-finite. Equivalently, any brick-infinite algebra admits infinitely many bricks which are not τ-rigid.","pith_inferences":["τ-rigidity functions as a coarse but effective filter separating algebras with finite brick sets from those with infinite ones.","Analogous statements might be examined for other module properties such as rigidity or finite projective dimension."],"forward_implications":["Any algebra possessing infinitely many bricks must possess infinitely many that are not τ-rigid.","The result confirms a weaker form of the conjecture that almost all bricks being rigid implies the algebra is brick-finite.","The statement holds for arbitrary finite-dimensional algebras and removes the E-tameness restriction used in earlier work."],"fun_headline_variants":["Brick-infinite algebras have infinitely many non-τ-rigid bricks","Finitely many non-τ-rigid bricks imply brick-finite algebras","Every brick-infinite algebra has infinitely many non-τ-rigid bricks","Brick-infinite algebras yield infinitely many non-τ-rigid bricks"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A is a finite-dimensional algebra over an algebraically closed field.","fun_headline_variants_meta":{"raw":{"variants":["Brick-infinite algebras have infinitely many non-τ-rigid bricks","Finitely many non-τ-rigid bricks imply brick-finite algebras","Every brick-infinite algebra has infinitely many non-τ-rigid bricks","Brick-infinite algebras yield infinitely many non-τ-rigid bricks"]},"model":"grok-4.3","cost_usd":0.009242,"raw_usage":{"total_tokens":4102,"prompt_tokens":595,"num_sources_used":0,"completion_tokens":75,"cost_in_usd_ticks":92424500,"prompt_tokens_details":{"text_tokens":595,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3432,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":595,"tokens_out":75,"duration_ms":45672,"temperature":1.0,"reasoning_tokens":3432,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T02:30:28.581570+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit brick-infinite algebra in which all but finitely many bricks are τ-rigid would falsify the claim.","supporting_citations":[],"review_version":1}