{"id":"493a6bd5-49e9-4b1a-8d14-d8a10293520e","arxiv_id":"2511.08312","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"There are exactly 3044 chamber-regular lattices (up to isomorphism) on ~C2-buildings whose special-vertex links are the unique generalized quadrangle of order (3,5); assuming Kantor's conjecture, these are all such lattices.","lead":"This paper constructs the first examples of lattices acting regularly on the chambers of ~C2-buildings, using the unique generalized quadrangle of order (3,5) as the local data. It reports a computer-assisted count of 3,044 such lattices, which becomes a full classification if Kantor's conjecture on non-Moufang quadrangles is true.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness of the exhaustive local-action classifications (Lemmas 31–33) is unverifiable without shipped code; an error would directly change the 3044 count.","rationale":"The 3044 count is the paper's headline. The mathematical framework (triangles of groups, non-positive curvature, local-to-global) is standard, and the arithmetic summing to 3044 is internally consistent with the lemmas. However, the lemmas themselves are computational. The paper explicitly states 'These actions have been computed with GAP' and 'The tables these lemmas are referring to have been calculated with a computer', yet no code or certificates are provided. This makes the completeness claim unverifiable from the text. If a single edge-regular action were missed, the families in Lemmas 37–38 would expand, the double-coset counts would change, and the exact total would differ. The presentations in the appendix at least allow checking existence, but not uniqueness. Proposition 26 (group determines action) is also used to pass from triangle isomorphism classes to group isomorphism classes; its proof is sketchy, but the reader's concern about the computational enumeration is more directly load-bearing because it affects the existence of the 3044 as stated, not just the uniqueness. The paper is honest about the Kantor conjecture dependence and the non-Moufang nature. Overall, a conditional accept pending independent computational verification is appropriate.","tokens_in":21971,"tokens_out":26356,"duration_ms":252026,"concrete_test":"Run an independent GAP enumeration: for each of Aut(Q(3,5)), Aut(K4,4), Aut(K6,6), compute all subgroups of order equal to the number of edges that act regularly on the edge set, up to conjugacy, and compare with Tables 2–4. This is feasible: |Aut(K6,6)| = 2·(6!)² ≈ 1.04e6; the other groups are smaller. Also recompute the type-preserving isomorphism class counts from Observation 20 and Lemma 40 for every family in Lemmas 41–43, and verify the reductions in Lemma 45, using the Σ_i data in Appendix B. Publish the GAP script so the classification is fully reproducible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem's exact count 3044 rests on the exhaustive enumeration of edge-regular actions on Q(3,5), K4,4, and K6,6 (Proposition 29 and Lemmas 31–33). The paper states these were 'computed with GAP' but ships no code, scripts, or certificates. The presentations in Tables 2–4 establish existence of the listed actions, but not completeness: a missed conjugacy class of an edge-regular subgroup would add new families to Lemmas 37–38 and change the double-coset counts in Lemmas 41–43, hence the 3044 total. The same issue affects the compatibility enumeration of the 163+232 families and the Σ_i computations in Appendix B: these are finite but nontrivial group calculations, and the proofs of Lemmas 37, 38, 41, 42, 43 are sketched ('straightforward', 'we just applied Lemma 40') rather than machine-checked. Because the central numerical claim is an exact integer, a single off-by-one in these computations is material.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper uses triangles of groups to construct the first examples of chamber-regular lattices on two-dimensional affine buildings of type ~C2. It starts from the unique generalized quadrangle Q of order (3,5), classifies edge-regular actions on Q, K4,4, and K6,6 by computer (Lemmas 31–33), combines compatible local actions into developable, non-positively curved triangles of groups (Lemmas 37–38), and counts isomorphism classes using double-coset methods (Lemmas 41–45). This yields 3144 type-preserving isomorphism classes and, after identifying mirror isomorphisms, 3044 isomorphism classes (Proposition 44, Theorem 46). The authors prove (Proposition 26) that the acting group determines the triangle of groups, so the count is interpreted as a count of distinct lattices, each with a unique chamber-regular action. Conditional on Kantor's conjecture, these are claimed to be the only chamber-regular lattices on locally finite ~C2-buildings. The paper includes explicit matrix generators for two local actions and presentations for all 35 local actions.","tokens_in":22184,"tokens_out":28091,"duration_ms":273770,"significance":"If the classification and the computer enumerations are correct, these are the first chamber-regular lattices on ~C2-buildings; they are exotic (non-Bruhat–Tits) because Q is non-Moufang, and under Kantor's conjecture they form a complete list. The general CAT(0)/triangles-of-groups framework is standard, and the internal arithmetic is consistent: the contributions in Lemmas 41–43 sum to 3144, and the reductions in Lemma 45 give 3044. The authors also provide concrete data (matrices in Table 1 and presentations in Appendix A). The main weakness is that the exhaustive classifications and the double-coset counts are asserted to be computer-checked without shipping code or certificates, so the exact numerical claims are not independently verifiable as submitted. A further issue is that the unqualified theorem statements appear to exceed the 'type-preserving' scope of the method.","major_comments":[{"comment":"The exhaustive classifications of edge-regular actions on Q(3,5), K_{4,4}, and K_{6,6} are load-bearing for the exact count, but they are asserted only as computer calculations ('The following proposition was checked with a computer'; 'The tables ... have been calculated with a computer'). No GAP code, scripts, certificates, or output are provided. A missed conjugacy class in any of the three graphs would change the allowed families in Lemmas 37–38 and the double-coset counts in Lemmas 41–43, hence the 3144/3044 totals and the uniqueness claims. The presentations in Tables 2–4 do not by themselves prove that the abstract groups act on the named graphs, nor do they prove completeness. Please supply the code and verification data, or replace these assertions by human-checkable enumerations.","section":"§5, Prop. 29 and Lemmas 31–33"},{"comment":"The counts of type-preserving isomorphism classes in Lemmas 41–43 (which sum to 3144) are justified only by 'We just applied Lemma 40 to each case.' The Appendix B tables of local automorphism groups Σ_i are not sufficient to reproduce the class numbers 1, 2, 3, 5, 6, 9, 12, 18, 24 without the intermediate double-coset computations. Because these counts are the core numerical output, the manuscript needs reproducible computational evidence (code, certificates, or explicit representative systems) to allow an independent check of the exact totals.","section":"§5, proofs of Lemmas 41–43, Appendix B"},{"comment":"The classification pipeline enumerates type-preserving actions: Definition 11 requires Γ to be a group of type-preserving isometries, and every triangle-of-groups action is type-preserving. A chamber-regular action that is not type-preserving need not have edge-regular vertex-stabilizer actions on links, so it would not be detected by Lemmas 31–33. The abstract correctly says 'type-preserving ..., chamber-regular ..., lattices', but Theorem 1 and the second sentence of Theorem 46 omit this qualifier and claim 'exactly 3044 chamber-regular lattices' / 'the only lattices that act chamber-regularly'. Either the theorems should be restricted to type-preserving actions, or a proof should be given that every chamber-regular lattice on a locally finite ~C2-building is type-preserving.","section":"§1, Theorem 1; §5, Theorem 46"},{"comment":"This proposition is the bridge from triangles of groups to pairwise non-isomorphic lattices with a unique action. The proof is too compressed at a load-bearing point: after the Flat Triangle Lemma, the claim 'If Δ is not a chamber, V1,V2,V3 do not generate Γ' is justified by saying that Γ' = ⟨V1,V2,V3⟩ acts regularly on the orbit of Δ and is therefore not as transitive as Γ. But when Δ is not a chamber it is not a fundamental chamber, and it is not shown that the orbit of Δ is a chamber complex on which Γ' acts regularly. Please expand this argument or supply a reference.","section":"§3.3, Proposition 26"}],"minor_comments":[{"comment":"The relator 'ab1a^{-1}b' contains an undefined generator b; the generating set is {a,b1,b2}. Please correct (presumably to 'ab1a^{-1}b1' or 'ab1a^{-1}b2') and re-check the other presentations in Tables 2–4 for similar transcription errors, since these tables are the encoded output of the classification.","section":"Appendix A, Table 3 (L19)"},{"comment":"Items 8 and 9 use the notation T^{(1)} for t∈{30,…,35}; by Notation 36 the superscript should be T^{(2)} for those vertex groups.","section":"§5, Lemma 38"},{"comment":"'monomorphisms from ∏_i E_i' should presumably read 'from ∏_i Aut(E_i)'.","section":"§2.2, Observation 20"},{"comment":"The generating set lists 'a1, a1, b1, b2'; the second generator should be a2.","section":"Appendix A, Table 2 (L2)"},{"comment":"There are several typos: 'develobality', 'simplical', 'regulary', 'previuos', 'cellar complexes', 'isomorphism of between'. Also the proof of Lemma 25 refers to 'the universal cover is of type ~C2' where the universal cover is the development; the terminology could be clarified.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious candidate for publication if the computational data is made available and the theorem statements are aligned with the type-preserving scope. The internal counts are consistent, and the framework is credible. I would not reject on the basis of computer use alone, but the current manuscript does not allow a referee to verify the exact 3044 count. I recommend requesting supplementary code/certificates and an expanded proof of Proposition 26."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First examples of chamber-regular lattices on ~C2-buildings, with a sharp classification count, but the count rests on computer enumerations the authors don't ship. The math architecture is sound; the verification gap is the only serious issue.\n\nWhat's new: the 11 chamber-regular actions on Q(3,5) are new, and they're used to build the first chamber-regular actions on a finite generalized quadrangle and the first chamber-regular ~C2-lattices. The move from local actions to global lattices via triangles of groups is standard, but working through it for this non-Moufang quadrangle is substantial work. The appendices list presentations and small-group IDs, which is useful.\n\nThe counting scheme is coherent. Observation 20 reduces type-preserving isomorphism classes to double cosets; Lemmas 40–45 then compute representative numbers. I spot-checked the arithmetic: 3144 type-preserving classes reduce to 3044 isomorphism classes, consistent with the stated isomorphism corrections. The reliance on Kantor's conjecture is honestly labeled and only affects the uniqueness clause, not the existence of the 3044 examples.\n\nThe soft spot is exactly what the stress-test note flags. Lemmas 31–33 assert exhaustive classifications of edge-regular actions on Q(3,5), K4,4, K6,6, and Lemmas 37–38 assert compatibility enumerations, all 'computed with GAP.' No code, scripts, or certificates are provided. The proofs of Lemmas 41–43 are sketched ('straightforward,' 'we just applied Lemma 40'). Because the main theorem states an exact integer, a single missed conjugacy class or an off-by-one in a double-coset count changes the result. This is a genuine reproducibility gap, not a fatal one. The structure of the argument is transparent enough that an independent implementation is feasible; the authors should be asked to provide it.\n\nThe paper is for geometric group theorists working on buildings and lattices, and for finite-geometry people interested in generalized quadrangles. It deserves a serious referee, conditional on shipping the computational evidence. I'd recommend sending it out with a request for the GAP code or a verification script, and for the enumeration lemmas to be expanded to the level where a referee can follow the counting without redoing the computation.","headline":"First chamber-regular ~C2-lattices with a sharp count, but the exact 3044 depends on unshipped GAP code.","tokens_in":22667,"tokens_out":2564,"would_cite":true,"duration_ms":26897,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E42","51E12"],"pacs":[],"model":"deepseek-v4-flash","headline":"There are exactly 3044 chamber-regular lattices on the new C2-buildings constructed here, the first known examples.","keywords":["chamber-regular lattices","affine buildings","C2-buildings","generalized quadrangles","triangles of groups","exotic buildings","Kantor's conjecture","lattice actions"],"falsifier":"Re-running the exhaustive search over the automorphism groups of Q(3,5), K4,4 and K6,6 and finding an edge-regular action not listed in the tables, or finding two triangle-of-groups classes in the same family that produce isomorphic lattices, would change the 3044 count.","tokens_in":21844,"feed_emoji":"🧩","tokens_out":4652,"duration_ms":38757,"temperature":0.7,"pith_summary":"The paper constructs the first chamber-regular lattices acting on two-dimensional affine buildings of type C2, and counts them: exactly 3044 isomorphism classes. Chamber-regular means the group permutes the chambers (triangular 2-simplices) freely and transitively. The buildings have all special-vertex links equal to the unique generalized quadrangle of order (3,5), which is not a Moufang polygon; hence none of these buildings is a Bruhat-Tits building, and all the lattices are exotic. The construction and classification are carried out by encoding the actions as triangles of groups and exhaustively classifying the local edge-regular actions on the quadrangle and on the complete bipartite graphs K4,4 and K6,6. If a standard conjecture in finite geometry holds, these 3044 lattices are in fact all chamber-regular lattices on locally finite C2-buildings.","feed_headline":"3044 chamber-regular lattices built on exotic affine C2 buildings","feed_subtitle":"First such lattices; if Kantor's conjecture holds, this list is complete for all locally finite C2-buildings.","key_machinery":"The key machinery is the triangle of groups: a commutative diagram of seven finite groups (one face group, three edge groups, three vertex groups) that encodes a group action on a simply connected 2-complex transitive on triangles. When the local actions are non-positively curved and the angle sum is exactly π, the development is an affine building. The vertex stabilizers are the 11 chamber-regular groups on Q(3,5), the 10 groups on K4,4, and the 14 groups on K6,6. Isomorphism classes of triangles of groups with fixed local actions correspond to double cosets in products of automorphism groups, and these the paper computes explicitly, yielding the count.","core_discovery":"The central discovery is that chamber-regular lattices on C2-buildings exist and are completely determined by finite combinatorial data: three edge-transitive local actions on the generalized quadrangle Q(3,5) and on the complete bipartite graphs K4,4 and K6,6, together with the gluing isomorphisms among edge stabilizers. The paper shows there are 11 chamber-regular actions on Q(3,5), and then classifies all compatible triangles of groups. After counting type-preserving isomorphism classes and then full isomorphism classes, exactly 3044 isomorphism classes of lattices survive, and each such group admits only one chamber-regular action on its building.","pith_inferences":["The same triangle-of-groups assembly method could be applied to other generalized polygons with chamber-transitive groups, such as the Lunelli-Sce quadrangle; testing whether any chamber-regular actions arise there would provide an independent check of Kantor's conjecture.","The exhaustive computer classification of local actions is the load-bearing step; releasing the search code or certificates would let the 3044 count be independently verified and reused on neighbouring problems.","The uniqueness of the action may have uses beyond enumeration: the building can be recovered group-theoretically from the lattice, which could feed into rigidity or quasi-isometric rigidity questions for these groups."],"forward_implications":["If correct, these are the first chamber-regular lattices on C2-buildings, settling an open existence question.","Under Kantor's conjecture, the list of 3044 lattices is a complete classification of type-preserving chamber-regular lattices on locally finite C2-buildings.","The 11 chamber-regular actions on Q(3,5) are the first (and conjecturally the only) chamber-regular actions on a finite generalized quadrangle.","Because Q(3,5) is not Moufang, the resulting buildings are not Bruhat-Tits; the lattices are the first non-algebraic, exotic building lattices of this kind.","Each of the 3044 groups admits a unique chamber-regular action, meaning the action can be reconstructed from the abstract group structure. "],"fun_headline_variants":["First chamber-regular lattices on C2-buildings: 3044 found","Exotic C2-building lattices: 3044 new chamber-regular actions","Chamber-regular C2 lattices exist: 3044 examples, maybe all","First exotic C2-building lattices: 3044 chamber-regular groups","3044 chamber-regular lattices on C2-buildings, first ever"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The exhaustive computer classification of edge-regular actions on Q(3,5), K4,4 and K6,6 is complete and correct, but the paper ships no code or certificates to check it.","fun_headline_variants_meta":{"raw":{"variants":["First chamber-regular lattices on C2-buildings: 3044 found","Exotic C2-building lattices: 3044 new chamber-regular actions","Chamber-regular C2 lattices exist: 3044 examples, maybe all","First exotic C2-building lattices: 3044 chamber-regular groups","3044 chamber-regular lattices on C2-buildings, first ever"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2622,"prompt_tokens":669,"completion_tokens":1953,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":413,"completion_tokens_details":{"reasoning_tokens":1848}},"tokens_in":413,"tokens_out":1953,"duration_ms":12196,"temperature":1.0,"reasoning_tokens":1848,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T22:49:58.533181+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-running the exhaustive search over the automorphism groups of Q(3,5), K4,4 and K6,6 and finding an edge-regular action not listed in the tables, or finding two triangle-of-groups classes in the same family that produce isomorphic lattices, would change the 3044 count.","supporting_citations":[],"review_version":1}