{"id":"cc4f3b37-dd75-42ae-80e3-33decbe4a623","arxiv_id":"2506.01315","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact regular genera are computed for S²×S¹×S¹ (6), the 4-torus (16), and seven small covers over Δ²×Δ² (8).","lead":"The paper resolves two long-open questions in PL topology by proving that the regular genus of S²×S¹×S¹ is 6 and that of the 4-torus is 16. It also classifies the seven small covers over Δ²×Δ² and shows all have regular genus 8.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unproved Observation in §2.2 is load-bearing: repeated applications justify the 120- and 52-vertex reductions, and a failure in any configuration would leave the final graphs uncertified as crystallizations.","rationale":"I read the paper as establishing exact regular genus values through explicit crystallizations together with the external lower bound of Proposition 1. The lower-bound arithmetic is correct: for S^2×S^1×S^1, χ=0 and m=2 give 6; for T^4, χ=0 and m=4 give 16; for the small covers, χ=1 and m=2 give 8. The main load-bearing point is the unproved Observation in §2.2. It is used centrally in the vertex reductions, and although it is likely true and probably provable from standard dipole moves, the paper does not supply that proof. The reader's conditional verdict identifies exactly this weakness, and I agree. The paper deserves credit for explicit constructions, Regina signatures, and the independent cube-based construction that partly supports Theorem 4; however, the signatures alone do not certify manifold recognition, and the invocations of the Observation are numerous enough that a single subtle failure would undermine the relevant upper bound. This is an addressable rigor gap rather than a demonstrated error, so the conditional verdict is appropriate.","tokens_in":28013,"tokens_out":24150,"duration_ms":265567,"concrete_test":"Write a script that, for each invocation of the Observation in Theorems 4 and 7, takes the graph before the move, deletes the chosen 2-dipole, checks that the remaining pair is a 3-dipole in the resulting graph, verifies that both intermediate graphs are contracted, and compares the graph obtained by the direct four-vertex deletion with the graph obtained by the two standard dipole cancellations. Run this for the moves with (Φ4,Λ4,Λ′4,{0,1}), (Φ5,Λ5,Λ′5,{1,2}), and (Φ6,Λ6,Λ′6,{0,2}) in Theorem 4 and for the three moves in Theorem 7 that use the Observation. If every check passes, the gap is expository; if any fails, the corresponding construction does not certify the claimed manifold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central upper bounds depend on the Observation in Subsection 2.2, which asserts that deleting two isomorphic k-colored edges connected by i- and j-colored pairs, via a 2-dipole followed by an induced 3-dipole, yields a crystallization of the same 4-manifold. This unproved move is used in Theorem 4 to reduce the 192-vertex gem of S^1×S^1×S^1×S^1 to 120 vertices, and in Theorem 7 to reduce the 96-vertex gems of the small covers to 52 vertices. The concern is not that the Observation is obviously false; it is that the proof of the central claims silently depends on a nonstandard composite move whose hypotheses must be rechecked after each 2-dipole cancellation. In particular, the four vertices being in distinct components of Γ_{∆4\\{i,j,k}} and the two edge-pairs lying in different components of Γ_{∆4\\{i,j}} are asserted, but the preservation of the 3-dipole condition after the first cancellation is not proved. If the condition fails in any one of the many applications, the reduced graph might no longer be a gem of the intended manifold, or might fail contractedness. The Regina isomorphism signatures identify the final graphs combinatorially but are not accompanied by Regina recognition certificates identifying the corresponding 4-manifolds; for Theorem 4 the later cube-based construction of Γ′ offers some independent support, but Theorem 2 and Theorem 7 have no such fallback. The cycle-count computations from the figures are also terse, but the structural validity of the constructed graphs is the more fundamental issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit edge-colored graphs (crystallizations) for S^2×S^1×S^1 (40 vertices), for S^1×S^1×S^1×S^1 (120 vertices), and for each of the seven Davis-Januszkiewicz equivalence classes of small covers over Δ^2×Δ^2 (52 vertices each). For each constructed graph the authors compute the regular genus and combine it with the external lower bound G(M) ≥ 2χ(M)+5m−4 from Proposition 1 (Basak–Casali) to conclude G(S^2×S^1×S^1)=6, G(S^1×S^1×S^1×S^1)=16, and that each of the seven small covers, including RP^2×RP^2, has regular genus 8. The paper also proposes a conjectural formula for the regular genus of the n-torus, verifies the n=4 case by an independent cube-based construction, and reports Regina isomorphism signatures for the principal crystallizations.","tokens_in":28311,"tokens_out":9166,"duration_ms":102021,"significance":"If the constructions are correct, the paper settles a conjecture from Basak's 2019 mapping-torus paper and the previously open value for the 4-dimensional torus, and it gives the first regular-genus computations for small covers over Δ^2×Δ^2. The main strengths are the fully explicit graph constructions, the absence of free parameters in the upper-bound constructions, and the provision of machine-readable Regina isomorphism signatures that allow independent combinatorial verification of the final graphs. These are concrete, checkable contributions to crystallization theory and to the classification of PL 4-manifolds by regular genus.","major_comments":[{"comment":"The Observation is load-bearing and is not proved. After canceling the 2-dipole {v1, v1′}, the cancellation joins, for every color c not in {i, j}, the former c-neighbor of v1 to the former c-neighbor of v1′. This can create new paths between v2 and v2′ in the color set Δ4\\{i,j,k}; the hypothesis that v1, v2, v1′, v2′ lie in distinct components of Γ_{Δ4\\{i,j,k}} does not by itself imply the required distinct-component condition for v2 and v2′ in (Γ1)_{Δ4\\{i,j,k}}. The assertion that v2 and v2′ form a 3-dipole after the first cancellation therefore needs a proof. The move is used repeatedly in Theorem 2 (moves Φ4, Φ5, Φ6) and Theorem 4 (moves Φ4 through Φ12); if the condition fails in any of these configurations, the reduced graphs may no longer be crystallizations of the claimed manifolds. Please supply a proof of the Observation or, failing that, a certified computational check of the dipole hypotheses after each intermediate step in Theorems 2 and 4.","section":"Subsection 2.2 (Observation)"},{"comment":"The Regina isomorphism signatures identify the final edge-colored graphs combinatorially, but an isomorphism signature alone does not certify which 4-manifold a graph represents; it certifies only that two graphs with the same signature are isomorphic. For Theorem 4, the later cube-based construction of Γ′ gives independent evidence that the 120-vertex graph represents the 4-torus, but for Theorem 2 (and for the identification of the small covers in Theorem 7) there is no independent recognition of the represented manifold. The manifold identification therefore rests entirely on the composition of moves in Sections 3 and 4, which is why the gap in the Observation is central. Please provide an independent recognition certificate for the 40-vertex graph of Theorem 2, for example by verifying that its associated colored triangulation is isomorphic to a known triangulation of S^2×S^1×S^1, or by constructing the final graph directly from a known gem.","section":"Theorems 2 and 4"},{"comment":"The proof of Theorem 7 is a long case analysis whose verification is delegated to Figure 9, with the sentence 'any statement whose justification is not explicitly provided can be understood to follow from Figure 9.' The final cycle counts—g{0,3}=g{0,4}=g{1,4}=g{2,3}=13 and g{1,2}=12 for the relevant permutations—are asserted after four polyhedral glue moves, but the reader cannot reconstruct all seven characteristic functions and all intermediate graphs from the figure without effectively repeating the whole computation. Since these numbers enter directly into the genus computation, please provide a systematic derivation or a machine-readable certificate (edge lists of all Γ′i, i=1,...,7, together with a script that computes the bi-colored cycle counts).","section":"Theorem 7"},{"comment":"The lower bound G(M(λ)) ≥ 8 uses Proposition 1 with m=2, the rank of the fundamental group. The statement 'it is easy to verify that the rank of the fundamental group of M^4(λ) is 2' is not backed by an argument or a reference. For the product RP^2×RP^2 the rank is 2, but for the six non-trivial RP^2-bundles the fundamental group should be computed explicitly or cited from a verifiable source. If any of these manifolds had rank 1, the lower bound would drop below 8 and the minimality conclusion would not follow. Please add the computation of π1 for each of the seven small covers.","section":"Theorem 7 and Lemma 6"}],"minor_comments":[{"comment":"There are typos: 'dimesional' should be 'dimensional' and the journal name in reference [20] is 'Aequationes', not 'Acquationes'.","section":"Section 1"},{"comment":"In the sentence introducing the sets for the second reduction, the last set is written as 'Λ′4 = {vA′6, vA′7}' but it should be 'Λ′5'; as written, Λ′4 is defined twice.","section":"Theorem 2"},{"comment":"The cycle count for G′2 is stated as 'all the cycles colored by {i,j} are of length 4 and thus g{i,j}=30' for five color pairs, but the counting argument from Figure 8 is not shown. Adding the explicit count of 4-cycles for each pair would make the computation reproducible.","section":"Theorem 4"},{"comment":"The sentence about {2,3}-colored cycles in Γ′i is initially confusing: it says there are 13 four-cycles for i∈{1,2,5} and then 9 four-cycles, 2 six-cycles, and 2 two-cycles for i∈{3,4,6,7}. Both statements give total g=13, but the wording suggests a contradiction; please rephrase to make the totals explicit.","section":"Theorem 7"},{"comment":"The figures are dense and use many labels; providing the edge lists of the final crystallizations in a supplementary file would make the Regina signatures and the cycle counts independently checkable without transcription from the figures.","section":"Figures 5–8 and 10–15"}],"recommendation":"major_revision","confidential_remarks":"The paper relies substantially on the authors' own preprint [1] for the construction method in Section 4. If that preprint is still under review, the editor may wish to confirm its availability and correctness, since Theorem 7 depends on the construction framework. The fit with a combinatorial-topology journal is good, and the explicit vertex counts and Regina signatures are valuable, but the unproved Observation in §2.2 is a genuine gap that should be closed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives exact regular genus values for three families of 4-manifolds: G(S²×S¹×S¹)=6, G(T⁴)=16, and G=8 for each of the seven small covers over Δ²×Δ². These are genuinely new results and they resolve a published conjecture plus a long-standing gap for the 4-torus. The authors build explicit crystallizations with vertex counts matching the lower bound from Proposition 1, so the upper and lower bounds meet. The Regina isomorphism signatures for the final graphs are real evidence that the constructions represent the claimed manifolds, and the small-cover classification in Lemma 6 is clean. I think the central claims are very likely correct.\n\nThe soft spot is exactly the one the stress-test flags: the Observation in Subsection 2.2 is unproved and load-bearing. It asserts that a 2-dipole removal followed by an induced 3-dipole removal preserves the manifold and yields a crystallization, and this composite move is used repeatedly in Theorems 4 and 7 to cut vertex counts from 192 to 120 and from 96 to 52. What is missing is a proof that after the first cancellation, the vertices forming the 3-dipole still satisfy the necessary conditions. If that condition fails in any of the configurations, the reduced graph might not be a gem of the intended manifold, or might not be contracted. The Regina signatures certify the final graphs but not the intermediate reductions. The cube-based construction of Γ′ for the 4-torus provides some independent support for Theorem 4, but Theorems 2 and 7 have no such fallback. Also, the bi-colored cycle counts are often asserted from figures; Theorem 7 even says that unproved statements follow from Figure 9. These are addressable gaps rather than fatal errors, but they are real and should be fixed before the paper is accepted.\n\nThe paper is written in the standard crystallization-theory style and the citation pattern is normal; self-citation to the authors' own prior construction is appropriate and not circular. The n-torus conjecture is clearly labeled as a conjecture, which is honest. The classification of the seven small covers is a nice byproduct, and the observation that the constructed crystallizations are weak semi-simple is a helpful connection to prior lower-bound results.\n\nThis deserves a serious referee. I would not desk-reject it, but I would ask the authors to prove the Observation in full generality, or at least verify its hypotheses case-by-case in the applications, and to provide more detailed cycle-count tables for the reduced graphs. With those additions it should be a solid contribution.","headline":"Solid new results on regular genus with a load-bearing unproved move; worth refereeing after the Observation in §2.2 is proved.","tokens_in":28858,"tokens_out":1308,"would_cite":true,"duration_ms":16389,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57Q15","57S25","52B11","52B70","05C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves exact regular genera for three families of 4-manifolds: $G(\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1)=6$, $G(\\mathbb{S}^1 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1)=16$, and $G=8$ for each of…","keywords":["$\\mathbb{Z}_2^n$-action","small cover","D-J equivalence","simple polytope","crystallization","regular genus","PL 4-manifold","torus"],"falsifier":"Take the intermediate graph produced after the first reduction in the 4-torus construction (the claimed 156-vertex gem) and compute its fundamental group; if it is not the free abelian group of rank 4, the unproved Observation fails and the 120-vertex graph would not be certified. Similarly, checking the 80-vertex and 64-vertex intermediate gems for each small cover against the known $\\mathbb{RP}^2$-bundle structure would settle whether the reduction to 52 vertices is valid.","tokens_in":27773,"feed_emoji":"🌀","tokens_out":8860,"duration_ms":81485,"temperature":0.7,"pith_summary":"Every closed PL 4-manifold can be encoded as an edge-colored graph called a crystallization, and the regular genus of the manifold is the smallest surface genus into which such a graph embeds regularly. This paper pins down that invariant for three families: it proves the regular genus of $\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1$ is 6, confirming a conjecture left open in the literature, and proves the regular genus of the 4-torus is exactly 16, closing a gap between previously known bounds. It then classifies the small covers over $\\Delta^2 \\times \\Delta^2$ up to D-J equivalence, showing there are exactly seven, and proves each has regular genus 8, including the product $\\mathbb{RP}^2 \\times \\mathbb{RP}^2$. The values are obtained through explicit vertex-minimal crystallizations with 40, 120, and 52 vertices, respectively, whose regular genera are computed from bi-colored cycle counts and matched against a known lower bound. The paper also proposes that the regular genus of the $n$-torus is $1+\\frac{(n+1)!(n-3)}{8}$ for $n \\ge 5$.","feed_headline":"Regular genus of the 4-torus is 16","feed_subtitle":"The same work fixes S²×S¹×S¹ at 6 and every Δ²×Δ² small cover at 8.","key_machinery":"The central objects are crystallizations: 5-regular edge-colored graphs on the color set $\\{0,1,2,3,4\\}$ dual to contracted triangulations of a 4-manifold. The regular genus of a crystallization is computed from the bi-colored cycle counts $g_{ij}$ through the formula $\\rho_\\varepsilon(\\Gamma) = 1 - \\tfrac{1}{2}\\big(-\\tfrac{3}{2}|V(\\Gamma)| + \\sum_i g_{\\varepsilon_i\\varepsilon_{i+1}}\\big)$ for a cyclic permutation $\\varepsilon$ of the five colors; minimizing over all $\\varepsilon$ and all crystallizations gives $G(M)$. The upper bounds come from explicit graphs, built from gems of $M\\times \\mathbb{S}^1$ using polyhedral glue moves and a two-stage dipole cancellation stated as an Observation in Section 2.2; the lower bound is the inequality $G(M) \\ge 2\\chi(M)+5m-4$, where $m = \\mathrm{rk}(\\pi_1(M))$.","core_discovery":"The paper's central claim is that the regular genus, a PL invariant measuring the minimal genus of a surface in which a manifold's crystallization embeds regularly, takes the values $G(\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1)=6$, $G(\\mathbb{S}^1 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1)=16$, and $G(M)=8$ for each of the seven small covers over $\\Delta^2 \\times \\Delta^2$. For each manifold it constructs a crystallization attaining the lower bound $2\\chi + 5m - 4$ of Proposition 1, where $m$ is the rank of the fundamental group, and then computes the regular genus of that crystallization by counting bi-colored cycles with respect to a chosen cyclic permutation of the five colors. The 40-vertex and 120-vertex graphs settle the previously open cases of $\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1$ and the 4-torus; the small-cover analysis classifies all $\\mathbb{Z}_2$-characteristic functions on $\\Delta^2 \\times \\Delta^2$ up to D-J equivalence, identifies one of the seven as $\\mathbb{RP}^2 \\times \\mathbb{RP}^2$ and the other six as $\\mathbb{RP}^2$-bundles over $\\mathbb{RP}^2$, and supplies a 52-vertex crystallization for each. Along the way the paper records that the seven covers collapse to at most four distinct PL homeomorphism types, with pairwise identical isomorphism signatures, and that all crystallizations constructed are weak semi-simple, a property equivalent to attaining the lower bound.","pith_inferences":["If the unproved Observation in Section 2.2 holds generally, the two-stage 2-dipole/3-dipole reduction is a reusable tool: other products $M \\times \\mathbb{S}^1$ with a matching base crystallization could receive the same vertex reduction, potentially yielding genus-minimal crystallizations for further 4-manifolds.","The pairwise agreement of isomorphism signatures shows that D-J equivalence is strictly finer than PL homeomorphism in dimension four; a natural next question, not answered here, is whether the four remaining classes are pairwise non-homeomorphic.","The $n$-torus conjecture is already testable at $n=5$: the candidate graph has 720 vertices and would have regular genus 361, so the bottleneck is finding a matching lower bound for manifolds whose fundamental group has rank greater than two.","Because weak semi-simplicity is equivalent to attaining the regular-genus lower bound and is closed under connected sum, every new weak semi-simple example enlarges a class whose additivity over connected sums would resolve the 4-dimensional smooth Poincar\\'e conjecture if it ever covered all simply-connected PL 4-manifolds."],"forward_implications":["The previously open conjecture that $\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1$ has regular genus 6 is settled, so this manifold joins the known orientable prime PL 4-manifolds of genus 6 alongside the mapping tori constructed in the literature.","The regular genus of the 4-torus is exactly 16, closing the interval between the earlier bounds of 4 and 28 and providing an explicit 120-vertex crystallization.","Every small cover over $\\Delta^2 \\times \\Delta^2$ has regular genus 8 and admits a 52-vertex genus-minimal crystallization; in particular $G(\\mathbb{RP}^2 \\times \\mathbb{RP}^2)=8$.","The seven D-J equivalence classes of small covers over $\\Delta^2 \\times \\Delta^2$ are at most four distinct PL homeomorphism types, with three pairs identified by identical isomorphism signatures.","All crystallizations constructed in the paper are weak semi-simple, so the class of closed PL 4-manifolds with known regular genus now includes these manifolds together with all of their connected sums.","The paper proposes that the regular genus of the $n$-torus is $1+\\frac{(n+1)!(n-3)}{8}$ for $n \\ge 5$, supported by the observed cycle structure of a candidate crystallization with $(n+1)!$ vertices."],"supporting_citations":[{"why":"Supplies Proposition 1, the lower bound $G(M) \\ge 2\\chi(M)+5m-4$ that every upper-bound construction in the paper is matched against.","marker":"[5]"},{"why":"Raises the conjecture that $G(\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1)=6$, which Theorem 2 resolves, and constructs the mapping tori with regular genus 6.","marker":"[2]"},{"why":"Introduces small covers and $\\mathbb{Z}_2$-characteristic functions, the classification framework used for $\\Delta^2 \\times \\Delta^2$.","marker":"[16]"},{"why":"Provides the gem construction over the $n$-simplex that the small-cover crystallizations are built from.","marker":"[1]"},{"why":"Defines the polyhedral glue moves and dipole cancellations used throughout the vertex-reduction sequences.","marker":"[18]"},{"why":"Extends the concept of genus to dimension $n$ and underlies the regular embedding and the bi-colored-cycle formula.","marker":"[22]"},{"why":"Characterizes regular genus zero as the sphere, fixing the base case of the invariant used in the paper's comparisons.","marker":"[19]"}],"fun_headline_variants":["Regular genus: T⁴=16, S²×S¹×S¹=6, small covers=8","All Δ²×Δ² small covers have regular genus 8","S²×S¹×S¹ genus 6, T⁴ genus 16, Δ²×Δ² covers 8","Regular genus numbers: T⁴=16, S²×S¹×S¹=6, small covers=8","All seven Δ²×Δ² small covers share regular genus 8"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on an unproved technical claim, stated as an Observation in Section 2.2, that certain graph-surgery moves used to shrink the crystallizations always preserve the underlying 4-manifold; if any of those moves goes wrong, the vertex reductions in the main theorems could produce graphs of the wrong manifold.","fun_headline_variants_meta":{"raw":{"variants":["Regular genus: T⁴=16, S²×S¹×S¹=6, small covers=8","All Δ²×Δ² small covers have regular genus 8","S²×S¹×S¹ genus 6, T⁴ genus 16, Δ²×Δ² covers 8","Regular genus numbers: T⁴=16, S²×S¹×S¹=6, small covers=8","All seven Δ²×Δ² small covers share regular genus 8"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001333,"raw_usage":{"total_tokens":5618,"prompt_tokens":1335,"completion_tokens":4283,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":951,"completion_tokens_details":{"reasoning_tokens":4154}},"tokens_in":951,"tokens_out":4283,"duration_ms":32056,"temperature":1.0,"reasoning_tokens":4154,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:45:07.417453+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the intermediate graph produced after the first reduction in the 4-torus construction (the claimed 156-vertex gem) and compute its fundamental group; if it is not the free abelian group of rank 4, the unproved Observation fails and the 120-vertex graph would not be certified. Similarly, checking the 80-vertex and 64-vertex intermediate gems for each small cover against the known $\\mathbb{RP}^2$-bundle structure would settle whether the reduction to 52 vertices is valid.","supporting_citations":[{"cited_title":"Basak and M","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 1, the lower bound $G(M) \\ge 2\\chi(M)+5m-4$ that every upper-bound construction in the paper is matched against."},{"cited_title":"Basak, Regular genus and gem-complexity of some mapping tori, Rev","cited_arxiv_id":null,"evidence_quote":"Raises the conjecture that $G(\\mathbb{S}^2 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1)=6$, which Theorem 2 resolves, and constructs the mapping tori with regular genus 6."},{"cited_title":"Davis and T","cited_arxiv_id":null,"evidence_quote":"Introduces small covers and $\\mathbb{Z}_2$-characteristic functions, the classification framework used for $\\Delta^2 \\times \\Delta^2$."},{"cited_title":"Agarwal and B","cited_arxiv_id":null,"evidence_quote":"Provides the gem construction over the $n$-simplex that the small-cover crystallizations are built from."},{"cited_title":"Ferri and C","cited_arxiv_id":null,"evidence_quote":"Defines the polyhedral glue moves and dipole cancellations used throughout the vertex-reduction sequences."},{"cited_title":"Gagliardi, Extending the concept of genus to dimension n, Proc","cited_arxiv_id":null,"evidence_quote":"Extends the concept of genus to dimension $n$ and underlies the regular embedding and the bi-colored-cycle formula."},{"cited_title":"Ferri and C","cited_arxiv_id":null,"evidence_quote":"Characterizes regular genus zero as the sphere, fixing the base case of the invariant used in the paper's comparisons."}],"review_version":1}