{"id":"ef8fbd89-4560-48e2-b00d-41d3dd7f1a6e","arxiv_id":"2506.18573","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For any group, the space of Conradian left-preorders is either finite or uncountable, and finiteness is characterized by the existence of a unique finite rational series without abelian jumps.","lead":"This paper defines Conradian left-preorders, a broad class of group relations, and proves the space of such preorders is either finite or uncountable, with exact counts in the finite case. A generalist might read it because it extends a central structural result in ordered group theory and connects to no-crossing actions and algebraic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main uniqueness proof relies on Proposition 5.9, which is false as stated: Z^2 admits two distinct maximal rational series.","rationale":"The reader's verdict is CONDITIONAL with moderate confidence, focusing on Proposition 4.5. The most load-bearing defect I find is different: the proof of the main theorem's uniqueness equivalence invokes Proposition 5.9, which is false without an extra hypothesis. This is not an objection to the theorem's plausibility; the intended application of Proposition 5.9 occurs in a context where no-abelian-jump and CO(G0)=∅ hold, and those hypotheses may make uniqueness true via Proposition 4.12. But as written, the argument contains a false statement and a non-applicable citation of Theorem 4.10. This supports the CONDITIONAL verdict: accept only after the lemma is corrected or the proof of Theorem 5.10 bypasses it. I do not see a fatal flaw in the main dichotomy.","tokens_in":39590,"tokens_out":19544,"duration_ms":175988,"concrete_test":"Settle the concern by checking Definition 4.1 against G=Z^2: verify that {0}≤⟨e1⟩≤Z^2 and {0}≤⟨e1+e2⟩≤Z^2 are both rational (rank-1 abelian quotients), that no rational series of length 3 exists (a length-3 series would require a strictly increasing chain of rank-1 subgroups of Z^2), and that the two length-2 series are distinct. If so, Proposition 5.9 fails exactly as stated, and the proof of Theorem 5.10 must be revised to add the no-abelian-jump hypothesis.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 5.10 uses Proposition 5.9 to prove the uniqueness clauses (b)↔(c). But Proposition 5.9 claims uniqueness of a maximal-length rational series without any no-abelian-jump hypothesis. That is false: for G=Z^2, both {0}≤⟨e1⟩≤Z^2 and {0}≤⟨e1+e2⟩≤Z^2 are rational series (each quotient is Z) of length 2, the maximum possible because a rational series of length 3 would force a strictly increasing chain of rank-1 subgroups of Z^2, which cannot exist. Hence uniqueness fails in general. The step 'COG0(G) is finite by Theorem 4.10' in the proof of Proposition 5.9 needs the extra hypothesis that the maximal series has no abelian jumps; without it, Theorem 4.10 does not apply. The central claim may still be true, but the proof as written depends on a false lemma and must be repaired, e.g. by using Lemma 5.6 and Proposition 4.12 under the no-abelian-jump hypotheses actually present in Theorem 5.10.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines Conradian left-preorders (total left-invariant preorders satisfying a Conrad-type condition) and studies the space CO(G) of such preorders on a group G. The main results are: a dichotomy theorem (CO(G) is either finite or uncountable, Corollary 5.13); a characterization of finiteness of CO(G) in terms of existence and uniqueness of a maximal-length rational series with no abelian jumps, together with the explicit cardinality formula |CO(G)| = 2^{n+1}-2 (Theorem 5.10); a relative version for CO_C(G) (Theorem 4.13); and characterizations of Conradian left-preorders via convex jumps, the inequality α ≺ βα², and a no-crossing dynamical condition (Theorems 2.10 and 6.2). The proofs follow and extend the strategy of Rivas for Conradian left-orders.","tokens_in":39741,"tokens_out":9370,"duration_ms":77429,"significance":"If the central results hold, the paper is a substantive contribution to the theory of left-preorders and Conradian orderability. The finiteness criterion and the dichotomy finite/uncountable are new for Conradian left-preorders, and the cardinality formula generalizes Rivas's results for Conradian left-orders. The paper also provides useful characterizations (convex jumps, n=2 condition, dynamical no-crossing) that are natural extensions of known results. The exposition is generally clear and the proofs are detailed; however, the current proof contains a false lemma (Proposition 5.9) that is load-bearing for the main theorem, so the manuscript requires revision before the central claims can be accepted.","major_comments":[{"comment":"Proposition 5.9 is false as stated. For G = Z², the series {0} ≤ ⟨e1⟩ ≤ Z² and {0} ≤ ⟨e1+e2⟩ ≤ Z² are both rational, both have maximal length 2, and are distinct, so uniqueness of a maximal-length rational series fails in general. The proof's step that \"CO_{G0}(G) is finite by Theorem 4.10\" is invalid because Theorem 4.10 requires the rational series to have no abelian jumps, and a maximal-length series need not satisfy this; indeed, in the counterexample the series has an abelian jump and CO_{G0}(Z²) = CO_{0}(Z²) is infinite. Since Proposition 5.9 is invoked in the proof of Theorem 5.10 to obtain the equivalence (b) ⇔ (c), the proof of the main theorem is currently incomplete. The statement can likely be repaired by proving uniqueness only for maximal rational series satisfying the additional hypotheses that appear in condition (b) (no abelian jumps and CO(G0)=∅), using Lemma 5.6 and Proposition 4.12, but as written the paper contains a false result that must be corrected.","section":"Proposition 5.9 and its use in Theorem 5.10"}],"minor_comments":[{"comment":"The displayed implication \"αC⪯βC =⇒ αK ˆ⪯βK\" is confusing and appears to be vacuous for α,β ∈ K, since then αK = βK; the intended statement likely concerns α,β ∈ L, and the quantifiers should be clarified.","section":"Remark 1.17"},{"comment":"In the proof of Lemma 2.6, the Conradian condition is applied to α^m with m = 0, where α^m = 1 is not strictly positive; the case m = 0 should be handled separately or the quantifier restricted to m > 0 to make the argument fully rigorous.","section":"Lemma 2.6"},{"comment":"The sentence \"the restriction of ⪯ on G0 give us a Conradian left-preorder on G relative to C0\" is garbled; it should state that the restriction gives a Conradian left-preorder on G0 relative to C0.","section":"Lemma 5.6"},{"comment":"In the statement of condition (d), the phrase \"There exists a finite rational series finite rational series starting on G\" contains a duplicated phrase that should be removed.","section":"Theorem 5.10, condition (d)"},{"comment":"The assertions that every left-preorder on T_n has stabilizer of the form ⟨a1,...,a_k⟩ and that the number of left-preorders is at most 2^{n-k} are stated without proof; since this example is used to identify all left-preorders, a proof or a precise reference to [8] would be helpful.","section":"Example 5.17"}],"recommendation":"major_revision","confidential_remarks":"The false Proposition 5.9 appears to be a localized error: the intended uniqueness statement for maximal rational series with no abelian jumps is plausibly correct and can be proved by combining Lemma 5.6 with Proposition 4.12. The rest of the paper, including Theorem 5.7 and the relative finiteness theorem, seems sound based on my reading. I recommend major revision so the author can correct Proposition 5.9 and revise the proof of Theorem 5.10 accordingly; rejection is not warranted given the likely fixability of the issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper defines Conradian left-preorders and proves a finite-vs-uncountable dichotomy for the space CO(G), with cardinality 2^{n+1}-2 in the finite case. That is a genuine extension of Rivas's results for left-orders, and the main theorem (5.10) looks right. But there is a real gap in Proposition 5.9 as stated: uniqueness of maximal-length rational series is false without the no-abelian-jump hypothesis.\n\nWhat's new and good: the notion of Conradian left-preorder is natural, and Theorem 2.10 giving equivalent characterizations (convex jumps, n=2 inequality, no-crossing dynamics) is solid. The proof structure follows Rivas closely but does real extra work for the preorder case. Proposition 4.6 and the counting argument are clean. The no-crossing theorem (6.2) is a nice bonus. The citation pattern is healthy: it builds openly on Rivas, Sikora, and Navas-Rivas.\n\nWhere it gets soft: Proposition 5.9 claims uniqueness of any maximal-length rational series. The stress-test's Z^2 example is correct: {0}≤⟨e1⟩≤Z^2 and {0}≤⟨e1+e2⟩≤Z^2 are both maximal-length rational series. The proof's line 'COG0(G) is finite by Theorem 4.10' does not follow from maximality alone; Theorem 4.10 requires a rational series without abelian jumps. So Proposition 5.9 is false as written. That said, in Theorem 5.10 the hypotheses include no abelian jumps, so the application can be repaired: under (b) one gets COG0(G) finite by Theorem 4.10, then Proposition 4.12 gives uniqueness. The main dichotomy and cardinality formulas should survive. Minor issues the reader flagged are real but small: a quantifier slip in Remark 1.17, a garbled sentence in Lemma 5.6, unproven claims in Example 5.17, and the m=0 edge in Lemma 2.6.\n\nWho is this for? People working on ordered groups and spaces of orderings. It deserves a serious referee: the core idea is good, the exposition is mostly careful, and the flaw is localized and fixable. I would send it out with a request to correct Proposition 5.9 (state and prove it under the no-abelian-jump hypothesis) and clean up the examples.","headline":"Genuine generalization of Rivas's theorem with a real but repairable flaw in Proposition 5.9; the main dichotomy and cardinality formulas survive.","tokens_in":40346,"tokens_out":4609,"would_cite":true,"duration_ms":43307,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06F15","20F60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Conradian left-preorder spaces are either finite or uncountable.","keywords":["Conradian left-preorders","space of left-preorders","rational series","convex subgroups","left-orderable groups","finite or uncountable dichotomy","no-abelian-jumps"],"falsifier":"Enumerate all Conradian left-preorders on the Klein-bottle group K=<a,b | $aba^{{-1}}$=$b^{{-1}}$>. The theorem predicts exactly 6, or $2^{3}$-2, with binary sign choices for the generators; finding any additional Conradian left-preorder, or showing that one of the six positive cones listed in the paper is not actually convex, would refute the counting part of the main theorem.","tokens_in":39340,"feed_emoji":"⚖️","tokens_out":6161,"duration_ms":58375,"temperature":0.7,"pith_summary":"This paper defines Conradian left-preorders, a weakening of Conradian left-orders where the strict cone can have a nontrivial kernel, and studies the space CO(G) of all such structures on a group G. Its main result is a dichotomy and a finiteness criterion: either CO(G) is finite, or it is uncountable; finiteness happens exactly when there is a unique finite rational series through the group, starting at the minimal subgroup C0 that supports relative Conradian preorders, with no abelian jumps at any two consecutive steps. In the finite case the count is forced to be 2^(n+1)-2, where n is the length of that series, and the space of relative Conradian preorders on C0 contributes exactly 2^n of those. The paper also characterizes Conradian left-preorders by convex-jump archimedean embeddings, by an inequality with exponent 2, and by a no-crossing dynamical condition on the action on cosets.","feed_headline":"Conradian preorders: finite or uncountable, never countable","feed_subtitle":"If finite, a unique rational series fixes the count at 2^(n+1)-2; otherwise the space of preorders is uncountable.","key_machinery":"The load-bearing objects are Conradian left-preorders: total, transitive, reflexive, left-invariant relations on G whose strict cone is such that for any two positive elements α and β some power α^n exceeds β. A rational series is a finite normal chain G0⊴...⊴Gn=G with each quotient torsion-free abelian of rank 1; an abelian jump is a two-step segment Gi⊴Gi+1⊴Gi+2 whose outer quotient Gi+2/Gi is abelian. The proof mechanism is the pair of mutually inverse gluing and splitting maps µ and ρ, which decompose a Conradian preorder on G into relative preorders on the quotient steps of a convex chain; the key theorem is that every term of a rational series without abelian jumps is convex in every Conradian left-preorder. That convexity theorem forces the finite case to look like a binary tree of choices, one sign per step, giving 2^n relative preorders and 2^(n+1)-2 global ones.","core_discovery":"The central claim is that the Conradian left-preorder space is a two-tier object. For any group with at least one Conradian left-preorder, define C0 as the intersection of all subgroups that support such a preorder; then CO(G) is finite if and only if there is a unique finite rational series C0=G0⊴...⊴Gn=G in which each successive quotient is torsion-free abelian of rank 1 and no two consecutive steps compose to an abelian quotient. When this happens, |CO(G)|=2^(n+1)-2 and |CO_C0(G)|=2^n; when it does not, CO(G) is uncountable. The proof builds a bijection between Conradian preorders respecting a convex chain and tuples of relative preorders on successive quotients, and shows that failure of the no-abelian-jumps condition leaves the space without isolated points, forcing uncountability (and, in the countable case, a Cantor space).","pith_inferences":["Because the finiteness criterion is phrased in terms of a unique rational series, finite CO(G) forces a rigid normal filtration; one natural test is whether any group with finite CO(G) must be poly-(torsion-free abelian of rank 1) with no abelian two-step quotients.","The dichotomy may extend beyond Conradian preorders: for any reasonable class of left-preorders defined by forbidding a localized dynamical pattern, a failure of finiteness should again produce uncountably many elements, mirroring Cantor-set phenomena in spaces of left-orders.","The cardinality formula suggests a direct enumerative algorithm for finitely presented groups: search normal chains, check the no-abelian-jumps condition, then count sign choices per step; verifying this on small examples such as the Klein-bottle group is a cheap computational check.","The no-crossing characterization may connect to harmonic actions on the line: crossing-free actions on coset spaces could supply a dynamical proof of local-indicability-type statements for Conradian preorders."],"forward_implications":["If CO(G) is infinite, it cannot be countably infinite: it is uncountable, and for countable G it is a Cantor set when it has no isolated points.","Finiteness of CO(G) is equivalent to all its elements being isolated, and also to the existence of a unique maximal-length rational series with no abelian jumps ending at C0.","If G admits a Conradian left-order, then CO(G) is finite exactly when there is a rational series from {1} to G with no abelian jumps, and in that case the number of Conradian left-preorders is 2^(n+1)-2 while the number of Conradian left-orders is 2^n.","For the groups T_n generated by the relations a_{i+1} a_i a_{i+1}^{-1} = a_i^{-1}, every left-preorder is Conradian and there are exactly 2^(n+1)-2 of them.","Conradian left-preorders admit a dynamical characterization: the left action on G/C has no crossing, which allows the Conradian property to be detected through an action rather than through cone inequalities alone."],"supporting_citations":[{"why":"Supplies Holder's archimedean embedding theorem and the classical Conradian left-order framework used in the convex-jump characterization.","marker":"[4]"},{"why":"Introduces Conrad's original right-ordered group notion whose left-order version is generalized here to left-preorders.","marker":"[5]"},{"why":"Provides the space-of-Conradian-left-orders blueprint and the T_n examples whose preorder counts are computed in the paper.","marker":"[17]"},{"why":"Proves that n=2 suffices in the Conradian condition for left-orders, a fact generalized to preorders in Theorem 2.10.","marker":"[12]"},{"why":"Gives the dynamical no-crossing characterization of Conradian left-orders that the paper adapts to actions on cosets.","marker":"[16]"},{"why":"Defines the space of relative orders and supplies topological properties of left-preorder spaces used as a baseline.","marker":"[1]"},{"why":"Introduces the product topology on spaces of orderings, used to define the space of Conradian left-preorders and its compactness.","marker":"[18]"},{"why":"Documents the groups T_n and their left-order structure, which the paper uses to compute the full space of left-preorders.","marker":"[8]"}],"fun_headline_variants":["Conradian preorder spaces: never countably infinite","Finite or uncountable: the only sizes for Conradian preorder spaces","Conradian preorders: finite or uncountable, nothing in between","No countable infinity in Conradian preorder spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The finiteness criterion rests on Proposition 4.5, which asserts that in a rational series without abelian jumps every intermediate subgroup is convex in every Conradian left-preorder; if the lemma supplying a non-commuting pair across adjacent quotients or the archimedean embedding of each convex jump fails, the formula |CO(G)| = 2^(n+1)-2 no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Conradian preorder spaces: never countably infinite","Finite or uncountable: the only sizes for Conradian preorder spaces","Conradian preorders: finite or uncountable, nothing in between","No countable infinity in Conradian preorder spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001185,"raw_usage":{"total_tokens":4812,"prompt_tokens":781,"completion_tokens":4031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":397,"completion_tokens_details":{"reasoning_tokens":3959}},"tokens_in":397,"tokens_out":4031,"duration_ms":28708,"temperature":1.0,"reasoning_tokens":3959,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:49:44.401748+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all Conradian left-preorders on the Klein-bottle group K=<a,b | $aba^{{-1}}$=$b^{{-1}}$>. The theorem predicts exactly 6, or $2^{3}$-2, with binary sign choices for the generators; finding any additional Conradian left-preorder, or showing that one of the six positive cones listed in the paper is not actually convex, would refute the counting part of the main theorem.","supporting_citations":[{"cited_title":"Grupos ordenables: estructura algebraica y din´ amica","cited_arxiv_id":null,"evidence_quote":"Proves that n=2 suffices in the Conradian condition for left-orders, a fact generalized to preorders in Theorem 2.10."},{"cited_title":"& Rivas, C","cited_arxiv_id":null,"evidence_quote":"Gives the dynamical no-crossing characterization of Conradian left-orders that the paper adapts to actions on cosets."}],"review_version":2}