{"id":"bfd7d5b2-7646-451c-81a0-d4b62090dddf","arxiv_id":"1908.01660","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two structures with identical one-variable definable sets can differ in pseudo-o-minimality, refuting an axiomatization by one-variable conditions and a pigeonhole principle conjecture.","lead":"The paper builds two ordered structures on the same set with the same order and the same one-variable definable subsets, where one satisfies the common theory of o-minimal structures and the other does not. This resolves two questions by Schoutens about whether one-variable definability determines pseudo-o-minimality and whether definable completeness plus type completeness forces the pigeonhole principle.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.21's displayed equivalence is false for '<' and '=', so the quantifier-elimination proof for T2 is not established.","rationale":"The paper's central claim, Theorem 6.27, requires that M0 and M2 have the same one-variable definable subsets, which is Corollary 6.25 and is proved via quantifier elimination for T2. The reader identified the QE proof in Section 6 as the weakest assumption, and my check confirms that the proof has a concrete flaw. I focused on the rank-reduction induction rather than Lemma 6.12. Lemma 6.21, handling rank (0,k+1), contains a displayed equivalence that is false for '<' and '=': Lemma 6.14 gives a conjunctive characterization for '<', while (21) is its disjunctive negation; inside Z equality never holds while (21) can be true; outside Z f is the identity so '<' and '>' are false, but (21) is true. Since Lemma 6.23 depends on Lemma 6.21 for the induction step, Theorem 6.24 and Corollary 6.25 are unsupported as written. This is a specific, checkable error rather than a vague concern about length or presentation. It is probably repairable by splitting the three order predicates and correcting the outside-Z disjunct, so the reader's CONDITIONAL verdict remains appropriate; my finding does not move the verdict.","tokens_in":19519,"tokens_out":22508,"duration_ms":224927,"concrete_test":"Run the finite approximation M_N from Proposition 3.3 with N≥2. Instantiate Lemma 6.21 with k=1, ψ(x)=x, □='<', and x=c1=(0,0). In M2, f maps the first column to the last column, so f^2(c1)=(N-1,0)>c1; hence f^2(c1)<c1 is false. The right-hand side of (21) evaluates to true: ψ(x)∈Z and c1≤f^0(c1)≤c2. This disproves the displayed equivalence for '<'. Repeat for □='>' and '=' to confirm no single choice of □ makes (21) correct, and then verify that splitting into three cases (using Lemma 6.14 for '<', its negation for '>', and ψ∉Z for '=') repairs Lemma 6.21.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 6.21 is the rank-reduction step for x-corrected formulas of rank (0,k+1). Its displayed equivalence (21) cannot hold for every □ in {<,>,=}: for ψ(x) in Z, Lemma 6.14 says f^{k+1}ψ < ψ iff all f^iψ > c2, while the disjunction in (21) is the negation of that condition. So for '<' the equivalence is inverted. For '=', inside Z equality f^{k+1}(z)=z is impossible by Axiom 12, but (21) is true whenever some iterate lands in [c1,c2]; outside Z, f is the identity, so '<' and '>' are false, yet the disjunct ψ∉Z ∧ ψ=ψ makes (21) true for every □. Thus two of the three order predicates are handled by a false equivalence. Lemma 6.23 and Theorem 6.24 rely on Lemma 6.21, so quantifier elimination for T2 is unproved; Corollary 6.25 (one-variable definable sets in M2 are definable in M0), hence Theorems 6.26 and 6.27, lose their support as written. The error is localized and likely repairable, but the proof has a genuine gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs two ordered structures M and N in the same language L1, on the same universe, with the same order and the same one-variable definable subsets, such that M is pseudo-o-minimal (a model of the common theory of o-minimal structures) while N has a definable closed bounded discrete set Z and a definable injection Z → Z that is not surjective, hence N fails the pigeonhole principle. This is claimed to answer negatively Schoutens' questions about whether pseudo-o-minimality can be axiomatized by first-order conditions on one-variable formulas alone, whether definable completeness plus type completeness implies the pigeonhole principle, and to partially answer a question of Fornasiero. The construction proceeds by building a theory T0 with a discrete closed bounded set Z, adding a bijection f (theory T1), then modifying f to be injective but not surjective (structure M2), while proving quantifier elimination for T2 to show that every one-variable definable set in M2 is already definable in M0.","tokens_in":19760,"tokens_out":3912,"duration_ms":38760,"significance":"If the main theorem is established, it is a significant negative result: it refutes any axiomatization of pseudo-o-minimality by one-variable definability conditions, and even any second-order theory in the language LDef. The paper also gives a useful concrete counterexample to the discrete pigeonhole principle in a definably complete, type complete structure. The consistency proofs via finite approximations, the quantifier-elimination proof for T0, and the careful modification of f are valuable contributions, and the main construction is elegant. However, the central preservation theorem for one-variable definability rests on a long quantifier-elimination proof for T2, and that proof currently contains a gap; the contribution is therefore conditional on repair.","major_comments":[{"comment":"The displayed equivalence used to reduce rank (0,k+1) is false as stated. For ψ(x) ∈ Z, Lemma 6.14 gives f^{k+1}(z) < z ↔ ⋀_{i=0}^k (f^i(z) > c2), whereas the disjunction in (21), namely ⋁_{i=0}^k (c1 ≤ f^i(ψ(x)) ≤ c2), is the negation of that conjunction. Thus (21) is not equivalent to the '<' case. For '=', the left side f^{k+1}(z) = z is false for all z ∈ Z by Axiom 12, while the right side can be true depending on the iterates; for ψ(x) ∉ Z, f^{k+1}(x) = x, so the left side is false for '<' and '>' and true for '=', while the disjunct (ψ(x) ∉ Z ∧ ψ(x) = ψ(x)) makes the right side true for every □. Consequently the equivalence fails for all three order predicates in general. Since Lemma 6.23 and Theorem 6.24 invoke Lemma 6.21, the quantifier elimination for T2 and hence Corollary 6.25 are not established as written.","section":"§6, Lemma 6.21, Eq. (20)–(21)"},{"comment":"The same gap propagates to the g-case in Lemma 6.21, where the proof substitutes g^{k+1}(ψ(x)) into the false equivalence (20). Because (20) is invalid, the rank reduction for formulas of the form g^{k+1}(ψ(x)) □ ψ(x) is not justified. Since this is the only reduction step for rank (0,k+1), the induction in Lemma 6.23 is incomplete. The error appears localized and likely repairable by giving separate treatments of '<', '>', and '=', and by properly handling the cases ψ(x) ∈ Z and ψ(x) ∉ Z, but the current proof is not sound.","section":"§6, Lemma 6.21, case (2)"},{"comment":"The proof of Lemma 6.16 also depends on an unstated uniformity: the constant terms τ_i are chosen before fixing x, and the case split on ψ_1(x) and ψ_2(x) being equal to these constants is used to reduce all terms to a common rank. This is plausible, but when Lemma 6.21 is repaired, the interaction between these constant-term exceptions and the rank reduction must be re-checked, because the exceptional points are not themselves in Z and the order comparisons there require the 'outside Z' branch. The current text does not fully spell out this uniformity, and the gap in Lemma 6.21 makes it impossible to verify the induction.","section":"§6, Lemma 6.16 and Lemma 6.21"}],"minor_comments":[{"comment":"The text has several OCR-style artifacts: 'By /suppress Los’ Theorem' in the introduction, 'if Sn(x) =c4' with missing spacing, and 'rank(φ,x )< rank(ϕ,x )' with inconsistent spacing. These should be cleaned up.","section":"Throughout"},{"comment":"The displayed formula (20) writes f^k ◦ ψ(x) □ ψ(x) although the lemma concerns rank (0,k+1) and the subsequent text uses f^{k+1}; the index should be made consistent throughout.","section":"Lemma 6.21, equation (20)"},{"comment":"In the definition of rank, the cases for atomic formulas involving F(x) ∈ Z and F(x) □ τ use the same notation (−∞, deg(F)); this is fine, but it would help to explicitly say that the second coordinate is the degree of the outermost term, since the lexicographic ordering is then not immediately transparent from the notation.","section":"Definition 6.17"},{"comment":"The axiom for π is written '(∀x)(π(x) ∈ Z) ∧ (∀y ∈ Z (¬C<(x,y,π(x))))', but the variable x in the second conjunct is not quantified; it should presumably be '(∀x)(∀y ∈ Z)(¬C<(x,y,π(x)))'.","section":"§3, Axiom 7"}],"recommendation":"major_revision","confidential_remarks":"The central construction is interesting and, if the quantifier-elimination gap is repaired, would answer the stated questions. The error in Lemma 6.21 is concrete and load-bearing; it is not merely a presentation issue. I would not recommend rejection, because the gap is localized and the surrounding framework (T0 QE, consistency arguments, the construction of M2) is solid. However, the revised version must provide a correct rank-reduction argument for all three order predicates in Lemma 6.21, and must re-verify Lemma 6.23 and Corollary 6.25 afterward."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: if the Section 6 gap is fixed, this is a nice result. The construction of M2 is genuinely new: it shows that even naming all one-variable definable sets in a second-order way cannot axiomatize pseudo-o-minimality, which is strictly stronger than Rennet's non-axiomatizability result. The tail-shift modification of f is clever, and the paper is honest about what remains open (the real closed field case is still open).\n\nWhat is solid: the consistency proof by finite approximations is plausible, Lemma 6.1 rigorously establishes the failure of the pigeonhole principle in M2, and the T0 quantifier elimination looks essentially right. The citations to Schoutens, Novak, and Rennet are appropriate, and the paper does not force constants or normalize to get its answer.\n\nThe soft spot is Section 6. Lemma 6.21 is not correct as written. For rank (0,k+1), Lemma 6.14 gives f^{k+1}(z)<z iff all f^i(z)>c2, but (21) gives the disjunction of c1≤f^i(z)≤c2, which is the negation of that condition. So the displayed equivalence fails for '<'. It also fails for '=' inside Z, where iterates cannot equal the starting point. For '>' the disjunction happens to match 'not <', but that does not save the lemma. Lemma 6.23 and Theorem 6.24 rely on Lemma 6.21, so Corollary 6.25 and Theorems 6.26–6.27 are unsupported as written.\n\nI want to keep this in proportion: the error looks localized and repairable. Replace (21) with the correct conjunction for '<', handle '=' separately, and let '>' be the negation of '<' or '='. The induction strategy still seems viable. A smaller gap is that Proposition 3.3 asserts the finite models are o-minimal without spelling out the verification; that is probably routine but should be written down.\n\nWho this is for: model theorists working on o-minimality, definable completeness, and tame structures. It deserves a serious referee despite the gap. I would not cite the main theorem until the QE proof is repaired, but I would encourage revision rather than rejection.","headline":"A genuinely stronger counterexample for pseudo-o-minimality, but the QE proof for T2 has a concrete error in Lemma 6.21 that needs repair before the main theorem is established.","tokens_in":20242,"tokens_out":9639,"would_cite":false,"duration_ms":90345,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03C64"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two ordered structures with the same order and the same one-variable definable subsets can still differ on the common theory of o-minimality, because a definable injective non-surjective shift can be hidden on a discrete set.","keywords":["pseudo-o-minimal","pseudo-finite sets","definably complete","type complete","pigeonhole principle","o-minimalism","quantifier elimination","cyclic order"],"falsifier":"Work in a model $M_0$ of $T_0$ and take any term $F$ built from $S$, $P$, and $\\pi$. Lemma 6.12 says there are finitely many constants $\\tau_1,\\dots,\\tau_k$ such that for every $x$ outside those constants, only finitely many elements of $Z$ lie between $x$ and $F(x)$. Finding one such $F$ and one input $x$ outside the constant exceptions with infinitely many $Z$-elements between $x$ and $F(x)$ would break the rank reduction and with it Corollary 6.25. More directly, exhibiting any one-variable $L_1$-definable subset of $M_2$ that is not definable in $M_0$ would refute the theorem.","tokens_in":19320,"feed_emoji":"🔄","tokens_out":10833,"duration_ms":112679,"temperature":0.7,"pith_summary":"The paper establishes that the common first-order theory of o-minimal structures cannot be recognized from one-variable definable sets alone. It constructs two ordered structures $M$ and $N$ on the same universe, with the same order and the same one-variable definable subsets, where $M$ is pseudo-o-minimal and $N$ is not: $N$ has a definable closed, bounded, discrete set $Z$ and a definable injective map $f:Z\\to Z$ that is not surjective. This gives negative answers to the question whether pseudo-o-minimality is axiomatizable by first-order conditions on one-variable formulas, and to the question whether definable completeness plus type completeness forces the pigeonhole principle. It matters because it delimits what one-variable definability can certify about an ordered structure and pinpoints the real-closed-field question as genuinely algebraic.","feed_headline":"One-variable definable sets can't tell o-minimality apart","feed_subtitle":"A cyclic shift on a discrete definable set breaks pigeonhole without altering any one-variable definable set.","key_machinery":"The machine is the cyclic-order structure on the distinguished set $Z$, with successor $S$, predecessor $P$, and circular projection $\\pi$, together with a shift $f$ that cyclically permutes two halves of $Z$. In $M_2$ the shift is changed on a tail so that $f(Z)=Z\\setminus\\{P(c_3)\\}$; nevertheless $f$ commutes with $S$ up to finitely many exceptions, so the semigroup generated by $f,g,S,P,\\pi$ is Abelian up to those exceptions. Quantifier elimination is then forced by a two-coordinate rank: terms with high $f,g$-degree are replaced by corrected terms of lower rank using one-variable comparisons on $Z$, and atomic formulas are eventually rewritten as comparisons of cyclic terms that involve no $f$ or $g$. The outcome is Corollary 6.25: the one-variable definable sets of $M_2$ coincide with those of $M_0$, while $f\\upharpoonright Z$ witnesses the failure of pigeonhole.","core_discovery":"The central discovery, stated as Theorem 6.27, is a pair of ordered structures in one language that are indistinguishable by order and by one-variable definability but differ in pseudo-o-minimality. The pseudo-o-minimal member is a model $M_0$ of a theory $T_0$ that contains the common theory of o-minimal structures and carries a closed, bounded, discrete set $Z$ equipped with a cyclic successor structure. The second member $M_2$ is obtained by modifying a bijective cyclic shift $f$ on $Z$ so that $f$ becomes injective but omits one point; then $Z$ is a pseudo-finite set with a definable injective non-surjective self-map, so the pigeonhole principle fails, and $M_2$ is not pseudo-o-minimal. The proof that the two structures nevertheless have the same one-variable definable sets runs through a quantifier-elimination theorem for $M_2$: every $L_1$-definable subset in one free variable reduces, outside finitely many constants, to an $L_0$-definable set. Consequently no first-order scheme over one-variable formulas, and no second-order theory over the one-variable definable subsets, can axiomatize pseudo-o-minimality.","pith_inferences":["A natural extension, not pursued in the paper, would be to preserve definable subsets in any fixed finite number of variables while still changing pigeonhole behavior; that would show no fixed arity of definable-set data can axiomatize pseudo-o-minimality.","The construction suggests a general coding phenomenon: on a definably complete structure with a definable discrete set, an injective non-surjective definable shift can be rendered invisible to one-variable definability by cyclic-order bookkeeping.","If the real-closed-field conjecture is true, its proof must use field-specific properties such as multiplication, because the construction removes the field operations and immediately produces a definably complete, type complete counterexample."],"forward_implications":["No first-order scheme ranging over all one-variable formulas can axiomatize the common theory of o-minimal structures; any axiom system for pseudo-o-minimality must use several variables or higher-order data.","Definable completeness together with type completeness does not imply the pigeonhole principle: the class contains a pseudo-finite set with a definable injective non-surjective self-map.","There is no second-order axiomatization of pseudo-o-minimality in the language $\\{<,\\mathrm{Def}\\}$ where $\\mathrm{Def}$ names the one-variable definable sets.","The equivalence between pseudo-finite and pseudo-o-finite sets fails in the definably complete, type complete setting: a pseudo-finite set can behave unlike a finite set once a predicate for it is added.","The conjecture for definably complete expansions of real closed fields remains open, but the obstruction cannot be purely order-theoretic, since the ordered counterexample here refutes the analogous statement without field operations."],"supporting_citations":[{"why":"Supplies the definitions of pseudo-o-minimality, type completeness, pseudo-o-finite sets, and the pigeonhole principle, along with the two questions answered negatively here.","marker":"[Sch14]"},{"why":"Introduces pseudo-finite sets and poses the conjecture about definably complete real closed fields and the pigeonhole principle.","marker":"[For10]"},{"why":"Continues the study of pseudo-finite sets and locally o-minimal structures that motivates the terminology and the conjecture.","marker":"[For13]"},{"why":"Gives the cut characterization of cyclic orders used to prove that the shift $f$ preserves the cyclic order on $Z$, a step underpinning the commutation machinery.","marker":"[Nov84]"}],"fun_headline_variants":["Same one-variable sets, different pigeonhole principle","Two structures, same one-variable sets, different pigeonhole","One-variable definable sets cannot detect pigeonhole failure","Pseudo-o-minimality not determined by one-variable definability","Cyclic shift on a discrete set hides from one-variable view"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the claim that every cyclic-order term moves any input past only finitely many points of $Z$, except at finitely many exceptional constants; if that claim fails, the mutated structure may acquire new one-variable definable sets, and the counterexample evaporates.","fun_headline_variants_meta":{"raw":{"variants":["Same one-variable sets, different pigeonhole principle","Two structures, same one-variable sets, different pigeonhole","One-variable definable sets cannot detect pigeonhole failure","Pseudo-o-minimality not determined by one-variable definability","Cyclic shift on a discrete set hides from one-variable view"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000982,"raw_usage":{"total_tokens":4174,"prompt_tokens":957,"completion_tokens":3217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":3131}},"tokens_in":573,"tokens_out":3217,"duration_ms":22315,"temperature":1.0,"reasoning_tokens":3131,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:07:52.386575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work in a model $M_0$ of $T_0$ and take any term $F$ built from $S$, $P$, and $\\pi$. Lemma 6.12 says there are finitely many constants $\\tau_1,\\dots,\\tau_k$ such that for every $x$ outside those constants, only finitely many elements of $Z$ lie between $x$ and $F(x)$. Finding one such $F$ and one input $x$ outside the constant exceptions with infinitely many $Z$-elements between $x$ and $F(x)$ would break the rank reduction and with it Corollary 6.25. More directly, exhibiting any one-variable $L_1$-definable subset of $M_2$ that is not definable in $M_0$ would refute the theorem.","supporting_citations":[],"review_version":1}