{"id":"90456228-0a94-48b2-b9d3-12e394c76c4b","arxiv_id":"1908.06542","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For complex affine Poisson algebras, the Poisson Dixmier-Moeglin equivalence holds exactly when |C|-separability of Poisson prime ideals forces finite separability, and exactly when all symplectic cores are locally closed.","lead":"This paper gives new topological tests for the Poisson Dixmier-Moeglin equivalence: the equivalence holds precisely when every Poisson prime ideal that is covered by fewer than continuum many minimal primes is actually covered by finitely many, and precisely when all symplectic cores are locally closed. The results generalize earlier model-theoretic and torus-action theorems and apply to commutative differential algebras over large fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.3(iii) is false and Theorem 6.7(ii) is false as stated; the core criterion needs repair. For A=C[x,y] with {x,y}=x, DME holds but the primitive core of 0 is open and not equal to V(0)∩max.","rationale":"The reader identified Lemma 6.5 and the equality C(m)={Q | P(m)⊆Q} as load-bearing. My analysis shows the concern is real and concrete: the equality is not true in general, and it is not actually what Brown-Gordon's closure statement provides. The failure is not merely an external citation risk; it produces a false Proposition 5.3(iii) and a false Theorem 6.7(ii). The central criterion that DME is equivalent to local closedness of symplectic cores in the maximal spectrum may still be correct, since a repair using \\overline{C(m)}=V(P(m))∩max and the Jacobson property appears to work. Therefore the paper needs a substantive revision: correct Proposition 5.3 by removing or replacing the equality, restrict or delete Theorem 6.7(ii), and rework the proof of Theorem 6.7(iii) through closure equality instead of actual equality. Because the main constructive claims are plausible and the flaw is localized and fixable, a conditional acceptance is still the appropriate verdict rather than rejection. I disagree in part with the reader's framing only in that the weak point is not an unverified external lemma but an internal misstatement that a simple example exposes.","tokens_in":24297,"tokens_out":56948,"duration_ms":537727,"concrete_test":"Explicitly compute the counterexample A=C[x,y], {x,y}=x. Verify that the Poisson prime ideals are 0, (x), and the maximal ideals (x,y−c); that the primitive and rational ideals are {0}∪{(x,y−c)}; that 0 is open in P.spec; and that C(0)={maximal ideals with x−a, a≠0} is locally closed in max A but not equal to {Q∈max A | 0⊆Q}. This refutes Proposition 5.3(iii) while DME holds. Then compute the zero bracket on C[x,y] and check Cspec(0)={(0)} is not locally closed in spec C[x,y], refuting Theorem 6.7(ii). These two checks settle the concern without relying on unverified external assumptions.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's symplectic-core characterization depends on Proposition 5.3, but condition (iii) of that proposition is false. In the proof of (i)⇒(iii), the authors prove that the intersection of the maximal ideals in C(P) equals P and then infer C(P)={Q∈max R | P⊆Q}. That inference requires C(P) to be closed; local closedness plus Jacobson does not force it. Concrete counterexample: let A=C[x,y] with Poisson bracket {x,y}=x, and take P=0. The Poisson prime spectrum is {0} ∪ {(x)} ∪ {(x,y−c): c∈C}; 0 is open, the points (x,y−c) are closed, and (x) is neither open nor closed. Thus the locally closed Poisson primes are exactly {0} and the points (x,y−c). The Poisson primitive ideals are also exactly these: the core of a generic maximal ideal (x−a,y−b), a≠0, is 0, while the cores of the singular maximal ideals (x,y−c) are themselves. The rational Poisson primes are the same set, so DME holds. However, for P=0, C(0) is the open set {maximal ideals with a≠0}, while {Q∈max A | 0⊆Q}=max A. Hence equality fails, so Proposition 5.3(iii) is false. The application to Poisson algebras claims the equality is supplied by Lemma 6.5(i)-(ii), but the Brown-Gordon lemma at most gives closure equality \\overline{C(m)}={Q | P(m)⊆Q}; in this example the closure of C(0) is max A while C(0) itself is a proper open set. A repaired proof can use local closedness together with closure equality, but the paper's stated argument does not go through. Separately, Theorem 6.7(ii) quantifies over all prime ideals and is false even for the zero bracket on C[x,y]: Cspec(0)={(0)} is not locally closed in spec C[x,y] although DME holds. The theorem should restrict (ii) to Poisson primitive ideals or delete it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a topological criterion for the Poisson Dixmier-Moeglin (DM) equivalence. It introduces κ-separability for points of a Zariski space and of a poset, and proves for noetherian commutative differential algebras over a large base field k that Δ-primitive, Δ-rational and |k|-separable Δ-prime ideals coincide (Theorem 4.5), so that the Δ-DM equivalence is equivalent to the statement that |k|-separability implies ℵ0-separability in Δ-spec R (Theorem 4.6). It then applies this to complex affine Poisson algebras, obtaining the poset criterion (Theorem 6.2), a local detection criterion (Theorem 6.3), and a claimed characterization in terms of local closedness of symplectic leaf/core strata (Theorems 6.7 and 6.10). The abstract also promises a generalization of a weaker DM equivalence from Bell–Launois–Sánchez–Moosa to arbitrary commutative differential algebras.","tokens_in":24687,"tokens_out":16331,"duration_ms":162231,"significance":"Theorems 4.5, 4.6 and 6.2 are attractive and, if correct, constitute a genuinely parameter-free topological characterization of the Poisson Dixmier-Moeglin equivalence: no group action and no finite-strata assumption is needed. The treatment of Δ-cores is detailed, the relevant external results are cited, and the main chain from Section 4 appears to be proved step-by-step. However, the symplectic-core section contains a false proposition and a mis-stated theorem, so the paper needs substantial revision before the core-stratification claims (Theorem C of the introduction) can be accepted. The counterexamples below show that the errors are not merely cosmetic: they affect the exact content of Proposition 5.3 and Theorem 6.7.","major_comments":[{"comment":"The implication (i)⇒(iii) is false as stated. After proving ∩_{M∈C(P)} M = P, the proof writes C(P) = {Q ∈ max R | (∩_{M∈C(P)} M) ⊆ Q} = {Q ∈ max R | P ⊆ Q}. The first equality describes the Zariski closure of C(P), not C(P) itself, and a locally closed set need not equal its closure. Concretely, let A = C[x,y] with Poisson bracket {x,y} = x and take P = 0. The Poisson prime spectrum is {0} ∪ {(x)} ∪ {(x,y−c) : c ∈ C}; the locally closed, primitive and rational Poisson primes are exactly {0} and the points (x,y−c), so the Poisson DM equivalence holds. But C(0) = {maximal ideals (x−a, y−b) with a ≠ 0}, which is a proper open dense subset of max A, whereas {Q ∈ max A | 0 ⊆ Q} = max A. Hence the equality in condition (iii) fails. In the Poisson application, Lemma 6.5 supplies only closure equalities, namely \\overline{L(m)} = {q | P(m) ⊆ q} and \\overline{L(m)} = \\overline{C(m)}; it does not supply equality of C(m) with that closed set. The proof of Theorem 6.7 must therefore be revised: the correct input is local closedness together with closure equality, not the stronger set equality used in the paper.","section":"§5, Proposition 5.3, proof of (i)⇒(iii)"},{"comment":"The statement 'Cspec(p) is locally closed in spec A for any p ∈ spec A' is false. In the same example A = C[x,y] with {x,y} = x, take p = (x). The only prime ideal whose Poisson core is (x) is (x) itself, so Cspec((x)) = {(x)}. But (x) is the generic point of the irreducible curve V(x) and is not locally closed in spec C[x,y] (it is not open in its closure, since V(x) has infinitely many closed points). Since the Poisson DM equivalence holds in this example, (i) does not imply (ii) as stated. The intended statement is presumably Proposition 5.3(ii), namely local closedness of Cspec(P) for P ∈ P.prim A, or equivalently for primes p with Poisson primitive core. The theorem statement and its proof need to be corrected accordingly.","section":"§6, Theorem 6.7(ii)"}],"minor_comments":[{"comment":"There is a typo: 'Every every open subset' should read 'Every open subset'.","section":"§2, first paragraph"},{"comment":"The displayed expression C(P) \\setminus \\overline{C(P)} should read \\overline{C(P)} \\setminus C(P); as written it describes the empty set and makes the subsequent argument unreadable.","section":"§5, proof of Proposition 5.3(iii)⇒(i)"},{"comment":"The sentence 'A Poisson prime ideal of A is a prime ideal of A that is also a prime ideal' should end with 'also a Poisson ideal'.","section":"§1, definition of Poisson prime ideal"},{"comment":"The claim that the equality C(m) = {q ∈ max A | P(m) ⊆ q} is 'automatically satisfied by Lemma 6.5(i)&(ii)' is incorrect; Lemma 6.5 gives closure equality, not set equality. The proof should be rewritten using closure equality, as indicated in the major comments.","section":"§6, proof of Theorem 6.7"},{"comment":"Examples 6.11 and 6.12 contain typographical errors such as 'noncommuative', 'singluar' and 'M oeglin'; these should be corrected in the final version.","section":"Global"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the defects are localized to Section 5 and Section 6.7, and they appear repairable without changing the main topological criterion in Theorem 6.2. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things worth knowing. The κ-separability/poset part is the real contribution. Theorems 4.5–4.6 and their Poisson consequence Theorem 6.2 are proved in detail, they generalize Bell–Launois–Sánchez–Moosa in a clean way, and the adaptation of Bell–Wang–Yee's base-field trick is careful. That half looks solid and is new.\n\nThe symplectic-core half is not. Proposition 5.3(iii) is false as stated. In the proof of (i)⇒(iii), from ∩_{M∈C(P)}M = P the authors conclude C(P) = {Q ∈ max R | P ⊆ Q}. That conclusion confuses a set with its Zariski closure. The intersection equality only gives the closure. Concrete example: A = C[x,y] with {x,y} = x. The DME holds: primitive, rational, and locally closed Poisson primes are {0} together with the points (x, y−c). But for P = 0, C(0) = {max ideals with a ≠ 0} = max∖V(x), while {Q | 0 ⊆ Q} = max. So the equality in (iii) fails while DME holds. The same closure confusion appears in the appeal to Lemma 6.5; as typeset, that lemma cannot supply the equality used in Proposition 5.3 and Theorem 6.7.\n\nTheorem 6.7(ii) is also false as stated. On C[x,y] with the zero Poisson bracket, DME holds but Cspec(0) = {0} is not locally closed in spec C[x,y]. The theorem should quantify over primitive ideals, or over maximal cores only, not over every prime ideal.\n\nI want to be clear about proportion: I found no gap in the Section 4 chain, and the poset criterion is substantial. But the abstract and Theorem C overstate the symplectic-core criterion, and the proof does not go through as written. The gap is repairable, probably by using local closedness together with closure equality, but right now the Section 5–6 arc is unsound.\n\nThis paper deserves a serious referee; the good half is worth referee time. I would not desk-reject, but I would not accept until the core equality is fixed or the claims are weakened.","headline":"The κ-separability/poset criterion is a genuine contribution, but the symplectic-core half is not sound as written: Proposition 5.3(iii) and Theorem 6.7(ii) are false, and the proof of Theorem C needs repair.","tokens_in":25341,"tokens_out":16623,"would_cite":true,"duration_ms":179891,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16D60","17B63","13N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a complex affine Poisson algebra satisfies the Poisson Dixmier-Moeglin equivalence exactly when every Poisson prime ideal lying over fewer than continuum many minimal Poisson primes lies over only finitely many.","keywords":["Poisson Dixmier-Moeglin equivalence","Poisson prime spectrum","symplectic cores","symplectic leaves","kappa-separability","commutative differential algebras","Poisson primitive ideals","Zariski topology"],"falsifier":"Look for a maximal ideal $\\mathfrak{m}$ in a singular complex affine Poisson algebra for which the Zariski closure of the symplectic leaf $L(\\mathfrak{m})$ is a proper subset of the symplectic core $C(\\mathfrak{m})$; the paper's Lemma 6.5 asserts these always coincide, so such an example would falsify the symplectic-core criterion in Theorem 6.7.","tokens_in":24063,"feed_emoji":"📐","tokens_out":6908,"duration_ms":63281,"temperature":0.7,"pith_summary":"This paper claims that for a complex affine Poisson algebra, the whole Poisson Dixmier-Moeglin equivalence—the coincidence of Poisson primitive, Poisson rational, and locally closed Poisson prime ideals—is a purely topological phenomenon, readable from the Zariski topology of the Poisson prime spectrum. Specifically, the equivalence holds exactly when every Poisson prime ideal with fewer than continuum many minimal Poisson primes over it has only finitely many. The same topology also detects the equivalence through the symplectic core stratification on the maximal spectrum: all symplectic cores must be locally closed. Because the result is formulated for commutative differential algebras, it also extends a previously known weaker version of the equivalence from Poisson algebras to that broader setting.","feed_headline":"Poisson Dixmier-Moeglin equivalence is purely topological","feed_subtitle":"Locally closed symplectic cores decide when primitive, rational, and locally closed Poisson primes coincide.","key_machinery":"The central object is $\\kappa$-separability for points of a Zariski space: a point $p$ is $\\kappa$-separable when the complement of $\\{p\\}$ in its closure can be covered by fewer than $\\kappa$ closed irreducible subsets, equivalently when $p$ has fewer than $\\kappa$ covers in the specialization poset. The paper proves that $\\aleph_0$-separability is exactly local closedness, and that, over large base fields, $|k|$-separability is exactly $\\Delta$-rationality and $\\Delta$-primitivity. Passing to Poisson algebras is done by taking the derivations $\\Delta$ to be the Hamiltonian derivations of the Poisson bracket, so that Poisson ideals, Poisson cores, and the Poisson prime spectrum are exactly the $\\Delta$-objects.","core_discovery":"For any complex affine Poisson algebra $A$, the paper establishes that $A$ satisfies the Poisson Dixmier-Moeglin equivalence if and only if, in the poset $(\\mathrm{P.spec}\\, A, \\subseteq)$, every Poisson prime ideal that is $|\\mathbb{C}|$-separable is also $\\aleph_0$-separable. Concretely, a Poisson prime ideal is $|\\mathbb{C}|$-separable when fewer than continuum many minimal Poisson prime ideals lie over it, and $\\aleph_0$-separability means that there are only finitely many such minimal primes. The paper also proves an equivalent geometric formulation: the equivalence holds exactly when every symplectic core in the maximal spectrum is locally closed. These results are obtained through a broader theorem for commutative differential algebras $R$ over a base field $k$ satisfying $\\dim_k R < |k|$ and $|\\Delta| < |k|$, where $|k|$-separability in the $\\Delta$-prime spectrum is shown to be equivalent to $\\Delta$-rationality.","pith_inferences":["Because the criterion in Theorem 6.2 is phrased only in terms of the specialization poset of $\\mathrm{P.spec}\\, A$, one can in principle verify the Poisson Dixmier-Moeglin equivalence from a finite or countable description of the poset, without computing Poisson brackets on large subalgebras; this is an editorial extension, since the paper does not develop such a computational procedure.","The paper's use of the cardinality bound $\\dim_k R < |k|$ suggests that the same topological criterion should be testable for Poisson algebras over other uncountable base fields, replacing $|\\mathbb{C}|$ with $|k|$ and $\\aleph_0$ with the corresponding threshold; the paper does not state such a generalization for Poisson algebras.","One way to search for a counterexample to the Poisson Dixmier-Moeglin equivalence would be to construct an affine Poisson algebra whose Poisson prime poset has a prime ideal with countably many but not finitely many minimal primes over it and then check whether that prime is rational; the paper's theorem predicts such a prime cannot be rational."],"forward_implications":["If a complex affine Poisson algebra's Poisson prime spectrum is a union of locally closed pieces each homeomorphic to the Poisson spectrum of another affine Poisson algebra, the algebra satisfies the Poisson Dixmier-Moeglin equivalence if and only if each piece does.","If the Poisson bracket is algebraic, meaning every symplectic leaf is locally closed, then the equivalence holds; this recovers earlier finite-leaf results as a special case.","If an algebraic group acts rationally by Poisson automorphisms and there are only finitely many orbits of symplectic leaves or symplectic cores, the equivalence holds.","For commutative differential algebras satisfying $\\dim_k R < |k|$ and $|\\Delta| < |k|$, the paper's Theorem 4.6 gives an analogous topological criterion, so the result is not special to Poisson structures."],"supporting_citations":[{"why":"Supplies the Amitsur-trick argument and the Dixmier-Moeglin equivalence for noetherian algebras over large base fields; Lemma 4.3 is adapted from its Lemma 2.3.","marker":"[8]"},{"why":"Establishes the weaker Poisson Dixmier-Moeglin equivalence via model theory that the paper generalizes, and whose Theorem 7.1 is extended by Theorem 4.5.","marker":"[4]"},{"why":"Provides Lemma 6.5 on symplectic leaves and symplectic cores, the notion of algebraic Poisson bracket, and the finite-primitive-ideals result generalized in Corollary 4.7.","marker":"[11]"},{"why":"Supplies the framework of $\\Delta$-Dixmier-Moeglin equivalence, the retraction $\\pi$ from the prime spectrum to the $\\Delta$-prime spectrum, and the torus-action results that Theorem 5.1 and Proposition 5.6 extend.","marker":"[16]"},{"why":"Supplies the Jacobson ring and Nullstellensatz results used in Lemma 3.1(iv) and Proposition 3.5, together with standard facts about prime spectra.","marker":"[10]"},{"why":"Gives the original proof for complex affine Poisson algebras that locally closed Poisson primes are Poisson primitive and Poisson rational, which Proposition 3.5 generalizes to differential algebras.","marker":"[28]"},{"why":"Provides the result that the $\\Delta$-core of a prime ideal is prime, used repeatedly in Lemma 3.1(i) and in the proof of Theorem 4.5.","marker":"[15]"}],"fun_headline_variants":["Locally closed cores decide Poisson Dixmier-Moeglin equivalence","Topology alone settles Poisson Dixmier-Moeglin","Poisson Dixmier-Moeglin equivalence: a topological test","Symplectic cores pin down Poisson Dixmier-Moeglin equivalence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The symplectic-core characterization rests on the Brown-Gordon lemma that for every maximal ideal $\\mathfrak{m}$ the Zariski closure of the symplectic leaf $L(\\mathfrak{m})$ is exactly the symplectic core $C(\\mathfrak{m})$; if that equality fails on singular affine Poisson varieties, the geometric criterion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Locally closed cores decide Poisson Dixmier-Moeglin equivalence","Topology alone settles Poisson Dixmier-Moeglin","Poisson Dixmier-Moeglin equivalence: a topological test","Symplectic cores pin down Poisson Dixmier-Moeglin equivalence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001424,"raw_usage":{"total_tokens":5737,"prompt_tokens":927,"completion_tokens":4810,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":4745}},"tokens_in":543,"tokens_out":4810,"duration_ms":33650,"temperature":1.0,"reasoning_tokens":4745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:42:00.929624+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a maximal ideal $\\mathfrak{m}$ in a singular complex affine Poisson algebra for which the Zariski closure of the symplectic leaf $L(\\mathfrak{m})$ is a proper subset of the symplectic core $C(\\mathfrak{m})$; the paper's Lemma 6.5 asserts these always coincide, so such an example would falsify the symplectic-core criterion in Theorem 6.7.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Amitsur-trick argument and the Dixmier-Moeglin equivalence for noetherian algebras over large base fields; Lemma 4.3 is adapted from its Lemma 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the weaker Poisson Dixmier-Moeglin equivalence via model theory that the paper generalizes, and whose Theorem 7.1 is extended by Theorem 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Lemma 6.5 on symplectic leaves and symplectic cores, the notion of algebraic Poisson bracket, and the finite-primitive-ideals result generalized in Corollary 4.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the framework of $\\Delta$-Dixmier-Moeglin equivalence, the retraction $\\pi$ from the prime spectrum to the $\\Delta$-prime spectrum, and the torus-action results that Theorem 5.1 and Proposition 5.6 extend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Jacobson ring and Nullstellensatz results used in Lemma 3.1(iv) and Proposition 3.5, together with standard facts about prime spectra."},{"cited_title":"Oh, Symplectic ideals of Poisson algebras and the Poisson structure associated to quantum matrices, Comm","cited_arxiv_id":null,"evidence_quote":"Gives the original proof for complex affine Poisson algebras that locally closed Poisson primes are Poisson primitive and Poisson rational, which Proposition 3.5 generalizes to differential algebras."},{"cited_title":"Dixmier, Enveloping algebras","cited_arxiv_id":null,"evidence_quote":"Provides the result that the $\\Delta$-core of a prime ideal is prime, used repeatedly in Lemma 3.1(i) and in the proof of Theorem 4.5."}],"review_version":1}