{"id":"49eb0f3d-dfe2-4728-91b2-a8e5f178446e","arxiv_id":"2502.00155","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Given any ordering of the first half of a Hilbert series, there exists an artinian Gorenstein algebra realizing it, proved via Nagata idealization of whiskered graphs and the graph Roller Coaster theorem.","lead":"The paper constructs artinian Gorenstein algebras whose Hilbert series can be made to rise and fall in any prescribed order, by importing a graph-theory result about independent sets. It also exhibits a wide family of Koszul Gorenstein algebras failing the weak Lefschetz property, a standard property in commutative algebra.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.6's claimed 'iff' is false: for q=3, π=(2,3), k=2, ℓ=3 gives a2+a2=22 > a3+a1=15 while π(2)<π(3). Theorem 5.7's proof therefore does not establish the Roller Coaster theorem for all d.","rationale":"The reader's verdict flagged Lemma 5.6 as the weakest point, specifically the sentence 'we can check computationally' for q<10. My stress-test agrees that Lemma 5.6 is the load-bearing step, but identifies a more serious problem: the lemma's second equivalence is demonstrably false for q=3, independently of any computational check. The proof of Lemma 5.6 implicitly assumes that in a_k+a_{q−k+1}, the second index lies in the binomial range (i.e., q−k+1 < ⌈q/2⌉), an assumption that fails at the boundary k=⌈q/2⌉. Since Theorem 5.7's proof relies on this equivalence to claim that every permutation is realized, the main theorem is not established for small d. The concrete counterexample with q=3 directly shows the construction cannot produce the reverse ordering h_2<h_1 for d=4. This is a genuine mathematical flaw, not a missing finite check. The central claim might still be true by other constructions, but as written the proof is invalid. I therefore recommend moving the verdict from CONDITIONAL to REJECT, or at minimum to UNVERDICTED until Lemma 5.6 is corrected or the small cases are handled separately.","tokens_in":17400,"tokens_out":13969,"duration_ms":130663,"concrete_test":"Instantiate Lemma 5.6 with q=3 and π=(2,3): c=(3 choose 3)=1, a_1=3, a_2=11, a_3=12. For k=2, ℓ=3, the claimed equivalence requires a_2+a_2 < a_3+a_1, i.e. 22 < 15, which fails despite π(2)<π(3). This single computation decisively refutes the lemma as stated. Optionally, compute the Hilbert functions of the Perazzo algebras from Theorem 5.7 for d=4 in Macaulay2 and verify that both possible π yield h_1<h_2, so the reverse order is absent from the construction.","verdict_should_be":"REJECT","load_bearing_attack":"The central existence theorem (Theorem 1.1) rests on Lemma 5.6, which asserts that for ⌈q/2⌉ ≤ k < ℓ ≤ q, the inequality a_k+a_{q−k+1} < a_ℓ+a_{q−ℓ+1} holds if and only if π(k)<π(ℓ). This is not merely missing a computational check; it is false. Take q=3, so ⌈q/2⌉=2 and c=(3 choose 3)=1. For π=(2,3), a_1=3, a_2=9+2=11, a_3=9+3=12. With k=2, ℓ=3, we have a_2+a_{3−2+1}=a_2+a_2=22 and a_3+a_{3−3+1}=a_3+a_1=15, so the claimed inequality is false even though π(2)<π(3). The proof's displayed substitution replaces a_{q−k+1} by (q choose k−1); this is valid only when q−k+1 < ⌈q/2⌉, which fails for k=⌈q/2⌉. Because Theorem 5.7 applies Lemma 5.6 to order the entries h_k = i_k+i_{d−k}, the construction cannot realize every permutation for small d. For d=4 (so q=3), the two choices of π give h_1≈a_1+a_3=15 or 14 and h_2≈2a_2=22 or 24, so h_2<h_1 is never produced. Thus the 'for every d' claim is unproved for small d, and the reader's identified missing computation is not the core defect: the lemma itself is wrong.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Lefschetz properties of artinian algebras associated to whiskered graphs and uses them to construct Perazzo Gorenstein algebras via Nagata idealization. Its main advertised result, Theorem 1.1, claims that for every positive integer d and every permutation of {1,...,floor(d/2)}, there is an artinian Gorenstein algebra whose first half of the Hilbert series is ordered according to that permutation. The proof passes through an 'approximate well-covered polynomial' construction adapted from Cutler and Pebody, then transfers coefficient inequalities to the Hilbert functions of simplicial Perazzo algebras. The paper also proves WLP failure for whiskered graph algebras with large independence number and constructs a large family of G-quadratic Gorenstein algebras failing the WLP.","tokens_in":17812,"tokens_out":16097,"duration_ms":155367,"significance":"If Theorem 1.1 were established, it would be a substantial strengthening of Boij's valley theorem, showing that the first half of Gorenstein Hilbert functions is combinatorially unconstrained. The paper's general strategy—combining approximate well-covered independence polynomials with simplicial Perazzo forms—is attractive and connects graph theory directly to Gorenstein Hilbert functions. The results on WLP failure for whiskered graphs and their Perazzo algebras, especially the G-quadratic examples of Corollary 4.7 and Theorem 4.6, are valuable and appear to be largely unaffected by the problem discussed below. However, the central Roller Coaster theorem currently rests on a false lemma, so the main claim is not established as stated.","major_comments":[{"comment":"Lemma 5.6 is false as stated. For q=3 and the permutation pi=(2,3), one has ceil(q/2)=2, c=binom(3,3)=1, and the definition gives a_1=3, a_2=11, a_3=12. Taking k=2 and ell=3, we have pi(2)<pi(3), but a_2+a_{3-2+1}=a_2+a_2=22, while a_3+a_{3-3+1}=a_3+a_1=15; the asserted strict inequality 22<15 is false. The source of the error is visible in the proof: the replacement of a_{q-k+1} by binom(q,k-1) is valid only when q-k+1 < ceil(q/2), but at k=ceil(q/2) the index q-k+1 equals ceil(q/2), which falls in the second case of the definition of a_i. Since Theorem 5.7 applies Lemma 5.6 to obtain the Roller Coaster ordering, Theorem 1.1 is not proved for every d; for example, when d=4 the construction cannot produce the order h_2<h_1 required by the transposition on {1,2}. The lemma must be repaired, or the statement of Theorem 1.1 must be restricted to the range in which the construction actually works, with small d handled separately.","section":"Section 5, Lemma 5.6"},{"comment":"The proof of the first part of Lemma 5.6, i.e. the verification of the growth condition (5.1), contains an undocumented finite check: the sentence 'we can check computationally (for q<10)' is not accompanied by code, data, or a precise description of the computation. Since Theorem 5.7 quantifies over every d, this finite check is load-bearing if the lemma is to be used in its full generality. Moreover, the displayed chain of inequalities in that paragraph appears garbled (for example, the expressions involving 1+2q(q-1)/(3q) and 1+(2/3)^q/q do not cohere as written), so the proof is not verifiable in its current form.","section":"Section 5, Lemma 5.6"}],"minor_comments":[{"comment":"In case (2) of the proof of Lemma 5.5, the displayed expression writes i_{k+1}(G)+i_{q-k}(G), but the intended quantities are i_ell(G)+i_{q-ell+1}(G); the indices in that line should be corrected.","section":"Section 5, Lemma 5.5"},{"comment":"The reduction by symmetry from a permutation of {1,...,floor(d/2)} to a permutation of {ceil(d/2),...,d-1} should be spelled out explicitly, especially for odd d, because the index range supplied by Lemma 5.6 with q=d-1 is {ceil((d-1)/2),...,d-1}, which contains an extra element when d is odd.","section":"Section 5, Theorem 5.7"},{"comment":"The statement that Conjecture 3.15 has been verified computationally for all smaller bipartite graphs and for all graphs on at most 7 vertices is cited without accompanying data or code; please supply reproducible verification or reduce the claim accordingly.","section":"Remark 3.16"},{"comment":"The displayed direct sum decomposition in Lemma 3.1 contains several typographical artifacts, such as missing braces and stray characters around the variables y_i, which make the proof harder to read.","section":"Section 3, Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"The central advertised theorem depends on Lemma 5.6, which is false as stated; this is not a mere presentation issue. The rest of the paper, particularly the WLP results for whiskered graphs and the construction of G-quadratic Perazzo algebras failing WLP, appears sound and of independent interest. I would encourage the authors to repair or replace the approximate-polynomial construction, or to weaken the statement of Theorem 1.1 to the range where the construction is valid, and to supply the missing computational checks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, Sections 3 and 4 deliver real results on Lefschetz properties for whiskered graphs and simplicial Perazzo forms. Second, the flagship Roller Coaster theorem is not established by this paper: Lemma 5.6, which is load-bearing for Theorem 5.7, is false.\n\nThe counterexample is small. For q=3, ceil(q/2)=2, c=C(3,3)=1. Take pi=(2,3). The definition in Lemma 5.6 gives a1=3, a2=11, a3=12. For k=2, ell=3, the claimed inequality a_k+a_{q-k+1}<a_ell+a_{q-ell+1} becomes 11+11<12+3, i.e. 22<15. Yet pi(2)<pi(3). So the 'if' direction is just wrong. This is not the missing computational check the reader flagged; it is a false statement. Because Theorem 5.7 applies Lemma 5.6 to arbitrary permutations, the 'for every d' claim is unproved; for d=4 the construction never gives h2<h1. The proof of condition (5.1) is also garbled, and the q<10 check is not supplied, but those are secondary.\n\nThe WLP part of the paper is much healthier. Proposition 3.2 (SLP for whiskered complete graphs), Proposition 3.5 (WLP for whiskered graphs with independence number at most two and odd D), Corollary 3.12 (failure of WLP when alpha(G)>=n/3+2), and the Perazzo transfer in Theorem 4.6 are coherent and appear correct. The bipartite n>=12 statement in Corollary 4.7 is a genuine new family of G-quadratic Gorenstein algebras failing WLP. There is some small-case computational residue in Remark 3.16, but it is not central to the main WLP applications. Citations are honest and non-circular.\n\nWho gains: people working on Lefschetz properties of monomial and edge ideals, and on Perazzo algebras, should read Sections 3 and 4. The Roller Coaster problem itself remains open for small socle degrees under this construction. I would send the paper to a serious referee because the WLP material is strong and the roller coaster question is important, but I would flag Lemma 5.6 for the referee and expect major revision. I would not cite it in its current form.","headline":"The WLP results for whiskered graphs and Perazzo forms are solid and worth knowing, but the advertised Roller Coaster theorem is not proved: Lemma 5.6 is false.","tokens_in":18319,"tokens_out":11949,"would_cite":false,"duration_ms":111740,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13E10","13H10","13F55","05C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Gorenstein Hilbert series can realize any ordering of their first half, matching every permutation of {1,...,⌊d/2⌋}.","keywords":["weak Lefschetz property","artinian Gorenstein algebras","whiskered graphs","Koszul algebras","Hilbert series","Nagata idealization","Perazzo forms","roller coaster theorem"],"falsifier":"Take q = 9 and the permutation π of {5,6,7,8,9} that reverses the order; compute the sequence a_i from Lemma 5.6 and check whether a_k / C(9,k) ≤ a_{k+1} / C(9,k+1) for every k. If it fails, Theorem 5.7 is false for d = 10; if it holds, the same check must still be completed for the remaining q < 10.","tokens_in":17214,"feed_emoji":"🎢","tokens_out":9156,"duration_ms":75149,"temperature":0.7,"pith_summary":"This paper proves that the first half of the Hilbert series of an artinian Gorenstein algebra is unconstrained: for any socle degree d and any permutation π of {1,...,⌊d/2⌋}, there exists a Gorenstein algebra whose h-vector satisfies h_{π(1)} < ... < h_{π(⌊d/2⌋)}. This strengthens Boij's valley theorem, which had shown only that many valleys are possible. The construction builds the algebras as Nagata idealizations (trivial extensions that turn level algebras into Gorenstein ones) of monomial algebras coming from independence complexes of whiskered graphs, so the resulting Gorenstein algebras are Perazzo algebras and, in a large family, G-quadratic and hence Koszul. If the claim is right, unimodality is a rare accident among Gorenstein Hilbert functions, and the symmetry h_i = h_{d-i} is the only constraint on the first half.","feed_headline":"Gorenstein Hilbert series can roller-coaster in any order","feed_subtitle":"Any permutation of the first half of a Gorenstein Hilbert series is realized by an explicit construction.","key_machinery":"The load-bearing object is the simplicial Perazzo form F(∆) = Σ x_i u_{F_i}, where F_1,...,F_m are the facets of a pure simplicial complex ∆; its Macaulay dual generator (the inverse-system form whose annihilator is the Gorenstein ideal) is an artinian Gorenstein algebra that is the Nagata idealization (the trivial extension R ⋉ ω(−d)) of the Stanley–Reisner ring A(∆). When ∆ is the independence complex of a whiskered graph it is shellable, so the Gorenstein algebra is G-quadratic and Koszul. The roller-coaster argument uses the approximate well-covered polynomials of [8]: a coefficient sequence a_i is chosen so that the sums a_k + a_{d-k} obey the prescribed permutation, the sequence is realized up to scaling by the independence numbers of a well-covered graph, and the symmetric sums h_k = i_k + i_{d-k} of the Gorenstein algebra inherit the inequalities through Lemma 5.5.","core_discovery":"The central discovery is that the symmetry condition h_i = h_{d-i} is the only constraint on the first half of an artinian Gorenstein Hilbert series: for every d and every permutation π of {1,...,⌊d/2⌋}, the paper constructs an artinian Gorenstein algebra with Hilbert series ∑_{i=0}^d h_i t^i such that h_{π(1)} < ... < h_{π(⌊d/2⌋)}. The algebras are simplicial Perazzo algebras, obtained as Nagata idealizations of the Stanley–Reisner rings A(Ind(w(G))) of independence complexes of whiskered graphs, and the construction transfers the Roller Coaster theorem for well-covered graphs to this algebraic setting via approximate well-covered polynomials.","pith_inferences":["If Conjecture 5.8 holds, the Koszul property is compatible with arbitrarily shaped first halves of Gorenstein Hilbert series, so failure of unimodality would not be an artifact of non-Koszul presentations; a testable route is to check whether the complexes produced in Theorem 5.7 can be chosen vertex-decomposable, upgrading G-quadratic to a stronger combinatorial condition.","The quantitative threshold α(G) ≥ n/3 + 2 for failure of the weak Lefschetz property suggests a sharper conjecture: that α(G) ≥ 3 already suffices. This is computationally checkable for all graphs on at most 8 vertices, and the bipartite case in the paper already covers all n ≥ 12.","The approximate-polynomial transfer used here may apply to other symmetry-constrained Hilbert-function problems, such as level algebras or Gorenstein algebras with prescribed socle type, where symmetry could again be shown to be the only obstruction."],"forward_implications":["For every socle degree d, the collection of possible first halves of Gorenstein Hilbert series is unconstrained (Theorem 5.7).","Whiskering any graph whose independence number is at least n/3 + 2 yields an artinian algebra A(w(G)) that fails the weak Lefschetz property (Corollary 3.12).","For every bipartite graph on n ≥ 12 vertices, the whiskering produces a G-quadratic Gorenstein Perazzo algebra failing the weak Lefschetz property (Theorem 1.2).","A large family of G-quadratic, hence Koszul, Gorenstein algebras fails the weak Lefschetz property, and the paper conjectures that unconstrained Hilbert-series shapes persist among Koszul Gorenstein algebras of large socle degree (Conjecture 5.8).","Whiskered complete graphs, by contrast, give algebras with the strong Lefschetz property under any monomial artinian reduction (Proposition 3.2)."],"supporting_citations":[{"why":"Supplies the Roller Coaster theorem for well-covered graphs and the approximate well-covered polynomial machinery that the paper transfers to Gorenstein algebras.","marker":"[8]"},{"why":"Introduces Nagata idealization and the first nonunimodal Gorenstein Hilbert series; the paper's construction is built on this idealization.","marker":"[36]"},{"why":"Boij's valley theorem, the earlier result showing Gorenstein Hilbert functions can have many valleys, which Theorem 1.1 strengthens to arbitrary first-half order.","marker":"[3]"},{"why":"Shows the algebra defined by a simplicial Perazzo form is Koszul and G-quadratic when the complex is Cohen–Macaulay and shellable, giving the constructed algebras their Koszul property.","marker":"[9]"},{"why":"Proves that independence complexes of whiskered graphs are shellable and Cohen–Macaulay, which makes the resulting Gorenstein algebras G-quadratic.","marker":"[11]"},{"why":"Provides the dimension inequality for very well-covered graphs used to force failure of surjectivity of multiplication by the linear form (Lemma 3.11).","marker":"[23]"},{"why":"Identifies the multiplication map on the Stanley–Reisner ring with the log-matrix of the facets, used to detect when the simplicial form is Perazzo (Lemma 4.3).","marker":"[21]"},{"why":"Shows whiskered independence complexes have more facets than vertices, ensuring the form is Perazzo in the bipartite case (Corollary 4.7).","marker":"[22]"}],"fun_headline_variants":["Any order allowed for first half of Gorenstein Hilbert series","Roller-coaster Gorenstein algebras: any permutation of Hilbert series","Gorenstein algebras with Hilbert series in any order, as long as symmetric","New Gorenstein algebras: Hilbert series first half can be arbitrary","From whiskered graphs to Gorenstein algebras with wild Hilbert series"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that the construction works for every socle degree rests on the assertion, in Lemma 5.6, that a certain ratio inequality holds for q < 10 'as we can check computationally' — a finite check that is not printed, so the fully verified statement is for d ≥ 11 unless that check is supplied.","fun_headline_variants_meta":{"raw":{"variants":["Any order allowed for first half of Gorenstein Hilbert series","Roller-coaster Gorenstein algebras: any permutation of Hilbert series","Gorenstein algebras with Hilbert series in any order, as long as symmetric","New Gorenstein algebras: Hilbert series first half can be arbitrary","From whiskered graphs to Gorenstein algebras with wild Hilbert series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000704,"raw_usage":{"total_tokens":3102,"prompt_tokens":797,"completion_tokens":2305,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":413,"completion_tokens_details":{"reasoning_tokens":2212}},"tokens_in":413,"tokens_out":2305,"duration_ms":17270,"temperature":1.0,"reasoning_tokens":2212,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T20:00:16.114434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take q = 9 and the permutation π of {5,6,7,8,9} that reverses the order; compute the sequence a_i from Lemma 5.6 and check whether a_k / C(9,k) ≤ a_{k+1} / C(9,k+1) for every k. If it fails, Theorem 5.7 is false for d = 10; if it holds, the same check must still be completed for the remaining q < 10.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Roller Coaster theorem for well-covered graphs and the approximate well-covered polynomial machinery that the paper transfers to Gorenstein algebras."},{"cited_title":"Stanley, Hilbert functions of graded algebras , Advances in Math","cited_arxiv_id":null,"evidence_quote":"Introduces Nagata idealization and the first nonunimodal Gorenstein Hilbert series; the paper's construction is built on this idealization."},{"cited_title":"Algebra 23 (1995), no","cited_arxiv_id":null,"evidence_quote":"Boij's valley theorem, the earlier result showing Gorenstein Hilbert functions can have many valleys, which Theorem 1.1 strengthens to arbitrary first-half order."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the algebra defined by a simplicial Perazzo form is Koszul and G-quadratic when the complex is Cohen–Macaulay and shellable, giving the constructed algebras their Koszul property."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that independence complexes of whiskered graphs are shellable and Cohen–Macaulay, which makes the resulting Gorenstein algebras G-quadratic."},{"cited_title":"Levit and Eugen Mandrescu, Independence polynomials and the unimodality conjecture f or very well-covered, quasi-regularizable, and perfect graphs, Graph theory in Paris, 2007, pp","cited_arxiv_id":null,"evidence_quote":"Provides the dimension inequality for very well-covered graphs used to force failure of surjectivity of multiplication by the linear form (Lemma 3.11)."},{"cited_title":"Algebraic Combin","cited_arxiv_id":null,"evidence_quote":"Identifies the multiplication map on the Stanley–Reisner ring with the log-matrix of the facets, used to detect when the simplicial form is Perazzo (Lemma 4.3)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows whiskered independence complexes have more facets than vertices, ensuring the form is Perazzo in the bipartite case (Corollary 4.7)."}],"review_version":1}