{"id":"bf9f94ad-e86c-45bb-9d10-06708406a4d9","arxiv_id":"1909.00081","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors disprove the Cuttler-Greene-Skandera conjecture by showing H_44 - H_521 is nonnegative on R^3_{≥0} via an explicit sum of 41 squares, despite 44 and 521 being incomparable in dominance order.","lead":"The paper gives a counterexample to a 2011 conjecture about inequalities between homogeneous symmetric functions, using a computer-generated sum of squares certificate. It shows that two incomparable partitions can still satisfy the inequality in three variables, breaking the pattern seen for other symmetric function bases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 3-variable SOS certificate does not establish the stated all-n inequality: term-normalization depends on n, and no lifting argument is given.","rationale":"Close reading confirms the reader's concern. Section 1 defines G_lambda >= G_mu with the explicit quantifier 'any number of variables n', while the proof certifies only n=3. The normalization constants make the n>3 statement genuinely different: D_n with extra variables set to zero is not the certified D_3. Remark 6 acknowledges the n->infinity question, so the gap is not an artifact of the presentation. I do not see a second independent flaw: Proposition 1 and Lemma 1 are standard, the rational SOS data is checkable in principle, and the symmetry and real-zero methodology is sound. Therefore the verdict remains conditional: the paper should either state the fixed-n theorem precisely or supply a lifting argument from three to arbitrarily many variables.","tokens_in":749,"tokens_out":1444,"duration_ms":143051,"concrete_test":"Test the n=4 case: form D_4 = h_4^{(4),2}/35^2 - h_5^{(4)} h_2^{(4)} h_1^{(4)}/(56*10*4), and use the same symmetry-reduced SDP plus rational-rounding pipeline (or a rigorous global optimizer) to decide whether D_4(x1^2,x2^2,x3^2,x4^2) is nonnegative on R^4_{>=0}. A negative value would disprove Theorem 2 as stated; a rational SOS certificate for n=4 and n=5 would show the 3-variable gap is repairable and force a rewritten all-n statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's stated target, defined in Section 1, is G_lambda >= G_mu for any number of variables n. The proof of Theorem 2 in Section 4, however, only produces an SOS representation for (H44-H521)(x1^2,x2^2,x3^2), i.e., nonnegativity on R^3_{>=0}. This does not imply the same normalized difference is nonnegative for n>3, because the normalization constants change with n. For example, setting extra variables to zero gives a different polynomial, not the certified three-variable one. No monotonicity or reduction from n=3 to all n is supplied, and Remark 6 explicitly leaves open whether counterexamples persist for large n. Thus the main claim as stated, a universal counterexample to the Cuttler-Greene-Skandera conjecture, is not proven; the established result is a fixed-3-variable SOS counterexample.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the Cuttler-Greene-Skandera conjecture on term-normalized homogeneous symmetric functions. It claims that H44−H521 is nonnegative on the nonnegative orthant even though the partitions (4,4) and (5,2,1) are incomparable in dominance order, and that this gives a degree-minimal counterexample to the conjecture. The proof is computational: an SDP is run in three variables, symmetrized, constrained using real zeros, and then rounded to exact rational data, yielding a claimed representation of (H44−H521)(x1^2,x2^2,x3^2) as a sum of 41 rational squares. The full certificate is not printed; the paper refers to an external GitHub repository and a Max Planck mathrepo page. Section 3 reports additional numerical SOS certificates for incomparable pairs in degrees 8, 9, and 10.","tokens_in":10441,"tokens_out":27062,"duration_ms":250899,"significance":"If the three-variable certificate is correct, it is a nontrivial computational result: it is a clean example of exact rational SOS certification for a degree-16 polynomial, and it demonstrates the practical value of symmetry reduction and real-zero constraints in semidefinite programming. However, the advertised significance as a disproof of the CGS conjecture depends on a quantifier that the proof does not deliver. As written, the paper establishes a fixed-n=3 SOS statement, not the universal inequality required by the paper's own definition in Section 1. The computational certificate, if fully supplied with a permanent location, could still be a useful benchmark contribution, but the main claim needs substantial revision.","major_comments":[{"comment":"The paper defines Gλ ≥ Gμ as the inequality holding for any number of variables n, and the abstract and Theorem 2 claim to disprove the CGS conjecture. The proof in Section 4 only certifies nonnegativity on R^3_{\\ge 0}, via the SOS decomposition of (H44−H521)(x1^2,x2^2,x3^2). No argument is given that this three-variable certificate extends to n>3; setting extra variables to zero does not reduce the n-variable inequality to the certified one because the normalization denominators hλ(1^n) and hμ(1^n) depend on n. Remark 6 explicitly leaves open whether counterexamples persist for large n. Thus the stated universal counterexample is not established; the proven statement is a fixed-n=3 SOS result.","section":"Section 1 and Section 4"},{"comment":"The claimed explicit SOS certificate is not self-contained. The proof prints the first row of the Gram matrix and the first square, and states that the remaining squares are in a .txt file on a GitHub repository and a mathrepo page. Since Theorem 2 rests entirely on this certificate, the full list of 41 squares, or the verifying code, should be included as supplementary/ancillary material with a permanent identifier. As submitted, the reader cannot verify Theorem 2 from the text alone.","section":"Section 4 and reference [14]"}],"minor_comments":[{"comment":"The abstract says the paper provides a sums-of-squares decomposition of H44−H521, but the certificate is for (H44−H521)(x1^2,x2^2,x3^2); this wording should be corrected.","section":"Abstract and Section 4"},{"comment":"The phrase \"nonnegative octant\" should be \"nonnegative orthant\".","section":"Section 4"},{"comment":"The notation in Theorem 2, \"Hµ−Hλ≥ 0 ... provided by H44−H521,\" should specify explicitly which of the two partitions is λ and which is μ.","section":"Theorem 2"},{"comment":"The text states that a poset of SOS certifications is provided below, but the figure or diagram is not present in this version; it should be included.","section":"Section 3"},{"comment":"The definition of term-normalized symmetric functions should state the condition on the number of variables n needed so that g(1) is nonzero, since for l(λ)>n the normalized polynomial is not defined.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"The key tension is between the universal definition in Section 1 and the three-variable proof. If the CGS conjecture in [10] is intended per fixed n, the authors should say so explicitly and reconcile their Section 1 wording; if it is intended universally, the main theorem is not proved. I would ask the editor to check the exact statement in [10]. In either case, the paper needs a substantive revision: either add a lifting argument for all n, or reframe the result as a fixed-three-variable counterexample and remove the claim that the conjecture is disproved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The SOS certificate is real; the headline is not. The genuinely new thing is an explicit 41-square rational decomposition certifying (H44−H521)(x1^2,x2^2,x3^2) ≥ 0 on R^3, with code and data shipped so the certificate can be checked by expansion. That is a solid computational result, and the method—symmetry reduction plus constraints from real zeros to make rational rounding work—is interesting and likely reusable. Prior proofs up to degree 7 make the pair genuinely degree-minimal.\n\nThe problem is the advertised conclusion. The abstract and Theorem 2 say this disproves the Cuttler–Greene–Skandera conjecture. But the conjecture, as the paper itself defines it in Section 1, is about the inequality holding for any number of variables n. The proof only gives three variables. Setting extra variables to zero does not reproduce the certified polynomial because the normalization constants hλ(1^n) change with n. So no lifting from n=3 to all n is supplied, and Remark 6 even suggests the counterexample may disappear for large n. As written, the paper proves a fixed-n counterexample, not the universal disproof it advertises.\n\nThis is a load-bearing gap, not a cosmetic one. The fix could be to prove a lifting theorem, which is implausible given the asymptotic behavior, or to explicitly state the result as a three-variable counterexample to the per-n version of the conjecture. That more modest claim is still interesting and new. The numerical poset in Section 3 is a nice complement.\n\nEverything else checks out: the SOS certificate is exact and machine-checkable, the real-zero argument is legitimate, the references are appropriate, and the exposition is clear. The paper deserves a serious referee; the referee should ask for a corrected claim or an all-n proof. I would not cite it as a disproof of CGS until that is resolved.","headline":"A real SOS certificate and a clever method, but the advertised disproof covers only n=3 and the universal CGS conjecture remains open.","tokens_in":10953,"tokens_out":7850,"would_cite":false,"duration_ms":66280,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper refutes the 2011 converse conjecture for homogeneous symmetric functions by giving a sum-of-squares certificate that $H_{44}-H_{521}$ is nonnegative despite incomparable dominance.","keywords":["homogeneous symmetric functions","dominance order","sums of squares","semidefinite programming","nonnegative orthant","rational SOS certificate","counterexample"],"falsifier":"Expand the 41 listed square summands symbolically and compare with the displayed polynomial: exact equality verifies the three-variable certificate. To test the universal claim, run the same semidefinite search for $(H_{44}-H_{521})(x_1^2,\\ldots,x_n^2)$ with $n=4$; a negative value or an infeasible SOS program at any point would show the three-variable proof does not extend to the definition's 'any number of variables'.","tokens_in":10093,"feed_emoji":"🧮","tokens_out":9380,"duration_ms":85343,"temperature":0.7,"pith_summary":"The paper sets out to refute a 2011 conjecture: for term-normalized homogeneous symmetric functions, a difference $H_\\mu - H_\\lambda$ should be nonnegative on the nonnegative orthant only when $\\mu$ dominates $\\lambda$, matching the pattern known for monomial, elementary, power-sum, and Schur functions. The authors exhibit a specific pair of incomparable partitions, $\\lambda = 44$ and $\\mu = 521$, and prove that $H_{44} - H_{521}$ is nonnegative by writing $(H_{44} - H_{521})(x_1^2, x_2^2, x_3^2)$ as an explicit sum of 41 rational squares. Since a sum of squares cannot take negative values, this is a direct algebraic certificate that the homogeneous basis breaks the dominance-order rule. The proof establishes the three-variable version of the inequality; the paper remarks that the large-$n$ behavior remains open.","feed_headline":"41 squares break the dominance rule for symmetric means","feed_subtitle":"A difference of homogeneous symmetric functions is nonnegative despite incomparable partitions, disproving the 2011 conjecture.","key_machinery":"The load-bearing mechanism is the Gram-matrix characterization of sums of squares: a homogeneous polynomial $h$ of degree $2d$ is a sum of squares iff $h = m^T A m$ for some positive semidefinite matrix $A$ indexed by monomials of degree $d$. The paper turns this into a semidefinite program whose solution must exactly reproduce $h = (H_{44}-H_{521})(x_1^2,x_2^2,x_3^2)$, then uses three structural reductions: symmetrization with respect to the variable-permutation group, linear constraints forced by real zeros of $h$, and rational rounding to convert floating-point output into an exact rational matrix. Factoring this $45\\times 45$ matrix yields the 41 squares.","core_discovery":"The central claim, Theorem 2, is that $H_{44}-H_{521} \\geq 0$ provides a degree-minimal counterexample to the proposed converse: the partitions $44$ and $521$ are incomparable in dominance order, yet the normalized homogeneous symmetric function difference is nonnegative after the substitution $x_i \\mapsto x_i^2$, which certifies nonnegativity on the nonnegative orthant. The nonnegativity is not asserted abstractly: the paper gives a rational sum-of-squares decomposition into 41 squares, produced by solving a semidefinite program, imposing $S_3$-symmetry and real-zero constraints, rounding the numerical Gram matrix to exact rational entries, and factoring it. Because the certificate is explicit and checkable by squaring and summing, the counterexample is a proof rather than a numerical heuristic. The paper also reports many further numerical counterexamples in degrees 8, 9, and 10 and organizes them in a poset showing how dominance order would need to be modified.","pith_inferences":["If the same symmetry-plus-real-zeros rounding pipeline were run in four or more variables, it would either extend the counterexample or show that the three-variable certificate is an artifact of low dimension, directly addressing the paper's open large-$n$ question.","The appearance of a rational SOS certificate at degree 16 with large denominators suggests that exact rational certificates may be common in the interior of the SOS cone, so symmetry reduction may be more important than raw degree for finding them.","The modified poset in Section 3 could be tested against candidate partial orders generated by weighted cumulative sums; the blue arrows would provide immediate falsification data for any proposed order."],"forward_implications":["The 2011 converse conjecture is false as stated for the homogeneous basis, so dominance order cannot fully classify normalized homogeneous symmetric function inequalities.","The explicit certificate upgrades the counterexample from a numerical find to a checkable algebraic proof, and it shows the existence of rational SOS certificates at degree 16 despite known obstructions at degree 4.","The paper's poset of numerically certified differences suggests that many more incomparable pairs satisfy the inequality; converting those to exact certificates would map the true nonnegativity order for fixed small $n$.","For the paper's stated definition, which demands the inequality for any number of variables, the proof covers only $n=3$; whether the counterexample survives as $n$ grows is explicitly left open."],"supporting_citations":[{"why":"States the known Muirhead-type classification and poses the converse for homogeneous symmetric functions that the paper refutes.","marker":"[10]"},{"why":"Provides the standard equivalence between sums of squares and positive semidefinite Gram matrices used to set up the semidefinite program.","marker":"[5]"},{"why":"Supplies the symmetry-reduction theorem restricting the Gram matrix search to the invariant subspace, which makes exact rational rounding possible.","marker":"[11]"},{"why":"Gives the classical sum-of-squares proof for the arithmetic-geometric mean case and anchors the substitution technique used here.","marker":"[16]"},{"why":"Establishes that rational SOS certificates need not exist, motivating why the exact certificate here is a substantive step.","marker":"[23]"},{"why":"Stores the explicit rational matrix and the 41 square summands that constitute the certificate for Theorem 2.","marker":"[14]"}],"fun_headline_variants":["41 squares disprove dominance conjecture for symmetric functions","Incomparable partitions yet nonnegative: SOS counterexample","Homogeneous symmetric functions break dominance rule","SOS polynomial disproves symmetric function conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof certifies nonnegativity only in three variables, while the statement being refuted requires the inequality to hold for every number of variables $n$, and no argument in the paper bridges that gap.","fun_headline_variants_meta":{"raw":{"variants":["41 squares disprove dominance conjecture for symmetric functions","Incomparable partitions yet nonnegative: SOS counterexample","Homogeneous symmetric functions break dominance rule","SOS polynomial disproves symmetric function conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000449,"raw_usage":{"total_tokens":2251,"prompt_tokens":920,"completion_tokens":1331,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":1273}},"tokens_in":536,"tokens_out":1331,"duration_ms":10137,"temperature":1.0,"reasoning_tokens":1273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:03:34.514647+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand the 41 listed square summands symbolically and compare with the displayed polynomial: exact equality verifies the three-variable certificate. To test the universal claim, run the same semidefinite search for $(H_{44}-H_{521})(x_1^2,\\ldots,x_n^2)$ with $n=4$; a negative value or an infeasible SOS program at any point would show the three-variable proof does not extend to the definition's 'any number of variables'.","supporting_citations":[{"cited_title":"Inequalities for symmetric means","cited_arxiv_id":null,"evidence_quote":"States the known Muirhead-type classification and poses the converse for homogeneous symmetric functions that the paper refutes."},{"cited_title":"Parrilo, and Rekha R","cited_arxiv_id":null,"evidence_quote":"Provides the standard equivalence between sums of squares and positive semidefinite Gram matrices used to set up the semidefinite program."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the symmetry-reduction theorem restricting the Gram matrix search to the invariant subspace, which makes exact rational rounding possible."},{"cited_title":"¨Uber den Vergleich des arithmetischen und des geometrischen Mittels","cited_arxiv_id":null,"evidence_quote":"Gives the classical sum-of-squares proof for the arithmetic-geometric mean case and anchors the substitution technique used here."},{"cited_title":"Sums of squares of polynomials with rational coefficients","cited_arxiv_id":null,"evidence_quote":"Establishes that rational SOS certificates need not exist, motivating why the exact certificate here is a substantive step."},{"cited_title":"Sos counterexample","cited_arxiv_id":null,"evidence_quote":"Stores the explicit rational matrix and the 41 square summands that constitute the certificate for Theorem 2."}],"review_version":1}