{"id":"eb5618f2-c838-4e3f-ac59-d8cc25aa9a56","arxiv_id":"2502.08948","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The gamma-polynomial of a symmetric polynomial being log-concave (with no internal zeros) implies the polynomial itself is log-concave, answering a question of Branden, Ferroni, and Jochemko.","lead":"This paper proves that if the gamma-polynomial of a symmetric polynomial is log-concave, then the original polynomial is also log-concave, settling a question from 2024. The proof uses a new combinatorial inequality about binomial coefficients, established by counting lattice paths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3's path-counting proof uses an invalid double-counting identity: for r<i it counts lattice points on PQ that are absent from the LHS of (4.2), so the proof of the main theorem fails.","rationale":"The reader's accepted verdict and weakest-assumption analysis focus on Claim 4.5 and the implicit reversal in the path involution. That issue is real but repairable. The decisive problem sits earlier in the same proof: the double-counting step that rewrites the LHS as a sum over all lattice points of PQ. For r<i, points on PQ corresponding to j>r have k=r-j<0 and are absent from the LHS, yet they contribute nonzero path products. The n=8, i=3, r=2 example is a concrete counterexample to the asserted equality (673 vs 883). Consequently the path-counting representation of the LHS, and all subsequent min/max and cancellation arguments, apply to a different quantity. Because Theorems 4.2 and 4.3 are invoked for all r>=0 to prove Theorem 1.3, the central claim is not established by the written proof. This is a correctness risk independent of any failure of Claim 4.5, so my recommendation is to reject the paper in its current form, or at minimum require a corrected proof of Theorem 4.3.","tokens_in":12079,"tokens_out":39616,"duration_ms":332866,"concrete_test":"Independently compute both sides of the identity 'LHS of (4.2) = sum_{A in PQ} #(O->A) * #(A->D)' for n=8, i=3, r=2. The left side is 673; the right side is 883. More generally, run a short script evaluating both sides for random parameters with 0 <= r < i; the equality fails whenever r < i. If it fails, the path-counting proof of Theorem 4.3 as written does not establish (4.2).","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing flaw is in Theorem 4.3 (Section 4), on which Theorem 1.3 depends. The proof identifies the LHS of (4.2), a sum over j,k in {0,...,i} with j+k=r, with the sum over all lattice points A=(n-i-j,i-j) in PQ of (#paths O->A)(#paths A->D). But the LHS only admits j with r-i <= j <= min(i,r). When r<i, points with j>r (so k<0) contribute nonzero products. Example: n=8, i=3, r=2 gives LHS = C(8,3)C(4,1)+C(6,2)^2+C(4,1)C(8,3)=224+225+224=673, while summing over all four PQ points adds the j=3 term C(2,0)C(10,4)=210, giving 883. Thus the asserted equality LHS = sum_alpha #{alpha intersect PQ} is false, and equations (4.3)-(4.5) compute the wrong quantity. Since Theorem 4.2 and Theorem 1.3 require every r>=0, this is not a minor gap: the proof of the central theorem is invalid for r<i. The reader's concern about Claim 4.5 is secondary; the implicit reversal in the involution is repairable, but the double-counting error is not.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the question, raised by Brändén, Ferroni, and Jochemko, whether log-concavity of the γ-polynomial of a symmetric polynomial h implies log-concavity of h. The main theorem (Theorem 1.3) asserts this implication, together with preservation of the no-internal-zeros property. The proof is elementary and divides into three parts: an explicit formula for the coefficients c^{(i)}_{jk} of h_i^2-h_{i-1}h_{i+1} in the monomial basis of the γ variables (Lemma 3.1); a sign-structure lemma (Lemma 3.3) showing that negative coefficients on each anti-diagonal can only occur as a final block; and a reduction to a binomial inequality (Theorem 4.3), which is attacked by a lattice-path double-counting and injection argument. If the proof of Theorem 4.3 is repaired, the paper would answer the motivating question and provide a self-contained combinatorial proof.","tokens_in":12354,"tokens_out":13762,"duration_ms":137544,"significance":"If correct, Theorem 1.3 is a natural and valuable completion of the known hierarchy: it extends the Brändén–Gal real-rootedness theorem and the ultra-log-concavity theorem of Brändén–Ferroni–Jochemko to ordinary log-concavity, using elementary methods. The paper is clearly written and includes instructive examples (Examples 2.3, 3.4, 4.6) and a self-contained derivation of the necessary binomial inequality. However, the current proof of Theorem 4.3 contains a double-counting error that is load-bearing: the lattice-path computation proves a different inequality than (4.2), so the main theorem is not established as written. The defect appears local and repairable, so the paper merits a major revision rather than rejection.","major_comments":[{"comment":"The asserted equality LHS = sum_{A∈PQ∩Z^2} #{paths O→A} #{paths A→D} is false when r<i. For A=(n-i-j,i-j) with j∈{0,...,i}, the product equals the summand of (4.2) only for k=r-j≥0. When j>r, the point A is still on PQ but the product equals C(n-2j,i-j) C(n-2r+2j,i-r+j), which is not a term of the LHS of (4.2). For example, n=8, i=3, r=2 has LHS=224+225+224=673, while the sum over all four lattice points of PQ includes the extra j=3 term C(2,0)C(10,4)=210, giving 883. Since equations (4.3)–(4.5) are derived from this equality, they concern a modified left-hand side and do not prove (4.2). The gap occurs exactly in the range r<i, which is needed in Theorem 4.2, so Theorem 1.3 is not established as written.","section":"Section 4, Theorem 4.3, display before Figure 1"},{"comment":"The rotation by 180 degrees about the center of the rectangle spanned by R and R′ maps a path from R to R′ to a path from R′ to R; to obtain an involution on paths from R to R′ one must also reverse the direction of traversal. This reversal is implicit in Figure 3 and in the equality of intersection counts, but it should be stated explicitly. This is a local correction, but it is needed for the cancellation leading to (4.5).","section":"Section 4, Claim 4.5"}],"minor_comments":[{"comment":"The first displayed summation in the proof of Lemma 4.1 should start at i=0, not i=1, since the subsequent identity includes the term a_0 b_0.","section":"Lemma 4.1"},{"comment":"The displayed identity in Remark 4.7 has garbled notation: the summation should be over j,k with j+k=r and the coefficient should read c^{(i)}_{j,k}.","section":"Remark 4.7"},{"comment":"The caption of Figure 4 states that there are 15 contributing paths; the reader would benefit from a brief explanation of how this number is obtained from the formula in (4.5).","section":"Example 4.6"}],"recommendation":"major_revision","confidential_remarks":"I see no indication that Theorem 1.3 is false; the problem is confined to the proof of the technical inequality Theorem 4.3. A correct path-counting argument, or a different proof of (4.2), would make the paper acceptable. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on the FPV paper. The main theorem—that log-concavity of γ_h with no internal zeros passes to h—is a good answer to a genuinely open question. The reduction in Sections 2–4 is elegant: rewriting h_i^2 − h_{i−1}h_{i+1} as nonnegative combinations of binomial terms and using Lemma 4.1 to reduce to the sum of coefficients on each anti-diagonal is clean. The path-counting idea is the right kind of machinery, and the paper is self-contained and honestly engaged with BFJ24.\n\nThat said, the path-counting proof has a load-bearing gap. In Theorem 4.3, the LHS sum over j,k ∈ {0..i} with j+k=r is identified with the sum over all lattice points A on the full segment PQ of (paths O→A)·(paths A→D). But the LHS only covers the points with j≤r; points with j>r give k<0 and are not summands, yet the path product is nonzero. For example, with n=8, i=3, r=2, the LHS equals 673, while summing over all four points on PQ gives 883 (the extra j=3 term is 1·C(10,4)=210). So the equality LHS = Σ_α #{α∩PQ} is false, and equations (4.3)–(4.5) compute a different quantity. This matters because Theorem 4.2 is needed for all r, and the range r<i is exactly where the error appears. The involution in Claim 4.5 does not rescue this; it assumes the false equality.\n\nThe reader's report called this a clean proof with minor exposition issues. I disagree. The reversal omission in Claim 4.5 and the strictness of B>0 in Lemma 3.3 are minor; the double-counting error is not. Maybe the proof can be repaired by restricting PQ (and the corresponding P′Q′) to the subsegments where the indices are valid, but that would change the decomposition and the involution, and the paper doesn't do it. So as written, Theorem 1.3 is unproved.\n\nMy recommendation: this deserves peer review—the question is important and the reduction is genuinely valuable—but a serious referee should be asked to check Theorem 4.3 line by line. Accept only after the path-counting is fixed or replaced. I would not cite it in its current form.","headline":"The result is plausible and the reduction is elegant, but the path-counting proof of the central inequality has a real double-counting error for r<i, so the paper needs major revision before acceptance.","tokens_in":12905,"tokens_out":6818,"would_cite":false,"duration_ms":62078,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05A19","05A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Log-concave gamma-polynomial forces log-concave polynomial","keywords":["log-concave polynomials","gamma-positivity","symmetric polynomials","unimodality","binomial inequalities","lattice paths","Lorentzian polynomials","gamma-polynomial"],"falsifier":"The theorem would be false if one could exhibit a symmetric polynomial with log-concave, zero-free gamma-polynomial whose coefficient sequence fails log-concavity. Since the proof reduces the theorem to inequality (4.2), a direct check of that finite binomial inequality for any concrete tuple $(n,i,r)$ — for instance by computer for small values — would settle the proof's correctness; a single failing tuple would locate the error precisely.","tokens_in":11882,"feed_emoji":"🧮","tokens_out":6465,"duration_ms":55573,"temperature":0.7,"pith_summary":"Every symmetric polynomial with center of symmetry $n/2$ can be written in the gamma basis $x^i(1+x)^{n-2i}$, and the coefficients form its gamma-polynomial. This paper proves that if that gamma-polynomial is log-concave and has no internal zeros, then the original polynomial is log-concave and has no internal zeros. The result answers a question raised in recent work on Lorentzian polynomials, which had already established the analogous statement for the stronger property of ultra log-concavity. The proof is combinatorial: it reduces the claim to a binomial-coefficient inequality and verifies that inequality by counting lattice paths.","feed_headline":"Log-concave gamma-polynomial forces log-concave polynomial","feed_subtitle":"A symmetric polynomial inherits log-concavity from its gamma-polynomial, settling an open question with a lattice-path proof.","key_machinery":"The gamma-basis expression $h(x)=\\sum_{j=0}^{\\lfloor n/2\\rfloor}\\gamma_j x^j(1+x)^{n-2j}$ is the setting; the sequence $\\gamma_0,\\ldots,\\gamma_{\\lfloor n/2\\rfloor}$ is the gamma-polynomial. The proof's load-bearing identity is inequality (4.2): for fixed $n$, $i$, and $r$, the sum of products of binomial coefficients $\\binom{n-2j}{i-j}\\binom{n-2k}{i-k}$ over $j+k=r$ is at least the corresponding sum with $i-1$ and $i+1$ in place of $i$. This is proved by double-counting north-east lattice paths from $O$ to $D$ through two diagonal segments, $PQ$ and $P'Q'$. The key step is Claim 4.5, where a 180-degree rotation about the center of the rectangle spanned by two lattice points is an involution that sends intersections with one segment to intersections with the other, preserving the total count; this makes the difference between the two sides a manifestly nonnegative sum.","core_discovery":"The central claim is Theorem 1.3: log-concavity of the gamma-polynomial, together with absence of internal zeros, is inherited by the original symmetric polynomial. Since a log-concave sequence without internal zeros is automatically unimodal, the theorem supplies a new sufficient condition for unimodality that applies whenever gamma-positivity can be certified in the stronger log-concave form. The paper also notes that the converse is false, through an example from the ultra-log-concave setting. The proof is elementary in spirit but relies on a delicate lattice-path bijection, and in Lorentzian terminology it establishes that the operator sending gamma-coefficients to $h$ preserves the denormalized Lorentzian property.","pith_inferences":["The geometric involution behind Claim 4.5 may generalize to other pairs of line segments or higher-dimensional settings, since it is a purely positional bijection rather than an algebraic identity.","A computational search over small $n$, $i$, $r$ could independently verify inequality (4.2); although the paper proves it, such a check would be a cheap way to build confidence in any implementation that relies on the theorem.","The manifestly positive decomposition of the difference in (4.2) might yield quantitative lower bounds on the log-concavity margins for concrete families, which would be a testable extension for classes such as $h$-vectors of polytopes."],"forward_implications":["Any symmetric polynomial whose gamma-polynomial is log-concave and zero-free is now known to be log-concave, hence unimodal.","The result completes a chain of transfer theorems: real-rootedness, ultra log-concavity, and now ordinary log-concavity, each passing from gamma-polynomial to polynomial.","In Lorentzian terms, the gamma-to-$h$ operator preserves the denormalized Lorentzian property, which is notable because denormalized Lorentzian polynomials lack the closure under nonnegative changes of variables that makes the Lorentzian case easy.","The binomial inequality (4.2) stands on its own as a combinatorial identity, with the proof giving a positive expression for the difference between its two sides."],"supporting_citations":[{"why":"Proved the ultra-log-concavity analogue and posed the question this paper answers; also contributes the example showing the converse fails.","marker":"[BFJ24]"},{"why":"Established the parallel real-rootedness transfer that motivates the present result.","marker":"[Br¨ a06]"},{"why":"Independently proved the same real-rootedness transfer, placing the gamma-polynomial method in context.","marker":"[Gal05]"},{"why":"Introduced Lorentzian polynomials and the denormalized class; the paper frames its theorem as preservation of the denormalized Lorentzian property.","marker":"[BH20]"},{"why":"Supplies the generating-function identity used in Remark 4.7 as an alternative verification of the boundary case in inequality (4.2).","marker":"[Wil94]"}],"fun_headline_variants":["Gamma log-concavity lifts to original polynomial","Log-concave gamma means log-concave polynomial","Symmetric polynomials inherit log-concavity from gamma","New proof: gamma log-concavity implies log-concavity","Lattice-path proof settles log-concavity inheritance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the 180-degree rotation in Claim 4.5 is a true involution on lattice paths between any two eligible points and that it exactly swaps the number of intersections with the two segments; if that geometric bijection fails, the nonnegativity of the anti-diagonal sums — and hence the whole theorem — is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Gamma log-concavity lifts to original polynomial","Log-concave gamma means log-concave polynomial","Symmetric polynomials inherit log-concavity from gamma","New proof: gamma log-concavity implies log-concavity","Lattice-path proof settles log-concavity inheritance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000492,"raw_usage":{"total_tokens":2387,"prompt_tokens":881,"completion_tokens":1506,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":1422}},"tokens_in":497,"tokens_out":1506,"duration_ms":10566,"temperature":1.0,"reasoning_tokens":1422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:07:49.968367+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem would be false if one could exhibit a symmetric polynomial with log-concave, zero-free gamma-polynomial whose coefficient sequence fails log-concavity. Since the proof reduces the theorem to inequality (4.2), a direct check of that finite binomial inequality for any concrete tuple $(n,i,r)$ — for instance by computer for small values — would settle the proof's correctness; a single failing tuple would locate the error precisely.","supporting_citations":[],"review_version":1}