{"id":"ca9068ca-e216-4f7a-9132-e28c3ee972e1","arxiv_id":"2607.24186","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Deleting carefully chosen points from finite projective geometries yields representable matroids whose Kazhdan–Lusztig polynomials are not unimodal, so they need not be log-concave or real-rooted.","lead":"Mathematicians built explicit matroids, over every finite field, whose Kazhdan–Lusztig polynomials rise, dip, then rise again. That kills two popular conjectures: that those polynomials are always log-concave, and that they always have only real roots.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No significant objection identified. The one genuinely load-bearing hypothesis — the half-rank degree bound on C_U — is discharged by Proposition 2.6 with a correct counting and complement argument, and the identification P=C rests on a valid palindromicity-uniqueness step plus an independent geomet","rationale":"The Reader identified the correct weakest point (the half-rank degree hypothesis and its verification for the constructed deletions), and on my independent pass that point holds up: the complement construction in Proposition 2.6 is valid, the fiber-counting bounds |D|<q^{r−2} vs. q^{dim U} and q^{r−1} are correctly applied, the induction's hypothesis inheritance via (D_U)_{W/U}=D_W is properly established in Lemma 2.3, and the final palindromicity-uniqueness appeal matches the published [BV20] characterization. The paper also carries two genuinely independent proofs of Theorem 1.1 (the characteristic-polynomial/finite-field convolution in Remark 2.4 and Appendix A, and the small-map decomposition-theorem argument in Section 3), which substantially lowers correctness risk for a purely combinatorial claim. The AI-generated origin of the rank-7 example is fully disclosed, its proof is reproduced in human-checkable form in Appendix A (and its discriminant computation verifies), and the main theorems do not depend on any unverified AI output. Since I find no load-bearing concern beyond the one the Reader already isolated and the paper itself discharges, the verdict should remain ACCEPT; the proposed direct recomputation of the rank-7 instance is a cheap, decisive belt-and-suspenders check rather than a response to an identified defect.","tokens_in":13480,"tokens_out":5485,"duration_ms":52527,"concrete_test":"Independently recompute one instance end-to-end: take the rank-7 binary example of Theorem A.4 (V=F_2^7, D as given, 114 points), construct the matroid in SageMath/Oscar, and compute P_M(t) directly from the defining recursion (i)–(iii) over the lattice of flats, without using any projective-deletion formula. If the result equals 1+13t+7t²+t³, the Theorem 1.1 mechanism (degree-criterion ⇒ P=C) is confirmed on the motivating instance; a mismatch would indicate a flaw in Lemma A.3/Theorem 1.1. A second new instance (q=2, k=3, s=1, r=7) would test Proposition 2.6's prediction P_M=1+8t+7t²+t³.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I probed the same location the Reader flagged: whether the half-rank degree condition deg C_U(t) < (r−dim U)/2 for all proper flat spans U is actually verified for the line-separated deletions, since the entire identification P_{M/F_U}=C_U (and hence the explicit non-unimodal polynomials) collapses if some quotient secretly contains a large deleted subspace.\n\nOn close reading the verification is sound. In Proposition 2.6: (a) any subspace of dimension ≥2 whose projective points lie in D=P(W)∪S lies in W, by line separation — correct, since a line through a point outside P(W) inside such a subspace would violate separation. (b) The key step ruling out deg C_U ≥ 2 uses the complement trick: from a complement L=⟨ℓ1,ℓ2⟩ to U in B with P(L)⊆D one gets L≤W, then L′=⟨ℓ1+u,ℓ2⟩ with u∈U∖W is again a complement with P(L′)⊆D but L′⊄W, contradiction. This is correct, and U⊄W follows from U being a flat span (P(W)∩E=∅). (c) The remaining low-codimension cases are handled by the fiber-size bound: a nonconstant C_U forces an entire π_U-fiber of size q^{dim U}≥q^{r−2} inside D, contradicting |D|<q^{r−2}; the spanning of E is likewise forced since otherwise D would contain the q^{r−1} points outside a hyperplane. (d) The induction's hypothesis inheritance via (D_U)_{W/U}=D_W (Lemma 2.3) and the simplification claim are correct, and the final step — C_0 plus the known contraction terms summing to the palindromic Gaussian polynomial, with deg C_0<r/2, forcing P_M=C_0 by the [BV20] uniqueness characterization — is a legitimate use of the axioms. The geometric Section 3 gives genuinely independent support (smallness ⇔ degree bound; decomposition theorem). Arithmetic checks also pass: [6 choose 2]_2=651, [6 choose 3]_2=1395, s=589, the rank-17 constraint comes from s≤q^{r−k−1}, and the Appendix A cubic discriminant is indeed −268. I could not find a gap that the stated hypotheses do not close.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript constructs, for every finite field F_q, an F_q-representable matroid whose Kazhdan–Lusztig polynomial is not unimodal, thereby disproving the log-concavity conjecture of Elias–Proudfoot–Wakefield and the real-rootedness conjecture of Gedeon–Proudfoot–Young. The examples are represented by P(V)\\D, where D consists of a deleted projective subspace together with suitably separated points. Theorem 1.1 shows that, under the half-rank condition deg C_U(t)<(r-dim U)/2 for every proper flat span U, each contraction polynomial P_{M/F_U} equals an explicit subspace-enumerating polynomial C_U, while Z_M is the corresponding Gaussian-binomial polynomial. The paper gives both an inductive proof from the characterizing axioms and a geometric proof using matroid Schubert varieties, a birational small map, and the decomposition theorem. Corollary 1.3 then produces a strict coefficient valley over every finite field.","tokens_in":13970,"tokens_out":7247,"duration_ms":72490,"significance":"If accepted, this paper settles two prominent conjectures in matroid Kazhdan–Lusztig theory in the negative and substantially changes the expected coefficient behavior of these polynomials. Its strengths are notable: the counterexamples are explicit and representable over every finite field; the relevant polynomials are given by direct subspace counts; and the manuscript supplies two independent proofs of the main theorem, one combinatorial and one geometric via a small map and the decomposition theorem. The potentially delicate half-rank hypothesis is verified in Proposition 2.6 by a correct line-separation, complement, and cardinality argument. The explicit q=2 polynomial and the general q-binomial construction make the result readily checkable.","major_comments":[],"minor_comments":[{"comment":"In the paragraph following axioms (i)–(iii), the sentence identifying Z_M is duplicated: “...is called the Z-polynomial of M. is called the Z-polynomial of M.” Please remove the repeated fragment.","section":"§1"},{"comment":"“For every sufficiently larger” appears to be missing the variable r. It would also help to state the sufficient conditions explicitly—k<r/2, [k]_q+s<q^{r-2}, and s≤q^{r-k-1}—rather than only inside the proof.","section":"Corollary 1.2"},{"comment":"The inference “Since every point of P(W) was deleted, one has U≰W” should explicitly use that U is a nonzero flat span: if U≤W, then E∩P(U) is empty and cannot span U.","section":"Proof of Proposition 2.6"},{"comment":"The specialization notation changes from t to q^m and later uses qm; please write q^m consistently and state at the first use that polynomial identity follows from equality at infinitely many m.","section":"Remark 2.4"},{"comment":"The letter k is reused for the base field after previously denoting dim W. A distinct symbol such as \\Bbbk would avoid confusion. When invoking the small-map consequence of the decomposition theorem, it would also be useful to write the smallness criterion as codim{y:dim π^{-1}(y)≥i}>2i and note that finitely many strata and upper semicontinuity reduce it to the displayed inequalities.","section":"§3"},{"comment":"The introduction gives the rank-17 nonunimodal example, while the smaller rank-7 binary example with nonreal zeros appears only in the appendix. A brief forward pointer would make the relationship between the two examples clearer.","section":"Appendix A.2"}],"recommendation":"accept","confidential_remarks":"The AI-discovery appendix is unusually detailed, but the provenance is disclosed transparently and the mathematical arguments are independent of it. I see no novelty or citation concern; the appendix explicitly places the finite-field counting lemma in the prior literature."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The headline is simple: over every finite field there are representable matroids whose Kazhdan–Lusztig polynomials are not unimodal, so both the EPW log-concavity conjecture and the GPY real-rootedness conjecture are false. That is the result people will remember.\n\nWhat is actually new is Theorem 1.1. Under a half-rank degree bound on the subspace polynomials C_U in every proper contraction quotient, one gets P_{M/F_U}=C_U for every flat and Z_M equal to the Gaussian generating function of the ambient projective geometry. The line-separated deletions (Proposition 2.6) discharge the bound by a short counting-plus-complement argument, and the resulting family produces the explicit non-unimodal polynomials (k=6 plus a carefully chosen s). The Bose–Burton remark shows the construction is a controlled perturbation of a known family rather than an ad-hoc gadget.\n\nThe paper does the technical work cleanly. There are two independent proofs of the identification: a short induction from the characterizing axioms of P and Z, and a geometric argument that the forgetting map of matroid Schubert varieties is small precisely when the degree bound holds, so the decomposition theorem supplies the same equality. The arithmetic checks (Gaussian binomials, the binary rank-17 example, the cubic discriminant in the appendix) are correct. AI discovery of the initial rank-7 cubic is fully disclosed and does not carry the general theorems.\n\nThe only load-bearing hypothesis is the half-rank bound itself. It is verified for the constructions that matter; if someone later finds a deleted set that secretly admits a large subspace in some quotient, the identification would fail for that set, but that does not touch the counterexamples already written down. No circularity, no fitted parameters, citations are appropriate.\n\nThis is for anyone working on matroid KL polynomials, Z-polynomials, or the combinatorial side of intersection cohomology of matroid Schubert varieties. It deserves a serious referee and should be engaged with immediately.","headline":"Clean, high-confidence refutation of the log-concavity and real-rootedness conjectures for matroid KL polynomials, via an explicit projective-deletion theorem that holds up under both combinatorial and geometric proofs.","tokens_in":15030,"tokens_out":509,"would_cite":true,"duration_ms":9677,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B35","05E14","14F43"],"pacs":[],"model":"grok-4.5","headline":"Kazhdan–Lusztig polynomials of matroids need not be unimodal; counterexamples exist over every finite field.","keywords":["Kazhdan–Lusztig polynomials","matroids","unimodality","log-concavity","real-rootedness","projective geometry","Z-polynomial","representable matroids"],"falsifier":"Build the explicit binary example of rank 17 obtained by deleting a 6-flat plus 589 line-separated points and compute its Kazhdan–Lusztig polynomial; the claimed coefficients are 1 + 652t + 651t^2 + 1395t^3 + 651t^4 + 63t^5 + t^6, which decrease then increase. Any mismatch, or any deleted configuration that secretly admits a degree-(r−dim U)/2 or larger subspace in some quotient, would refute the argument.","tokens_in":14691,"feed_emoji":"📉","tokens_out":1133,"duration_ms":26255,"temperature":0.7,"pith_summary":"Two long-standing conjectures said that every matroid Kazhdan–Lusztig polynomial has log-concave coefficients and only real negative roots. This paper disproves both by building, over every finite field, representable matroids whose polynomials are not even unimodal. The examples come from deleting carefully chosen points from a finite projective geometry so that the Kazhdan–Lusztig polynomial simply counts the subspaces whose points all lie in the deleted set. Under a half-rank size condition on those deleted subspaces in every contraction, that count is exactly the Kazhdan–Lusztig polynomial, while the companion Z-polynomial stays the same as for the full geometry. Explicit choices then force a local valley in the coefficients, killing unimodality and therefore log-concavity and real-rootedness.","feed_headline":"Matroid KL polynomials need not be unimodal","feed_subtitle":"Point deletions from projective geometries kill log-concavity and real-rootedness over every finite field","key_machinery":"Projective deletions together with the half-rank degree condition: for the deleted set D, the subspace polynomial C_U enumerates subspaces whose projective points lie in the deleted set of the corresponding quotient; when deg C_U < (r − dim U)/2 for every proper flat span U, one obtains P_{M/F_U}(t) = C_U(t) and Z_M(t) equal to the full Gaussian binomial generating function.","core_discovery":"Over every finite field F_q there exist F_q-representable matroids whose Kazhdan–Lusztig polynomials are not unimodal. More generally, if points are deleted from a projective geometry so that in every proper contraction the subspace-counting polynomial C_U has degree strictly less than half the quotient rank, then the Kazhdan–Lusztig polynomial of each contraction equals C_U and the Z-polynomial equals the Gaussian generating function of the full space. Choosing a six-dimensional deleted flat plus a suitable number of extra line-separated points produces an explicit non-unimodal polynomial, so the log-concavity and real-rootedness conjectures both fail.","pith_inferences":["Any future positive conjecture about matroid Kazhdan–Lusztig polynomials must either restrict the matroid class (for example to paving, uniform, or graphic matroids) or replace unimodality by a weaker shape constraint that still allows local valleys.","The same deletion technique may produce counterexamples to other proposed coefficient inequalities once the half-rank condition is checked, because the polynomials become literally enumerative.","Because the examples are representable, the failure already occurs inside the geometric setting where the polynomials count intersection cohomology, not only for abstract matroids."],"forward_implications":["The log-concavity conjecture for matroid Kazhdan–Lusztig polynomials is false for representable matroids over every finite field.","The real-rootedness conjecture is likewise false; non-unimodal nonnegative coefficient sequences cannot be real-rooted.","For line-separated deletions of a k-flat plus s extra points (with k < r/2 and |D| small), the Kazhdan–Lusztig polynomial is exactly the truncated Gaussian polynomial of the k-flat plus an extra st term.","The Z-polynomial of every such deletion equals the Z-polynomial of the undeleted projective geometry.","Bose–Burton geometries appear as the undeleted special case and recover the known Gaussian Kazhdan–Lusztig polynomials when k < r/2."],"fun_headline_variants":["Matroid KL polynomials need not be unimodal","Point deletions kill unimodality of matroid KL polynomials","Representable matroids can have non-unimodal KL polynomials","KL polynomials of matroids need not be log-concave","Deleted projective points break KL unimodality over every F_q"],"cache_read_input_tokens":128,"weakest_assumption_plain":"In every proper contraction, no collection of deleted points is allowed to contain a subspace that reaches half the dimension of that contraction; if any such large deleted subspace appears, the identification of the Kazhdan–Lusztig polynomial with the subspace count fails.","fun_headline_variants_meta":{"raw":{"variants":["Matroid KL polynomials need not be unimodal","Point deletions kill unimodality of matroid KL polynomials","Representable matroids can have non-unimodal KL polynomials","KL polynomials of matroids need not be log-concave","Deleted projective points break KL unimodality over every F_q"]},"model":"grok-4.5","effort":"low","cost_usd":0.004227,"raw_usage":{"total_tokens":1263,"prompt_tokens":729,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":42268000,"prompt_tokens_details":{"text_tokens":729,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":467,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":729,"tokens_out":67,"duration_ms":7875,"temperature":1.0,"reasoning_tokens":467,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T21:31:19.597257+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Build the explicit binary example of rank 17 obtained by deleting a 6-flat plus 589 line-separated points and compute its Kazhdan–Lusztig polynomial; the claimed coefficients are 1 + 652t + 651t^2 + 1395t^3 + 651t^4 + 63t^5 + t^6, which decrease then increase. Any mismatch, or any deleted configuration that secretly admits a degree-(r−dim U)/2 or larger subspace in some quotient, would refute the argument.","supporting_citations":[],"review_version":1}