{"id":"20b31f31-710f-4b41-aef1-211e36be6932","arxiv_id":"2505.18896","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit IDP lattice polytopes in dimensions 7 and 12 have h*-vectors that are unimodal but not log-concave.","lead":"The paper gives two explicit lattice polytopes with the integer decomposition property whose h*-vectors are not log-concave, answering a question of Ferroni and Higashitani. The main Ehrhart unimodality conjecture remains open because the examples are still unimodal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"IDP and unimodularity are asserted on software checks with no certificates; without an independent Normaliz/Hilbert-basis verification the counterexamples may not answer Question 3.9(b).","rationale":"The reader's weakest assumption matches the soft spot: IDP is the load-bearing property and it is delegated to unshown software verification. I checked the non-log-concavity from the displayed h*-vectors directly: both fail the condition at i=5 (5*2 = 10 > 3^2 = 9), so once IDP is granted the central claim follows. The polytopes are explicit, so the absence of code is a fixable omission rather than a mathematical error. Since the reader already recommends CONDITIONAL, my stress-test does not shift the verdict. One caveat: the 'every triangulation is unimodular' claim in Theorem 1.3 is stronger than needed and would require a total-unimodularity-type certificate; the Normaliz Hilbert-basis check for IDP is the minimal test that settles the main question.","tokens_in":3381,"tokens_out":11565,"duration_ms":70216,"concrete_test":"Run Normaliz on the exact vertex matrices listed for Theorems 1.2 and 1.3. For each polytope P, compute the Hilbert basis of the cone over P and the Ehrhart h*-vector: IDP holds exactly when every Hilbert basis element has height 1, and the computed h*-vectors should match (1,2,3,4,5,3,2,1) and (1,2,3,4,5,3,2,1,0,0,0,0,0). Separately, confirm Theorem 1.3's 'every triangulation is unimodular' claim with Polymake's unimodularity check on the 12-dimensional 0/1-vertex set; if all checks pass the theorems stand, and if any fails the relevant assertion must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1.2 and 1.3 present explicit polytopes and give h*-vectors, but the property that makes these examples relevant—the integer decomposition property—is never proved. The only support is the closing sentence: 'All properties in these results can be verified using software packages like Polymake, Normaliz, Magma, or SageMath.' No scripts, logs, or certificates are included. For Theorem 1.2 this means the IDP assertion is unverified; for Theorem 1.3 the stronger claim that every triangulation is unimodular is likewise unverified. Since non-log-concavity of an arbitrary lattice polytope's h*-vector is not new, the central claim collapses if either polytope fails IDP. This is a completeness/reproducibility gap rather than an internal contradiction: the vertex data are explicit, so the checks are feasible, but they have not been supplied in the current preprint.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper announces two explicit lattice polytopes whose Ehrhart h*-vectors are not log-concave: a 7-dimensional polytope in R^7 with 9 vertices and h*-vector (1,2,3,4,5,3,2,1), and a 12-dimensional 0/1-polytope in R^12 with 15 vertices and h*-vector (1,2,3,4,5,3,2,1,0,0,0,0,0). The authors assert that both polytopes have the integer decomposition property (IDP), and that the 12-dimensional example is unimodular (every triangulation is unimodular) and has no quadratic triangulation. If these assertions are correct, the examples answer Question 3.9(b) of Ferroni and Higashitani, which asks whether IDP polytopes must have log-concave h*-vectors. The paper also states, without proof, that the 12-dimensional example is unimodularly equivalent to the arc polytope of a directed bipartite graph.","tokens_in":3528,"tokens_out":3527,"duration_ms":32155,"significance":"The claimed result is significant: it would disprove a natural log-concavity strengthening of the unimodality conjecture for IDP polytopes, and the 0/1-polytope example in dimension 12 is particularly striking because 0/1-polytopes form a very restricted class. The explicit vertex data and displayed h*-vectors make non-log-concavity directly checkable by hand for the vectors shown. However, the central relevance of the examples depends entirely on the IDP assertion (and for Theorem 1.3, on the stronger unimodularity assertion), and the paper provides no proof or reproducible computational certificate for these properties. The only support is the closing sentence stating that the properties can be verified with software. As a result, the paper is best read as a research announcement rather than a complete proof. Independent verification is feasible because the polytopes are given explicitly, so the gap is a matter of missing evidence rather than a perceived internal inconsistency.","major_comments":[{"comment":"The central claims that the displayed polytopes have the integer decomposition property (and that the 12-dimensional polytope has the stronger property that every triangulation is unimodular) are asserted without proof. The only support is the final sentence: 'All properties in these results can be verified using software packages like Polymake, Normaliz, Magma, or SageMath.' For a mathematical theorem, this is insufficient: no scripts, logs, or certificates are provided, and the connection to Question 3.9(b) collapses if either polytope fails IDP. The authors should either give a human-readable proof (for example, exhibit a unimodular triangulation, or give an explicit Hilbert-basis argument, or in the 0/1 case provide a regular unimodular triangulation) or make the computation fully reproducible by supplying the exact inputs, outputs, and version information for one of the named software packages.","section":"Theorem 1.2(1) and Theorem 1.3(1)–(2)"},{"comment":"The assertion that the polytope in Theorem 1.3 is unimodularly equivalent to the arc polytope of the directed bipartite graph in Figure 1 is stated without proof and without a precise correspondence between the 15 listed vertices and the edges of the graph. Since this proposition is used to claim the example 'comes from a directed graph', it should be either proved explicitly or clearly marked as an unproved observation that is not needed for the main theorem.","section":"Proposition 1.4"},{"comment":"The claim that the 12-dimensional polytope has no quadratic triangulation is not substantiated anywhere in the text. This property is part of the theorem statement, so it requires either a proof or a reproducible computational check. If it is not yet verified, it should be removed from the theorem and deferred to the promised future update.","section":"Theorem 1.3(3)"}],"minor_comments":[{"comment":"The paper is explicitly labeled 'preliminary report on research in progress' and states that it 'will be updated', yet Theorems 1.2 and 1.3 are presented as finished results. This tension should be resolved, for example by adding a clear 'research announcement' banner or moving the caveat to the statement of the theorems.","section":"Abstract and title"},{"comment":"The h*-vector is written as (1,2,3,4,5,3,2,1,0,0,0,0,0). While this is a valid 13-term vector for a 12-dimensional polytope, the trailing zeros may confuse readers into thinking the h*-polynomial has degree 7. Consider writing it explicitly as 1+2t+3t^2+4t^3+5t^4+3t^5+2t^6+t^7 or adding a sentence clarifying that h*_8 = ... = h*_12 = 0.","section":"Theorem 1.3(5)"},{"comment":"The directed bipartite graph in Figure 1 is not described in the text; the vertex labels 1 through 14 are drawn, but the edges are not listed. A short list of edges or an explicit mapping to the three non-standard vertices of Theorem 1.3 would make Proposition 1.4 meaningful to the reader.","section":"Figure 1"},{"comment":"The title and opening line of the full text contain typographical spacing errors ('LA TTICE POL YTOPES', 'VADYM' is fine, but 'POL YTOPES' should be 'POLYTOPES'). These are likely artifacts of the TeX source and should be corrected.","section":"Title"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a very short research announcement. The core mathematical content—explicit polytopes with non-log-concave h*-vectors—is plausible and, if substantiated, would be a valuable counterexample. However, the missing IDP and unimodularity proofs or reproducible computational certificates are load-bearing, and the paper's own text acknowledges that details will only come in a later version. I would suggest asking the authors to submit a full version with complete proofs or signed-off computational certificates before acceptance can be considered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: yes, this is a real new result, and the main question it answers is one people care about. The authors give two explicit lattice polytopes with non-log-concave h*-vectors, and the non-log-concavity is immediately checkable from the displayed h*-vectors. The 12-dimensional example is especially strong: a 0/1-polytope with only d+3 vertices, claimed to be unimodular (every triangulation unimodular), which would put it at the very top of the hierarchy. If the IDP claims hold, this cleanly refutes the natural log-concavity strengthening of the Ehrhart unimodality conjecture for IDP polytopes. The paper is honest about being a preliminary report, and the citation pattern looks appropriate, including giving credit to Ferroni for crosschecking.\n\nNow the soft spot, and it is load-bearing: the IDP property for Theorem 1.2 and the unimodular-triangulation property for Theorem 1.3 are not proved. The only support is the closing line that these can be verified with Polymake, Normaliz, Magma, or SageMath. No scripts, logs, or certificates are provided. For the 12-dimensional example, claiming every triangulation is unimodular is a strong statement, and I would want at least a Normaliz computation or a short argument before believing it. Since non-log-concavity of arbitrary lattice polytopes is old news, the entire relevance of these examples depends on the IDP/unimodular claims. That is a reproducibility gap, not an internal contradiction. The vertices are explicit, so the checks are feasible, but they have not been supplied in this version.\n\nThere are also a couple of minor things. The h*-vector of the 12-dimensional example has trailing zeros; that is fine, but it means the non-log-concavity is a coefficient-wise comparison at the third-to-last positive entry, not a subtle phenomenon. The machine-learning search is described only vaguely, which is acceptable for a research announcement. The paper says it will be updated with more details, so the lack of proofs may be temporary.\n\nMy overall take: this is a promising note with a genuinely interesting potential counterexample, but as it stands the main assertion is not independently verified in the text. I would not cite it as a confirmed theorem until the IDP certificates are available; I would cite it as a preprint if I needed to point to the question. For peer review, a serious referee should be able to verify the IDP claims from the explicit vertex data, so the paper deserves referee time rather than a desk reject. The recommendation should be conditional: ask the authors to provide the computational certificates or proofs before publication.","headline":"Explicit IDP polytopes with non-log-concave h*-vectors, answering an open question, but the load-bearing IDP claim is only asserted via software; the explicit data makes this worth refereeing with a request for certificates.","tokens_in":4046,"tokens_out":2107,"would_cite":false,"duration_ms":20959,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52B20","05A20","68T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two explicitly listed lattice polytopes with the integer decomposition property have $h^*$-vectors that are not log-concave, answering the stronger form of the Ehrhart unimodality question.","keywords":["Ehrhart polynomials","lattice polytopes","integer decomposition property","h*-vector","log-concavity","unimodality","0/1-polytopes","unimodular polytopes"],"falsifier":"Run an independent exact-arithmetic computation on the two displayed vertex sets, checking whether every lattice point of $kP$ is a sum of $k$ lattice points of $P$ for every $k$, and recompute the coefficients of $h^*_P(t)$; any failure of IDP or any coefficient different from the displayed $h^*$-vectors would settle the central claim false.","tokens_in":3172,"feed_emoji":"📐","tokens_out":12776,"duration_ms":74145,"temperature":0.7,"pith_summary":"This report constructs two concrete lattice polytopes, one in dimension 7 and one in dimension 12, and claims that both have the integer decomposition property (IDP) while their $h^*$-vectors fail log-concavity. The $h^*$-vector is the coefficient sequence of the numerator of the Ehrhart series, an encoding of the lattice-point counts of all dilations of the polytope. If the claims are right, the natural guess that every IDP polytope has a log-concave $h^*$-vector is false, even though the weaker and older unimodality conjecture remains untouched. Log-concavity implies unimodality for these $h^*$-vectors, so the examples show the stronger condition cannot be imposed. The paper is a preliminary report and says the listed properties were verified with standard computational software rather than by an included proof.","feed_headline":"Two explicit polytopes break log-concavity for IDP h*-vectors","feed_subtitle":"Seven- and twelve-dimensional IDP examples answer the stronger Ehrhart unimodality question.","key_machinery":"The key object is the $h^*$-polynomial: the numerator in $\\sum_{k\\ge 0} E_P(k)t^k = h^*_P(t)/(1-t)^{d+1}$, where $E_P(k)$ counts lattice points in the $k$-th dilation of a lattice polytope $P$ of dimension $d$. The hypothesis that makes the question sharp is IDP, the integer decomposition property, which says every lattice point of $kP$ is a sum of $k$ lattice points of $P$. The load-bearing examples are two explicit convex hulls with $d+3$ vertices each, chosen by an ongoing machine-learning-guided search; the paper's verification consists of computing their Ehrhart data and checking the coefficient inequality. For the 12-dimensional example, an additional mechanism is the claim that all triangulations are unimodular, which places the polytope in a very restrictive class and rules out explanations based on triangulation pathology.","core_discovery":"The paper's central claim is that the two vertex sets displayed in Theorems 1.2 and 1.3 are actual IDP polytopes whose Ehrhart-series numerators are not log-concave. In dimension 7 the $h^*$-polynomial is $h^*(t)=1+2t+3t^2+4t^3+5t^4+3t^5+2t^6+t^7$, and at $i=5$ the inequality $h^*_4h^*_6\\le (h^*_5)^2$ reads $10\\le 9$, which fails. In dimension 12 the same initial coefficient pattern appears, followed by five zeros, for a 0/1-polytope with 15 vertices that is claimed to have every triangulation unimodular. That second polytope is also described as the arc polytope of a small bipartite directed graph, meaning it is a convex hull of columns of an incidence matrix. The paper states that all these properties can be verified with computational software and says the examples answer a question posed in a recent survey; this version is explicitly described as work in progress.","pith_inferences":["Beyond the paper's results, one can test whether the recurring coefficients $1,2,3,4,5,3,2,1$ appear for IDP polytopes in dimensions below 7; finding one would show that dimension 7 is not the threshold.","Because the 12-dimensional example is an arc polytope of a bipartite graph, arc polytopes of other small bipartite graphs form a natural search space for further counterexamples, including possible non-unimodal $h^*$-vectors.","The paper's machine-learning-guided search converged to $d+3$ vertices but does not claim this vertex count is necessary; a reader might investigate whether this count is sufficient in some family of lattice polytopes."],"forward_implications":["Question 3.9(b) of the cited survey has a negative answer: IDP polytopes need not have log-concave $h^*$-vectors.","The original unimodality conjecture is not disproved: both displayed $h^*$-sequences are unimodal, so the paper kills only the stronger log-concavity version.","The 12-dimensional example shows that even a 0/1-polytope whose every triangulation is unimodular can have a non-log-concave $h^*$-vector.","Both examples have $d+3$ vertices, connecting the failure to the low-vertex-count family of 9 and 15 vertices that the authors' search converged to."],"supporting_citations":[{"why":"States Question 3.9(b), the precise question answered by the two counterexamples, and surveys the surrounding unimodality problems.","marker":"[3]"},{"why":"Formulates the underlying unimodality question for integrally closed lattice polytopes that motivates the stronger log-concavity version.","marker":"[7]"},{"why":"Supplies the standard definitions of Ehrhart polynomials and $h^*$-polynomials used to state and check the examples.","marker":"[1]"},{"why":"Provides the arc and root polytope construction used to identify the 12-dimensional example as an arc polytope of a bipartite graph.","marker":"[4]"},{"why":"Defines the hierarchy of unimodular polytopes in which the 12-dimensional example is claimed to be very special.","marker":"[6]"}],"fun_headline_variants":["IDP polytopes fail log-concavity in dimensions 7 and 12","Explicit IDP polytopes with non-log-concave h*-vectors","Counterexamples to IDP log-concavity in dimensions 7 and 12","Answering Ferroni-Higashitani: IDP h*-vectors not log-concave","IDP polytopes: non-log-concave h*-vectors in dims 7 and 12"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the software computations certifying the integer decomposition property (and, in the 12-dimensional case, the unimodularity of every triangulation) are correct, since the paper provides no code, logs, or independent certificate for that assertion.","fun_headline_variants_meta":{"raw":{"variants":["IDP polytopes fail log-concavity in dimensions 7 and 12","Explicit IDP polytopes with non-log-concave h*-vectors","Counterexamples to IDP log-concavity in dimensions 7 and 12","Answering Ferroni-Higashitani: IDP h*-vectors not log-concave","IDP polytopes: non-log-concave h*-vectors in dims 7 and 12"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002312,"raw_usage":{"total_tokens":8909,"prompt_tokens":924,"completion_tokens":7985,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":7866}},"tokens_in":540,"tokens_out":7985,"duration_ms":49013,"temperature":1.0,"reasoning_tokens":7866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:22:51.988456+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent exact-arithmetic computation on the two displayed vertex sets, checking whether every lattice point of $kP$ is a sum of $k$ lattice points of $P$ for every $k$, and recompute the coefficients of $h^*_P(t)$; any failure of IDP or any coefficient different from the displayed $h^*$-vectors would settle the central claim false.","supporting_citations":[{"cited_title":"Unimodality questions for integrally closed lattice polytopes","cited_arxiv_id":null,"evidence_quote":"Formulates the underlying unimodality question for integrally closed lattice polytopes that motivates the stronger log-concavity version."},{"cited_title":"Computing the continuous discretely","cited_arxiv_id":null,"evidence_quote":"Supplies the standard definitions of Ehrhart polynomials and $h^*$-polynomials used to state and check the examples."},{"cited_title":"Piechnik, and Francisco Santos","cited_arxiv_id":null,"evidence_quote":"Provides the arc and root polytope construction used to identify the 12-dimensional example as an arc polytope of a bipartite graph."},{"cited_title":"Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem","cited_arxiv_id":"2405.13431","evidence_quote":"Defines the hierarchy of unimodular polytopes in which the 12-dimensional example is claimed to be very special."}],"review_version":1}