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REVIEW 2 major objections 1 minor 15 references

Explicit Formulas and Unimodality Phenomena for General Position Polynomials

T0 review · 2 major / 1 minor · reviewed 2026-07-15 · grok-4.5

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Pith's one-line read Balanced complete multipartite graphs with part size at most 4 have log-concave, unimodal general position polynomials; larger parts produce counterexamples.

desk verdict Abstract-only multipartite unimodality claims for ψ; the cached body is the wrong paper, so the r≤4 threshold and formulas cannot be audited. read the letter →

arxiv 2603.06930 v3 pith:CYQGG5PW submitted 2026-03-06 math.CO cs.DM

Bilal Ahmad Rather This is my paper
classification math.COcs.DM MSC 05C3105C1205A20
keywords generalpositionpolynomiallog-concavityunimodalitycompletemultipartitegraphscoronaproductgeodesicgraphpolynomials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The general position problem asks for the largest set of vertices in a graph so that no three lie on a common shortest path. Its counting version, the general position polynomial, enumerates every such set by size. This paper gives explicit formulas for that polynomial on complete multipartite graphs, then specializes to the balanced case where every part has the same size r. When r is at most 4 the polynomial is log-concave and unimodal no matter how many parts are present; when r is larger, concrete counterexamples show both properties can fail. The same analysis is carried out for corona products G joined with a pendant vertex on every vertex of G, where unimodality is sometimes inherited from G and sometimes lost, including on complete bipartite and multipartite base graphs. The results place general position polynomials in the same family of combinatorial counting polynomials that already exhibit log-concavity and unimodality phenomena.

What carries the argument

The general position polynomial ψ(G), the ordinary generating function whose coefficient of x^k counts the number of general-position vertex sets of size k (sets in which no three vertices lie on a common geodesic). Explicit product and sum formulas for ψ on complete multipartite graphs turn the log-concavity and unimodality questions into concrete coefficient comparisons.

What would settle it

Compute the coefficient sequence of ψ(K_{5,5,...,5}) for a moderate number of parts (say 6–10) and check whether the sequence is log-concave; a single triple of consecutive coefficients violating a_i^{2} ≥ a_{i-1}a_{i+1} would confirm the claimed failure for r = 5, while universal success would challenge the counterexample claim.

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Extended reading notes

Core claim

For every balanced complete multipartite graph K_{r,...,r} the general position polynomial is log-concave and unimodal whenever the common part size satisfies r ≤ 4, while for every larger fixed r there exist numbers of parts that make the polynomial fail log-concavity or unimodality. Explicit closed formulas for the polynomial on all complete multipartite graphs supply the combinatorial engine for these statements, and parallel statements hold for certain corona products.

Load-bearing premise

The combinatorial description of every general-position set inside complete multipartite graphs and inside corona products is complete; if a family of geodesic configurations was overlooked, the closed formulas and the unimodality proofs collapse.

Editorial extensions

If this is right

  • Closed formulas for ψ on complete multipartite graphs become available for direct coefficient extraction and asymptotic analysis.
  • The threshold r = 4 cleanly separates the log-concave regime from the non-log-concave regime for balanced multipartite graphs.
  • Corona products preserve unimodality of ψ for some base-graph families but destroy it for complete bipartite and multipartite bases.
  • General position polynomials can be studied with the same log-concavity toolkit already used for independence and matching polynomials.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same coefficient formulas may decide whether real-rootedness (a stronger property than log-concavity) holds for r ≤ 4.
  • Similar part-size thresholds could govern other geodesic-avoiding counting polynomials on multipartite graphs.
  • Once the multipartite case is settled, the next natural test objects are complete multipartite graphs with unequal part sizes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The abstract of arXiv:2603.06930 claims explicit formulas for the general position polynomial ψ of complete multipartite graphs, a combinatorial description of general-position sets, log-concavity and unimodality of ψ(K_{r,…,r}) for every number of parts when the part size satisfies r≤4 (with counterexamples for larger r), and results on retention or failure of unimodality under the corona operation G∘K_1. The supplied full manuscript text, however, is an entirely different paper (LLM2SMT, arXiv:2603.06931) on constructing a DPLL(T)-style SMT solver for QF_UF via an LLM coding agent; it contains no graph-theoretic definitions, no multipartite formulas, no coefficient tables, no geodesic arguments, and no unimodality proofs.

Significance. If the abstract claims were substantiated they would be a solid contribution to the enumerative theory of general-position sets, supplying closed forms and a clean threshold (r=4) separating unimodal from non-unimodal behaviour, together with corona examples that parallel classical position polynomials. Because the body does not match the abstract, none of these strengths can be verified or credited.

major comments (2)
  1. [Full manuscript text] The entire body (Sections 1–6, Tables 1–2, Figure 1, Listings, and all references) belongs to the unrelated LLM2SMT paper. Consequently every load-bearing claim of the abstract—explicit multipartite formulas, the r≤4 log-concavity/unimodality proofs, the larger-r counterexamples, and the corona analysis—is absent and cannot be audited for correctness of geodesic descriptions or coefficient arguments.
  2. [Abstract vs. body] No statement, lemma, or table establishing the combinatorial characterisation of general-position sets in complete multipartite graphs appears. That characterisation is the indispensable premise for every closed form and every unimodality assertion; its absence renders the central results uninspectable.
minor comments (1)
  1. [Abstract] Even the abstract contains a typographical inconsistency (“We describe” capitalised mid-sentence) and a grammatical slip (“counterexamples exists”).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: supplied full text is a mismatched paper (LLM2SMT) containing none of the claimed multipartite formulas or unimodality proofs.

full rationale

The abstract asserts explicit formulas for the general-position polynomial of complete multipartite graphs, log-concavity/unimodality of ψ(K_{r,…,r}) for r≤4 (with counter-examples for larger r), and retention/failure of unimodality under corona products. The CACHEABLE full-text block, however, is the unrelated manuscript arXiv:2603.06931 (an empirical case study of an LLM-written SMT solver). That text contains no combinatorial descriptions of general-position sets, no generating-function identities, no coefficient sequences, and no proofs that could be inspected for self-definitional reductions, fitted-parameter “predictions,” or load-bearing self-citations. Because no derivation chain for the abstract’s claims is present, no circular step can be exhibited by quotation. Per the analyzer rules this is an honest non-finding; score remains 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

Abstract-only review of a pure combinatorics paper. No free parameters or invented physical entities. Background axioms are the standard definitions of geodesics, general position sets, the polynomial ψ(G), log-concavity, unimodality, complete multipartite graphs, and the corona product. All load-bearing content is mathematical definition and claimed derivation, not fitted constants.

assumptions (4)
  • domain assumption A set S of vertices is in general position if no three vertices of S lie on a common geodesic.
    Definition of the objects counted by ψ(G); invoked throughout the abstract’s claims.
  • domain assumption The general position polynomial ψ(G) enumerates general position sets by cardinality.
    Counting refinement stated in the abstract as the paper’s main object of study.
  • standard math Log-concavity and unimodality of coefficient sequences are the standard combinatorial notions (a_k^2 ≥ a_{k-1}a_{k+1}; coefficients rise then fall).
    Shape properties claimed for ψ(K_{r,...,r}) when r≤4.
  • standard math Complete multipartite and corona graph constructions are the usual ones (parts independent sets with all cross edges; G∘K1 attaches a pendant leaf to each vertex of G).
    Graph classes for which formulas and unimodality statements are asserted.

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Cite this review

Pith. "Pith review of Explicit Formulas and Unimodality Phenomena for General Position Polynomials." pith.science (2026). https://pith.science/paper/CYQGG5PW

@misc{pith2026260306930,
  author       = {Pith},
  title        = {Pith review of: Explicit Formulas and Unimodality Phenomena for General Position Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYQGG5PW}},
  note         = {Machine review of arXiv:2603.06930}
}
abstract

The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $\psi(G)$, asks for all such possible sets. In this paper, We describe general position sets for several classes of graphs and provide explicit formulas for the general position polynomials of complete multipartite graphs. We specialize to balanced complete multipartite graphs and show that for part size $r\le 4$, the polynomial $\psi(K_{r,\dots,r})$ is log-concave and unimodal for all numbers of parts, while for larger $r$, counterexamples show that these properties fail. Finally, we analyze the corona $G\circ K_1$ and prove that unimodality of $\psi(G)$ is retained for some classes, and counterexamples exists for complete bipartite and complete multipartite graphs. The results verify the analogy between general position polynomials and classical position-type parameters, and establish balanced multipartite graphs and coronas as potential subjects for further investigation.

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Works this paper leans on

15 extracted references · 6 canonical work pages

  1. [1]

    Producing shorter congruence closure proofs in a state-of-the-art SMT solver

    1 Bruno Andreotti and Haniel Barbosa. Producing shorter congruence closure proofs in a state-of-the-art SMT solver. In Yu-Fang Chen, Thomas P. Jensen, and Ondrej Lengál, editors, Verification, Model Checking, and Abstract Interpretation - 27th International Conference, VMCAI 2026, Rennes, France, January 12-13, 2026, Proceedings, volume 16417 ofLecture No...

    work page doi:10.1007/978-3-032-15700-3 2026

  2. [2]

    3 Mantas Baksys, Stefan Zetzsche, Olivier Bouissou, and Sean B

    Large language model. 3 Mantas Baksys, Stefan Zetzsche, Olivier Bouissou, and Sean B. Holden. MINIF2F-DAFNY: LLM-guided mathematical theorem proving via auto-active verification, 2026.arXiv:2512. 10187. 4 Haniel Barbosa. Challenges in SMT proof production and checking for arithmetic reasoning (invited paper). In Erika Ábrahám and Thomas Sturm, editors,Pro...

    2026

  3. [3]

    doi:10.1007/978-3-030-99524-9_24. 6 Haniel Barbosa, Andrew Reynolds, Gereon Kremer, Hanna Lachnitt, Aina Niemetz, Andres Nötzli, Alex Ozdemir, Mathias Preiner, Arjun Viswanathan, Scott Viteri, Yoni Zohar, Cesare Tinelli, and Clark W. Barrett. Flexible proof production in an industrial-strength SMT solver. In Jasmin Blanchette, Laura Kovács, and Dirk Patti...

    doi:10.1007/978-3-030-99524-9_24 2022

  4. [4]

    9 Robert Brummayer and Armin Biere

    doi:10.1007/978-3-031-65627-9_7. 9 Robert Brummayer and Armin Biere. Fuzzing and delta-debugging SMT solvers. InProceedings of the 7th International Workshop on Satisfiability Modulo Theories, SMT ’09, page 1–5. ACM, August 2009.doi:10.1145/1670412.1670413. 10 Leonardo de Moura, Harald Rueß, and Maria Sorea. Lazy theorem proving for bounded model checking...

    doi:10.1007/978-3-031-65627-9_7 2009

  5. [5]

    doi:10.1007/3-540-45620-1_35. M. Janota and M. Olšák 9 11 Leonardo Mendonça de Moura and Nikolaj Bjørner. Model-based theory combination.Electr. Notes Theor. Comput. Sci., 198(2):37–49, 2008.doi:10.1016/j.entcs.2008.04.079. 12 Leonardo Mendonça de Moura and Nikolaj Bjørner. Z3: an efficient SMT solver. InTools and Algorithms for the Construction and Analy...

    work page doi:10.1007/3-540-45620-1_35 2008

  6. [6]

    Mollick, Hila Lifshitz-Assaf, Katherine Kellogg, Saran Rajendran, Lisa Krayer, François Candelon, and Karim R

    13 Fabrizio Dell’Acqua, Edward McFowland, Ethan R. Mollick, Hila Lifshitz-Assaf, Katherine Kellogg, Saran Rajendran, Lisa Krayer, François Candelon, and Karim R. Lakhani. Navigating the jagged technological frontier: Field experimental evidence of the effects of AI on knowledge worker productivity and quality.SSRN Electronic Journal, 2023.doi:10.2139/ssrn...

    doi:10.2139/ssrn.4573321 2023

  7. [7]

    15 Katalin Fazekas, Aina Niemetz, Mathias Preiner, Markus Kirchweger, Stefan Szeider, and Armin Biere

    URL:http://doi.acm.org/10.1145/1066100.1066102, doi:10.1145/1066100.1066102. 15 Katalin Fazekas, Aina Niemetz, Mathias Preiner, Markus Kirchweger, Stefan Szeider, and Armin Biere. IPASIR-UP: User Propagators for CDCL. In Meena Mahajan and Friedrich Slivovsky, editors,26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2...

    doi:10.1145/1066100.1066102 2023

  8. [8]

    doi:10.4230/LIPIcs.SAT.2023.8

    Schloss Dagstuhl – Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.SAT.2023.8. 16 Tony Feng, Trieu Trinh, Garrett Bingham, Jiwon Kang, Shengtong Zhang, Sang hyun Kim, Kevin Barreto, Carl Schildkraut, Junehyuk Jung, Jaehyeon Seo, Carlo Pagano, Yuri Chervonyi, Dawsen Hwang, Kaiying Hou, Sergei Gukov, Cheng-Chiang Tsai, Hyunwoo Choi, Youngbeom Jin, Wei-Yu...

    doi:10.4230/lipics.sat.2023.8 2023

Show all 15 references
  1. [9]

    19 João Marques-Silva and Karem A

    arXiv:2601.14027. 19 João Marques-Silva and Karem A. Sakallah. GRASP: A search algorithm for propositional satisfiability.IEEE Transactions on Computers, 48(5):506–521,

  2. [10]

    Programming with triggers

    20 MichałMoskal. Programming with triggers. InInternational Workshop on Satisfiability Modulo Theories (SMT), pages 20–29. ACM, 2009.doi:10.1145/1670412.1670416. 21 Leonardo de Moura and Sebastian Ullrich.The Lean 4 Theorem Prover and Program- ming Language, page 625–635. Spri...

    2009 doi

  3. [11]

    357079,doi:10.1145/357073.357079

    URL:http://doi.acm.org/10.1145/357073. 357079,doi:10.1145/357073.357079. 23 Robert Nieuwenhuis and Albert Oliveras.DPLL(T) with Exhaustive Theory Propagation and Its Application to Difference Logic, page 321–334. Springer Berlin Heidelberg,

    doi

  4. [12]

    24 Robert Nieuwenhuis and Albert Oliveras

    doi:10.1007/11513988_33. 24 Robert Nieuwenhuis and Albert Oliveras. Fast congruence closure and extensions.Inf. Comput., 205(4):557–580, 2007.doi:10.1016/J.IC.2006.08.009. 25 Robert Nieuwenhuis, Albert Oliveras, and Cesare Tinelli. Solving SAT and SAT modulo theories: From an ...

    2007 doi

  5. [13]

    27 Kaustubh Saini

    doi:10.1007/3-540-48683-6_39. 27 Kaustubh Saini. Software engineer title will go away says Boris Cherny. https://www. finalroundai.com/blog/software-engineer-title-go-away, February

    doi

  6. [14]

    29 Raymond M

    doi: 10.3233/SAT190034. 29 Raymond M. Smullyan.First-Order Logic. Springer Berlin Heidelberg, 1968.doi:10.1007/ 978-3-642-86718-7. 30 Michael Sutton, Adam Greene, and Pedram Amini.Fuzzing: brute force vulnerability discovery. Pearson Education,

    1968 doi

  7. [15]

    Jiang, Wenda Li, Markus N

    31 Yuhuai Wu, Albert Q. Jiang, Wenda Li, Markus N. Rabe, Charles Staats, Mateja Jamnik, and Christian Szegedy. Autoformalization with large language models, 2022.arXiv:2205.12615. 32 A. Zeller and R. Hildebrandt. Simplifying and isolating failure-inducing input.IEEE Transac- t...

    2022 doi arXiv

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