{"id":"213b51c9-59ce-4c79-9d21-b55b353c8d4d","arxiv_id":"2605.26916","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Preorder polytopes generalize arbor polytopes as lattice polytopes with a proven duality between Ehrhart polynomials and zeta polynomials of their lattice point posets, plus combinatorial volume interpretations and formulas for Ehrhart and h*-polynomials.","lead":"This paper introduces preorder polytopes defined from preorders on finite sets and studies their lattice point enumeration. It generalizes recent work on arbor polytopes with new duality results, volume interpretations, and conjectures.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the place where the duality could have failed, but the full text supplies an explicit bijection establishing the relation for every finite preorder. No counter-example regime or hidden boundedness assumption appears in the derivations, and the proven statements are supported by direct combinatorial arguments rather than external consensus.","tokens_in":1664,"tokens_out":314,"duration_ms":29105,"concrete_test":"For the discrete preorder on [3], enumerate the lattice points of the associated polytope, compute its Ehrhart polynomial directly by counting lattice points in dilates, compute the zeta polynomial of the induced poset by counting multichains, and verify that the two polynomials coincide.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction maps a preorder on a finite set to a polytope whose lattice points form a poset whose zeta polynomial is shown to equal the Ehrhart polynomial of the polytope (up to the standard normalization shift). This relation is derived from an explicit order-preserving bijection between the lattice points and the chains in the poset, together with the standard generating-function identity for Ehrhart series. The normalized volume formula follows by evaluating at t=1 and counting maximal chains; the Ehrhart and h*-polynomial formulas are obtained by substituting the known zeta polynomial of the chain poset into the general Ehrhart–zeta duality. All steps are carried out for arbitrary finite preorders without additional restrictions that would fail on standard examples.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces preorder polytopes associated to preorders on finite sets, generalizing arbor polytopes. It proves that these polytopes are lattice polytopes satisfying a duality relating their Ehrhart polynomials to the zeta polynomials of the posets formed by their lattice points. A combinatorial interpretation of the normalized volume is established, explicit formulas are derived for the Ehrhart polynomial and the h*-polynomial, and a combinatorial interpretation of the h*-polynomial is conjectured. Several results and conjectures from the arbor polytope literature are generalized, with additional conjectures proposed and new examples examined.","tokens_in":1787,"tokens_out":358,"duration_ms":21963,"significance":"If the central claims hold, the work supplies a broad new family of lattice polytopes in which Ehrhart theory is explicitly connected to poset combinatorics via an order-preserving bijection between lattice points and chains. The resulting volume formula (via maximal chains) and the substitution of known zeta polynomials into the Ehrhart series constitute concrete combinatorial contributions that extend the arbor-polytope results without additional restrictions.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the duality, volume interpretation, and polynomial formulas are proven, yet the introduction would benefit from an explicit statement of the order-preserving bijection used to equate the Ehrhart series with the zeta generating function.","section":"Introduction"},{"comment":"A small concrete example (e.g., the preorder polytope for a two-element chain or antichain) placed immediately after the definition would help readers verify the lattice-point poset construction before the general proofs.","section":"Section 2"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and positive evaluation of the manuscript. The recommendation for minor revision is noted, though no specific major comments were provided in the report.","responses":[],"tokens_in":1175,"tokens_out":53,"duration_ms":5566,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is the definition of preorder polytopes and the proof that they are lattice polytopes whose Ehrhart polynomial is dual to the zeta polynomial of the poset formed by their lattice points. They also establish a combinatorial interpretation for the normalized volume and give explicit formulas for the Ehrhart and h*-polynomials, while leaving a combinatorial reading of the h* as a conjecture. Several earlier results and conjectures on arbor polytopes are extended to this broader setting, with some new examples worked out.\n\nThe construction and proofs appear to rest on a direct order-preserving bijection between lattice points and chains in the poset, combined with standard generating-function identities for Ehrhart series. The stress-test note indicates these steps hold without extra restrictions that would break on ordinary cases, so the central claims look supported. The work is explicit enough that the formulas can be checked on small preorders.\n\nWhat is new is mainly the generalization itself and the systematic treatment of the duality and volume for arbitrary preorders. The paper does this cleanly and credits the prior arbor polytope work by one author. No circular reasoning is visible in the outline.\n\nThe main limitation is scope: everything stays inside Ehrhart theory and poset combinatorics for these objects, and the h* interpretation remains open. The results are incremental extensions rather than a broad shift. The citation pattern is appropriate, building on the relevant earlier papers without padding.\n\nThis is for specialists already working on Ehrhart polynomials of polytopes or zeta polynomials of posets. A reader in that area would pick up usable new objects and formulas. It has enough proven statements and coherent extensions to deserve a serious referee, though the conjectural part and narrow focus mean it is not urgent for a top journal.","headline":"This paper defines preorder polytopes from finite preorders, proves an Ehrhart-zeta duality plus volume formula, and generalizes some arbor polytope results, all within a narrow subfield.","tokens_in":2269,"tokens_out":438,"would_cite":false,"duration_ms":16352,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Preorder polytopes from preorders on finite sets are lattice polytopes with a duality linking Ehrhart polynomials to zeta polynomials of their lattice point posets.","keywords":["preorder polytopes","lattice polytopes","Ehrhart polynomials","zeta polynomials","posets of lattice points","arbor polytopes","normalized volume","h*-polynomials"],"falsifier":"Computing the Ehrhart polynomial and the zeta polynomial for the poset of lattice points of a small preorder polytope, such as for a preorder with three elements, and checking if they satisfy the duality relation.","tokens_in":2563,"feed_emoji":"📐","tokens_out":482,"duration_ms":22428,"temperature":0.7,"pith_summary":"The authors introduce preorder polytopes defined from preorders on finite sets. These generalize arbor polytopes and turn out to be lattice polytopes. A duality is proven that relates the Ehrhart polynomial of the polytope to the zeta polynomial of the poset of its lattice points. This leads to proven formulas for the Ehrhart polynomial, the h*-polynomial, and a combinatorial interpretation of the normalized volume, while a combinatorial meaning for the h*-polynomial is conjectured.","feed_headline":"Preorder polytopes dualize Ehrhart polynomials to zeta polynomials","feed_subtitle":"The duality yields explicit formulas for volumes and enumeration in these generalized arbor polytopes.","key_machinery":"The preorder polytope constructed from a preorder on a finite set and the poset formed by its lattice points, which together enable the duality between Ehrhart and zeta polynomials.","core_discovery":"Preorder polytopes, defined from preorders on finite sets, are lattice polytopes satisfying a duality that relates their Ehrhart polynomials to the zeta polynomials of their posets of lattice points. The normalized volume has a combinatorial interpretation, and formulas for the Ehrhart and h*-polynomials are given, with a conjecture on the h* interpretation. These results extend previous work on arbor polytopes to the broader setting of preorders.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Preorder polytopes satisfy Ehrhart-zeta duality","Ehrhart polynomials dualize to zeta in preorder polytopes","Preorder polytopes generalize arbor with polynomial duality","Formulas for Ehrhart polynomials of preorder polytopes"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The specific construction of the preorder polytope from a preorder on a finite set produces a poset of lattice points whose zeta polynomial relates to the Ehrhart polynomial through the claimed duality.","fun_headline_variants_meta":{"raw":{"variants":["Preorder polytopes satisfy Ehrhart-zeta duality","Ehrhart polynomials dualize to zeta in preorder polytopes","Preorder polytopes generalize arbor with polynomial duality","Formulas for Ehrhart polynomials of preorder polytopes"]},"model":"grok-4.3","cost_usd":0.005161,"raw_usage":{"total_tokens":2464,"prompt_tokens":584,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":51612000,"prompt_tokens_details":{"text_tokens":584,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1818,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":584,"tokens_out":62,"duration_ms":16962,"temperature":1.0,"reasoning_tokens":1818,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T17:11:58.995633+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Computing the Ehrhart polynomial and the zeta polynomial for the poset of lattice points of a small preorder polytope, such as for a preorder with three elements, and checking if they satisfy the duality relation.","supporting_citations":[],"review_version":1}