{"id":"3513fce6-b274-4765-8854-af896a1ac9a5","arxiv_id":"2607.26690","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Three inducibility conjectures fail: min FAS is always a minimal 3-cycle hitter; α*=2/3 does not imply 3-inducibility; Paley(43) has α*>3/5 yet is not 5-inducible.","lead":"The paper refutes three conjectures about which preference tournaments can arise from three or five voters, and gives the first modest-size explicit tournament that five voters cannot induce. It tightens the link between Kemeny ranking, feedback arc sets, and majority inducibility.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The Paley(43) non-5-inducibility claim rests on an unchecked billion-scale shell screen whose completeness is the single load-bearing residual risk.","rationale":"The reader correctly isolates the billion-scale level-≤1 screen (and its supporting MAS DP) as the weakest assumption behind the strongest claim. The rest of the paper—Thm 2.1, the explicit m=3/5/7/9 counter-examples to Milosz et al., the A(3) census, and the counting bounds—is either analytic or small-scale ILP with dual-solver cross-checks and does not threaten the headline. Because the authors already ship a reproducibility package, exact-arithmetic claims, and multiple internal sanity checks, the residual risk justifies CONDITIONAL rather than REJECT or ACCEPT; no stricter adjustment is warranted. My concrete test simply makes the same residual risk falsifiable by an independent party.","tokens_in":31507,"tokens_out":653,"duration_ms":39068,"concrete_test":"Independently re-implement the Aut-quotient meet-in-the-middle DP of App. E and confirm MAS(Paley(43))=543; then re-run (or spot-audit via the public repo) the razor+exact DB screen of App. F and verify that the minimum overlap over all reported candidate pairs is still ≥1 (ideally reproducing the stated min of 68). If either MAS differs or a DB-disjoint pair appears, Theorem 7.1 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 7.1’s non-inducibility half is analytic once two facts are granted: (i) MAS(P)=543 (orbit subset-DP, App. E), which forces slack5=6 and therefore at least two voters in the level-≤1 shell (§7.1), and (ii) the co-backing lemma (Lemma 7.2), which forces those two voters’ double-back sets to be disjoint. The screen (§7.3, App. F) then asserts that among the ~1.66×10^9 level-≤1 orders (1.84 M Aut-orbits), after razor restriction to the 538 triangles in W=[23] and one-sided Aut reduction, none of the ~4.38×10^9 candidate (representative, pool) pairs is DB-disjoint (reported min overlap 68). That negative exhaustive search is not machine-checked; it is certified only by the authors’ C++ pipeline, positive-detection count match, and smaller-q MAS gauntlet. A silent under-enumeration of the shell, a bug in DB-set construction, or an incomplete dangerous-pair filter would leave open the possibility that a DB-disjoint level-≤1 pair exists and that P is 5-inducible after all, collapsing the strict refutation of A(5) and the explicit N(5)≤43 bound. The analytic pillars (slack arithmetic, Lemma 7.2, razor soundness) are solid; the enumeration is the sole load-bearing soft spot.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies majority tournaments induced by few voters and the reduction of the Kemeny median problem to weighted feedback arc set. It strengthens Milosz–Hamel–Pierrot by proving that in every tournament a minimum FAS is a minimal 3-cycle hitting set (Theorem 2.1), then recovers their 3-voter weighted statement. It refutes both of their conjectures: the “only 3-cycle arcs reverse” property fails for every odd m≥5 (minimal counterexamples at (7,4), (9,4), (5,6)), and FAS=HS3 fails for a unique 3-inducible regular 11-vertex tournament T* (and six self-converse ones). It refutes the Shepardson–Tovey threshold conjecture A(m): for m=3 exactly on the boundary α*=2/3 (1,013 ten-vertex examples; unique regular cA3 on 11 vertices), and for m=5 strictly via Paley(43), which has α*=181/301>3/5 yet is not 5-inducible (Theorem 7.1). Counting via completion uniqueness improves N(5)≤38 non-constructively and sharpens Bachmeier et al.’s bounds for all odd m≤21; Paley(43) gives the first explicit modest-size witness N(5)≤43. Extensive ILP/LP censuses, obstacle catalogues at n=9, and a reproducibility package support the computational claims.","tokens_in":31845,"tokens_out":1578,"duration_ms":41866,"significance":"If correct, the work cleanly settles three open conjectures in tournament inducibility and Kemeny structure, supplies the first explicit non-5-inducible tournament near the counting scale (43 vs prior ~6×10^8), and improves all ten majoritarian-expressiveness bounds of Bachmeier et al. The analytic pillars—Theorem 2.1, co-backing (Lemma 7.2), slack forcing, completion uniqueness (Proposition 6.1)—are written proofs. Computational claims are backed by dual ILP encodings (CPLEX + CP-SAT), exact rational arithmetic, Aut-reduced orbit DP for MAS, a public GitHub verifier, and multiple sanity checks (MAS gauntlet, positive-detection count match, Aut-equivariance). That combination of refutation-by-counterexample, hereditary censuses, and reproducible large-scale search is a genuine advance for cs.DM / social-choice combinatorics.","major_comments":[{"comment":"Theorem 7.1’s non-inducibility half is analytic once MAS(P)=543 and Lemma 7.2 are granted, but the decisive step is the negative exhaustive screen (§7.3, Appendix F): ~1.66×10^9 level-≤1 orders, razor to 538 triangles in W=[23], one-sided Aut reduction, ~4.38×10^9 candidate pairs, reported minimum DB-overlap 68. Completeness is certified only by the authors’ pipeline plus sanity checks (positive-detection equality, level-0 exhaustive pair check, Aut-equivariance over 903 maps). For a load-bearing claim that strictly refutes A(5) and gives the explicit N(5)≤43 bound, the manuscript should state more explicitly what a third party must re-run or re-derive to accept the empty-screen result (e.g., independent recomputation of the orbit census identity, hash/checksum of the dangerous-pair stream, or a short machine-checkable certificate of shell cardinality). The analytic reductions and razor","section":"§7.3, Appendix E–F, Theorem 7.1"},{"comment":"Counterexample 4.2 and Theorem 4.1 assert FAS(T*)=17>16=HS3(T*) on a unique 3-inducible regular 11-vertex tournament, with FAS=HS3 for all n≤10. The ILP formulations (Appendix B.2–B.3) and Aut-orbit of the 9 minimum hitting sets are described, but the paper should record the explicit 16-arc hitting set (or a stable identifier into McKay’s catalogue plus the witness order already given) in the main text or a short appendix table so the FAS–HS3 gap is checkable without re-solving the ILP. The same applies to the six self-converse violators (Figure 4 / Appendix C): inducing profiles are given, but the numerical FAS and HS3 values per tournament are not tabulated.","section":"§4, Counterexample 4.2, Appendix B.2–B.3, C"}],"minor_comments":[{"comment":"Notation for Im,t and the margin hierarchy is introduced early and used in Conjecture 8.1; a one-line display of the chain I_{m,1}⊆⋯⊆I_{m,m} near the first use would help readers who skip the definitions block.","section":"§1 Definitions / Conjecture 8.1"},{"comment":"Figure 3 and Figure 5 are clear; Figure 4’s six panels are dense. Adding the FAS and HS3 numbers in each panel caption (or a small table) would make the Conjecture 2 refutation self-contained at a glance.","section":"Figure 4"},{"comment":"Table 2 improves Bachmeier et al.; citing the exact multiset count versus their Stirling estimate in one sentence in the table caption would clarify why three entries already improve under “exact” alone.","section":"Table 2, Theorem 6.3"},{"comment":"The phrase “strike-a-voter argument” (§8.2 item 4, Appendix D) is used before it is fully glossed; a brief parenthetical on first use would help.","section":"§8.2, Appendix D"},{"comment":"Minor typography: “aminimum-weight” (missing space) in Counterexample 3.1; “| solution” stray bar in the A(m) integrality-gap paragraph; “gained g≥ 4” double quote artifact in Appendix G.4.","section":"§3, §5, Appendix G.4"},{"comment":"The AI-use table is unusually transparent and welcome; consider moving the one-line “full responsibility” statement into the acknowledgments so the contribution table can stay in a supplement if the journal prefers.","section":"Motivation and statement of AI use"}],"recommendation":"minor_revision","confidential_remarks":"The central mathematical contributions are solid and the refutations are the right kind of result for a discrete-math / combinatorics venue. The only material residual is trust in a large negative enumeration; with the public repo and the checks already listed, minor revision asking for a clearer third-party verification path is proportionate—not a demand to replace the screen by a hand proof. Fit for cs.DM is excellent. No citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: they kill three published conjectures with explicit objects, strengthen the Milosz–Hamel–Pierrot 3-cycle theorem to every tournament (Thm 2.1 is short and clean), and replace a ~10^8-vertex non-5-inducible construction by Paley(43) sitting strictly above the 3/5 predictability line.\n\nWhat is new and solid: the unweighted FAS = minimal HS3 fact; concrete counterexamples to both MHP conjectures (including the unique regular T* at n=11 with FAS=17>HS3=16); the full A(3) boundary census (1013 tournaments at n=10, unique regular cA3); improved counting bounds on N(m) for all odd m≤21 via completion uniqueness; and the Paley argument that combines thin slack, the co-backing lemma, and a shell screen. The analytic pieces (slack forcing, Lemma 7.2, razor soundness, Prop 6.1) read correctly. Appendices give ILP formulations, dual solvers, exact arithmetic, and a public repo claim. Citation pattern is appropriate.\n\nThe soft spot is real but proportional: non-5-inducibility of Paley(43) rests on an exhaustive screen of ~1.66e9 level-≤1 orders (reduced by Aut and a window). That is not machine-checked; it is certified by their pipeline, positive-detection match, and smaller-q MAS checks. A silent undercount could in principle leave a DB-disjoint pair. Everything else in Thm 7.1 is analytic once MAS=543 and co-backing are granted. I would want the referees to poke the code and the orbit-DP, not to treat the claim as free-floating.\n\nThis is for people who care about Kemeny structure, McGarvey numbers, and tournament inducibility. Subfield-important, not a general-audience splash. It deserves serious referee time. I would bring it to reading group and cite the structural theorem and the Paley bound if I touch this area.","headline":"Clean structural theorem plus three real conjecture refutations, capped by an explicit Paley(43) non-5-inducibility proof whose only soft spot is a large but carefully cross-checked enumeration.","tokens_in":32564,"tokens_out":541,"would_cite":true,"duration_ms":18137,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","91B12","68Q17"],"pacs":[],"model":"grok-4.5","headline":"High predictability does not guarantee that a tournament is the majority of three or five voters.","keywords":["tournaments","Kemeny median","feedback arc set","McGarvey number","predictability","majority dimension","Paley tournament","inducibility"],"falsifier":"Exhibit either a five-voter profile whose majority tournament is Paley(43), or a pair of level-≤1 linear orders on 43 vertices whose sets of double-backed cyclic triangles are disjoint.","tokens_in":32338,"feed_emoji":"⚔️","tokens_out":1054,"duration_ms":18034,"temperature":0.7,"pith_summary":"The paper studies when a tournament (a complete set of pairwise rankings) can arise as the majority outcome of a few voters. A natural numerical condition, predictability, says every arc is supported by a large enough supermajority in some weighted population of rankings. The authors show this condition is not enough. For three voters it fails exactly on the boundary value 2/3, with more than a thousand ten-vertex counterexamples and a unique regular eleven-vertex witness. For five voters it fails strictly: the Paley tournament on 43 vertices has predictability above the 3/5 threshold yet is not inducible by any five linear orders. Along the way they strengthen a structural fact about feedback arc sets and refute two earlier conjectures that linked minimum reversals only to 3-cycles. The results give the first explicit tournament of modest size that five voters cannot realize, tightening the known bounds on the smallest order where five-voter inducibility fails.","feed_headline":"Five voters cannot realize the Paley tournament on 43 points","feed_subtitle":"Predictability above 3/5 is not enough; the first modest explicit counterexample appears","key_machinery":"The co-backing lemma plus a thin-slack shell argument: in any five-voter profile the double-back sets of cyclic triangles must be pairwise disjoint, while the predictability excess of Paley(43) forces at least two voters into the level-≤1 shell of near-maximum acyclic orders; an exhaustive automorphism-reduced screen shows no such pair is double-back-disjoint.","core_discovery":"Predictability at or above the natural majority threshold does not imply m-inducibility for m=3 or m=5. For three voters the implication fails exactly when predictability equals 2/3; for five voters the Paley tournament of order 43 supplies an explicit counterexample whose predictability 181/301 exceeds 3/5. In addition, every minimum feedback arc set of any tournament is already a minimal 3-cycle hitting set, yet the equality of their sizes fails for some 3-inducible tournaments at order 11, and the locality conjecture that medians reverse only 3-cycle arcs fails for every odd number of voters at least 5.","pith_inferences":["Because five-inducibility is upward-closed, any future refutation of the threshold at seven or more voters must begin among non-5-inducible tournaments; Paley(43) is currently the only explicit specimen of reasonable size.","The same thin-slack-plus-co-backing template may decide the still-open status of the smaller Paley tournaments of orders 23, 27 and 31.","Closing the gap 12≤N(5)≤38 would settle whether the counting bound or the explicit Paley witness is closer to the true threshold."],"forward_implications":["The threshold conjecture A(m) is false for m=3 (on the boundary) and for m=5 (strictly).","N(5), the smallest order of a non-5-inducible tournament, satisfies 12≤N(5)≤38 non-constructively and N(5)≤43 with an explicit witness.","FAS=HS3 holds for every tournament on at most 10 vertices but already fails for some 3-inducible tournaments on 11 vertices.","Medians of five or more odd numbers of voters can reverse arcs that lie in no directed 3-cycle.","The margin hierarchy I_{m,t} is already strict at m=3; the paper conjectures it remains strict for every odd m."],"fun_headline_variants":["Paley-43 beats 3/5 predictability but needs more than five voters","Five voters cannot induce Paley tournament on 43 vertices","Predictability ≥3/5 fails to ensure five-voter inducibility","m=3 and m=5 inducibility conjectures fall to explicit counterexamples","Every min FAS hits all 3-cycles; size equality fails already at n=11"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that Paley(43) is not five-inducible rests on a large computer enumeration that no two near-maximum orders have disjoint double-back sets, together with a dynamic-programming computation of its maximum acyclic subgraph size.","fun_headline_variants_meta":{"raw":{"variants":["Paley-43 beats 3/5 predictability but needs more than five voters","Five voters cannot induce Paley tournament on 43 vertices","Predictability ≥3/5 fails to ensure five-voter inducibility","m=3 and m=5 inducibility conjectures fall to explicit counterexamples","Every min FAS hits all 3-cycles; size equality fails already at n=11"]},"model":"grok-4.5","effort":"low","cost_usd":0.004934,"raw_usage":{"total_tokens":1460,"prompt_tokens":902,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":49344000,"prompt_tokens_details":{"text_tokens":902,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":474,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":902,"tokens_out":84,"duration_ms":8366,"temperature":1.0,"reasoning_tokens":474,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T23:52:44.542813+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit either a five-voter profile whose majority tournament is Paley(43), or a pair of level-≤1 linear orders on 43 vertices whose sets of double-backed cyclic triangles are disjoint.","supporting_citations":[],"review_version":1}