{"id":"66a6a1ff-f914-481d-beba-7818e5fc985b","arxiv_id":"2607.24738","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every oriented graph with minimum pseudo-semidegree at least k contains an antidirected path of length 2k−1, confirming Stein’s conjecture.","lead":"Every oriented graph whose vertices each have in-degree and out-degree at least k (or zero) contains an antidirected path of length 2k−1. The result settles Stein’s conjecture on antipaths and is tight.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No significant objection identified. The counting core is sound; the residual risk is only an unformalized missed rewiring in Lemma 2.4, especially boundary/degenerate index ranges.","rationale":"The reader’s ACCEPT/HIGH with medium correctness risk matches my read. The result is parameter-free, has an explicit sharpness construction, and the contradiction is a clean double count once the two-sided blow-up exists. The reader correctly located the two pressure points: dependence on Klimošová–Stein for parity/anticycle conversion, and the long Section 3 proof of Lemma 2.4. Those are real dependencies, but not objections: the former are published lemmas used in their stated regime, and the latter is detailed rather than circular. I would not move to CONDITIONAL because I cannot point to a concrete failing equation, missed case, or unjustified degree estimate; the only issue is that the most delicate rewirings are human-checked rather than machine-checked. The proposed test targets exactly that residual risk without demanding a full formalization of the paper.","tokens_in":19106,"tokens_out":13357,"duration_ms":472015,"concrete_test":"Write a small independent checker for Section 3: for ℓ=2..10 and all admissible i0,i (including i=i0−1, i=i0+2, i=2, i=ℓ−1), instantiate the listed sequences in Claims 3.4–3.5 and the four final blow-ups and verify (a) the singleton positions contain each index 1..2ℓ exactly once except one explicitly omitted index, (b) every consecutive pair is an edge with the orientation required for an antipath/blow-up, and (c) total length is 2ℓ+1. If any boundary instance duplicates/omits an index or reverses parity, Lemma 2.4 has a real gap; otherwise the main remaining risk is cleared. Optionally also SAT-enumerate k≤3, n≤10 for a counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I do not find a load-bearing flaw. The main line is tight: if a longest antipath has length 2ℓ+1<2k−1, Lemmas 2.1–2.2 legitimately forbid length-(2ℓ+2) antipaths/anticycles; Lemma 2.4 upgrades a one-sided blow-up to |V0|,|V2ℓ+1|≥2; Claim 2.5 then charges every surplus pair {v2i,v2i+1} to adjacent non-surplus pairs so e(V0,V′)+e(V′,V2ℓ+1)≤(ℓ+1)(|V0|+|V2ℓ+1|), while ¯δ0(G)≥k forces the same quantity to be at least k(|V0|+|V2ℓ+1|). Since ℓ≤k−2 this is contradictory. The endpoint terms are safely bounded by 2(|V0|+|V2ℓ+1|), and the pseudo-semidegree hypothesis is applied only to vertices already known to have nonzero relevant degree. The weakest point is not logical but epistemic: Lemma 2.4’s Section 3 rotations, especially Claim 3.5 and the final four blow-ups for i<i0 versus i>i0+1, rely on ellipsis notation where degenerate cases (ℓ small, i=i0−1 or i=i0+2) could in principle duplicate/omit an index or break alternation. I checked the index bookkeeping for these boundaries and it appears consistent: one internal index is intentionally omitted while xi,x are inserted, preserving length 2ℓ+1. External Lemmas 2.1/2.2 are load-bearing but are cited published results, not hidden assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript proves that every oriented graph G with minimum pseudo-semidegree δ̄⁰(G) ≥ k contains an antidirected path of length 2k−1, confirming Stein's Conjecture 1.2 and thereby Conjecture 1.1 for antipaths; Proposition 1.4 shows the bound is tight. The proof is by contradiction: assuming a longest antipath has length 2ℓ+1 < 2k−1 (parity via a lemma of Klimošová–Stein), the authors construct a two-sided \"antipath blow-up\" V₀v₁…v_{2ℓ}V_{2ℓ+1} with |V₀|, |V_{2ℓ+1}| ≥ 2 (Lemma 2.4, proved in Section 3), and then charge each \"surplus\" index i ∈ [ℓ−1] to adjacent non-surplus indices (Claim 2.5), giving e(V₀,V′)+e(V′,V_{2ℓ+1}) ≤ (ℓ+1)(|V₀|+|V_{2ℓ+1}|), while the degree condition forces the same quantity to be at least k(|V₀|+|V_{2ℓ+1}|). Since ℓ ≤ k−2, this is a contradiction. The core counting argument is clean, and the structural heavy lifting is confined to Lemma 2.3 (local forbidden configurations) and Lemma 2.4 (existence of the two-sided blow-up).","tokens_in":19456,"tokens_out":2747,"duration_ms":106046,"significance":"If correct, this settles Stein's Conjecture 1.2 (and hence Conjecture 1.1 for antipaths), closing a problem with a documented line of partial progress (Klimošová–Stein; Chen–Hou–Zhou; Skokan–Tyomkyn; Grzesik–Skrzypczyk), and the bound 2k−1 is shown best possible (Proposition 1.4). Strengths worth naming: the proof is self-contained modulo two published lemmas; it is parameter-free — no regularity, no ε's, no asymptotics — and yields the exact constant; the extremal construction in Proposition 1.4 is verified for all k; and the two-sided blow-up idea (Lemma 2.4 plus the charging scheme of Claim 2.5) is a genuine methodological advance over the one-sided blow-up of Grzesik–Skrzypczyk that powered all previous bounds. The result is definitive for its question and will likely be the reference point for the remaining cases of Conjecture 1.1.","major_comments":[{"comment":"Lemma 2.4, final displayed list of four antipath blow-ups (after Claim 3.5, p. 13, with Figure 3.5): the four cases are written in ellipsis notation, e.g. 'U v2i xi v2i+1 ... v2i0−1 v2ℓ ... v2i0 v2i−2 x v1 ... v2i−3'. The case split is i < i0 versus i > i0+1, but the degenerate near-boundary subcases (i = i0−1 and i = i0+2, where the '...' segments collapse to zero or one vertex) are not written out, and small ℓ (ℓ = 2 or 3, where [ℓ−1] and the ranges of I are nearly empty) are not separately discussed. My own index bookkeeping for these boundaries indicates the constructions remain valid — exactly one internal index is omitted while xi and x are inserted, preserving length 2ℓ+1 and alternation — but this is the load-bearing step of the whole paper, and the manuscript should not leave it to the reader. Please add a short verification of the boundary cases, or restructure the display so t","section":"Section 3, proof of Lemma 2.4"},{"comment":"The proofs of Lemma 2.3(h) and Claim 3.5 both estimate |N^-(v)\\(V0 ∪ V')| by d^-(v) − d^-(v, V'\\V_{2I}) − d^-(v, V_{2I}), and then lower-bound this using |V'\\V_{2I}| − d^+(v, V'\\V_{2I}). This step silently uses that for an oriented graph restricted to a fixed vertex set, d^-(v,W) + d^+(v,W) ≤ |W|, which is fine, but the arithmetic chain '≥ k − (2ℓ − |I| − k) − 1 = |I| + 2(k−ℓ) − 2' in Claim 3.4 (p. 11) deserves one line of justification, since the analogous computation appears three times (Lemma 2.3(h), Claim 3.4, Claim 3.5) with slightly different constants (−1 vs −2, |I|+3 vs |I|+2) and an off-by-one error here would propagate into the maximality contradictions |U| > |V0|. I checked the constants and believe they are consistent (the differing offsets come from d^-(v_{2j}, V_{2I}) ≤ 1 via Lemma 2.3(j) versus = 0 via (2.1)/(3.4)), but the manuscript should make the source of each offset","section":"Claim 3.4 and Lemma 2.3(h)"}],"minor_comments":[{"comment":"Reference [12]: 'R´edei, Ein kombinatoricher satz, Acta Lott. Szeged' should read 'Ein kombinatorischer Satz, Acta Litt. Sci. Szeged' (or Acta Sci. Math. (Szeged)) 7:39–43, 1934.","section":"References"},{"comment":"Reference [5] (Grzesik–Skrzypczyk) is cited only as an arXiv preprint; please update the bibliographic data if it has appeared, since Lemma 2.3(b) and Lemma 3.1 explicitly build on its Claim 12 and Lemma 10.","section":"References"},{"comment":"In Claim 2.5(d), the bound '|N^+(v_{2i−3})\\(V_{2ℓ+1} ∪ V')| ≥ 4 by Lemma 2.3(h) (applied to the reverse of G)' deserves a half-line of explanation: (h) gives |I|+3 where |I| counts surplus indices, and the ≥4 conclusion uses that i−2 itself contributes to that count. This is correct but not immediate.","section":"Claim 2.5(d)"},{"comment":"Notation §1.1: 'We simply write v for {v}' is used implicitly from Section 2 onwards (e.g. e(V0, v1)); consider stating explicitly that singletons are identified with their elements in e(·,·) and N±(·,·).","section":"Section 1.1"},{"comment":"The abstract and first sentence of Section 1 say 'minimum semidegree' while Theorem 1.3 is stated for minimum pseudo-semidegree δ̄⁰; the abstract should match the theorem, since the pseudo-semidegree formulation is strictly stronger and is the paper's actual contribution.","section":"Abstract"},{"comment":"Figures 2.1–3.5 are helpful, but the vertices V0/V2ℓ+1 are drawn identically to singleton vertices in some panels; a brief caption note that U, W, W± denote sets while xi, x denote vertices would ease reading.","section":"Figures"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is short, self-contained modulo two published Klimošová–Stein lemmas, and resolves a named conjecture; I see no concerns about citation pattern or novelty. My only reservation is that Lemma 2.4's long case analysis has not been independently verified by me in full; I recommend at least one referee or the authors themselves re-run the boundary cases of the four terminal blow-ups in Section 3 before final acceptance. Otherwise this is a clean fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This settles Stein’s antipath conjecture at the sharp threshold: every oriented graph with minimum pseudo-semidegree at least k has an antipath of length 2k-1. That is the first exact result after a sequence of successive coefficient improvements, and it finishes the antipath case of the broader path-orientation programme.\n\nWhat is new is the two-sided antipath blow-up (both ends enlarged simultaneously) together with the surplus-control claim that lets them charge every high-degree pair against neighbouring non-surplus pairs. Once Lemma 2.4 supplies a blow-up with |V0|,|V2ℓ+1|≥2, the counting in the proof of Theorem 1.3 is short and transparent: the surplus indices partition into disjoint blocks whose ϕ-sums are at most the block size times (|V0|+|V2ℓ+1|), the endpoint terms are bounded by 2, and the degree hypothesis then forces ℓ≥k-1, a contradiction. Sharpness is given by an explicit construction (Proposition 1.4). The local forbidden configurations in Lemma 2.3 are all witnessed by concrete longer antipaths or anticycles, so the bookkeeping is visible.\n\nThe soft spot is exactly where the stress-test puts it: Lemma 2.4 is a long rotation-and-extension argument (Section 3) that relies on index ellipses. I checked the boundary cases (small ℓ, i near i0) and the length and alternation appear to hold, but a missed rewiring remains the only realistic way the proof could fail. The two Klimošová–Stein lemmas used for parity and anticycle-to-path conversion are published and load-bearing only in the usual citation sense; nothing circular is going on.\n\nThis is for people who work on degree conditions in oriented graphs or digraph path problems. The argument is self-contained combinatorial reasoning with no free parameters. It deserves a serious referee who will walk through Section 3 carefully; I would accept it for peer review and expect it to clear after that check.","headline":"Exact confirmation of Stein’s antipath conjecture via a clean two-sided blow-up counting argument; the only real residual risk is a long but checkable case analysis in Lemma 2.4.","tokens_in":19990,"tokens_out":530,"would_cite":true,"duration_ms":14262,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","05C38"],"pacs":[],"model":"grok-4.5","headline":"Every oriented graph with minimum pseudo-semidegree at least k contains an antidirected path of length 2k−1.","keywords":["antidirected path","antipath","minimum semidegree","minimum pseudo-semidegree","oriented graph","Stein conjecture","antipath blow-up"],"falsifier":"Exhibit an oriented graph with minimum pseudo-semidegree at least k that has no antipath of length 2k−1, or find a flaw in the existence of a two-sided blow-up with both end-sets of size at least 2 (Lemma 2.4) or in the subsequent edge-counting contradiction.","tokens_in":19759,"feed_emoji":"🔀","tokens_out":979,"duration_ms":18879,"temperature":0.7,"pith_summary":"An antidirected path (antipath) is a path in a directed graph where each vertex has only incoming or only outgoing edges along the path. This paper proves that any oriented graph in which every nonzero in- or out-degree is at least k must contain an antipath with 2k−1 edges. That bound is tight: there are examples with the same degree condition that have no antipath of length 2k. The result settles Stein’s conjecture for antipaths and thereby confirms the antipath case of a broader conjecture that every oriented graph should contain every orientation of a path of length 2δ⁰(G)−1. The argument works by taking a longest antipath, expanding its ends into independent sets (an “antipath blow-up”), and deriving a degree contradiction unless the path is already long enough.","feed_headline":"Oriented graphs hide antipaths of length 2k−1","feed_subtitle":"Minimum pseudo-semidegree k forces an antidirected path of length 2k−1, settling Stein’s conjecture for antipaths.","key_machinery":"Antipath blow-up: a longest antipath whose two end vertices are replaced by independent sets V₀ and V_{2ℓ+1} of size at least 2, with all edges from V₀ into the next vertex and from the penultimate vertex into V_{2ℓ+1}. Counting edges between these ends and the internal path, together with surplus indices and non-adjacency lemmas, produces an upper bound that contradicts the minimum pseudo-semidegree unless the path already has length 2k−1.","core_discovery":"Every oriented graph G with minimum pseudo-semidegree δ̄⁰(G) ≥ k contains an antipath of length 2k−1. Equivalently, Stein’s Conjecture 1.2 is true, so the antipath case of Conjecture 1.1 also holds. The bound is best possible: for every k there exist oriented graphs with δ⁰ = δ̄⁰ = k that contain neither an antipath nor an anticycle of length 2k.","pith_inferences":["The concluding remarks note that the pseudo-semidegree analogue fails for any orientation containing a directed path of length 2, so further progress on the full conjecture will need the stricter ordinary minimum semidegree.","A natural next test is whether the same blow-up technique can force antipaths of length roughly 2δ⁰ under weaker local degree conditions or in tournaments with restricted cycle types.","If the two external Klimošová–Stein lemmas can be strengthened to even lengths or to longer cycles, the blow-up argument might yield longer guaranteed antipaths in denser oriented graphs."],"forward_implications":["Stein’s minimum-semidegree conjecture is settled for all antipaths.","The same bound is tight for both antipaths and anticycles of length 2k.","Directed paths and paths with one direction change were already known; only more mixed orientations remain open under the same degree hypothesis.","The blow-up-and-surplus counting method supplies a template for attacking other fixed orientations of paths."],"fun_headline_variants":["Min semidegree k forces antipaths of length 2k−1","Oriented graphs of semidegree k hold 2k−1 antipaths","Stein conjecture confirmed for antipaths of length 2k−1","Every oriented graph with δ̄⁰≥k has a 2k−1 antipath","Semidegree k yields antidirected paths reaching length 2k−1"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The argument depends on two prior facts: that a longest antipath shorter than 2k−1 must have odd length, and that a short anticycle of even length forces an antipath of the same length; if either fails, the parity and blow-up counting collapse.","fun_headline_variants_meta":{"raw":{"variants":["Min semidegree k forces antipaths of length 2k−1","Oriented graphs of semidegree k hold 2k−1 antipaths","Stein conjecture confirmed for antipaths of length 2k−1","Every oriented graph with δ̄⁰≥k has a 2k−1 antipath","Semidegree k yields antidirected paths reaching length 2k−1"]},"model":"grok-4.5","effort":"low","cost_usd":0.004084,"raw_usage":{"total_tokens":1123,"prompt_tokens":609,"num_sources_used":0,"completion_tokens":88,"cost_in_usd_ticks":40844000,"prompt_tokens_details":{"text_tokens":609,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":426,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":609,"tokens_out":88,"duration_ms":7376,"temperature":1.0,"reasoning_tokens":426,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T06:22:57.081274+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit an oriented graph with minimum pseudo-semidegree at least k that has no antipath of length 2k−1, or find a flaw in the existence of a two-sided blow-up with both end-sets of size at least 2 (Lemma 2.4) or in the subsequent edge-counting contradiction.","supporting_citations":[],"review_version":1}