{"id":"1f182675-0684-448d-9a43-dda20de663b0","arxiv_id":"1908.07155","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every shellable d-dimensional simplicial complex with at most d+3 vertices is extendably shellable, proved through exposed-edge deletions in chordal graphs.","lead":"The authors prove that every shellable d-dimensional simplicial complex on at most d+3 vertices is extendably shellable: any valid starting sequence of facets can always be completed. The proof rephrases shelling as deleting exposed edges from chordal graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 invokes Lemma 2.2 on the wrong graph: it calls G(X) chordal, but Lemma 2.2 applies to the complement G(X)^C; as printed the reduction fails, though the fix is straightforward.","rationale":"I read the paper in good faith and found a genuine, load-bearing error in the written proof of Theorem 1.1. The graph-theoretic core, Proposition 1.2, is proved cleanly with exposed edges and appears sound. Lemma 2.2 is imported from prior work and is not proved here, but even granting that lemma, the reduction in Theorem 1.1 applies it to the wrong graphs: the chordal graph associated to a shelling by the erasure dictionary is the complement of G(X), not G(X) itself. This is not a cosmetic typo, because G(X) can be non-chordal for a shellable X, and the inclusion needed for Proposition 1.2 is reversed in the printed argument. The fix is elementary: replace G(X) with G(X)^C and G(Y) with G(Y)^C throughout the reduction. This makes the proof go through, assuming Lemma 2.2 is correct, and the theorem itself is plausible and at the sharp threshold d+3. I therefore agree with the reader's conditional verdict: the paper should be accepted only after the complement-direction correction is made explicit in the proof of Theorem 1.1.","tokens_in":6774,"tokens_out":25537,"duration_ms":250209,"concrete_test":"Recompute the reduction in Theorem 1.1 with H := G(X)^C and G := G(Y)^C. Verify that (i) H is chordal by applying Lemma 2.2 to the shelling of X; (ii) G is chordal by applying Lemma 2.2 to the partial shelling of Y; (iii) H ⊂ G because Y ⊂ X means X has more erased edges; and (iv) concatenating the erasure sequence from K_n to G with the exposed-edge sequence from Proposition 1.2 translates via Lemma 2.2 into a shelling extension. As an additional check, construct a shellable 2-complex on 5 vertices whose G(X) is K_{3,2} (take X with facets complementing the edges of the chordal graph K_3 ∪ K_2) and confirm that G(X) is not chordal while G(X)^C is chordal, demonstrating that the printed choice H = G(X) is inconsistent with Lemma 2.2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 1.1 (Section 3), the authors set H = G(X), the graph whose edges are the facet complements V \\ F_i, and assert by Lemma 2.2 that H is chordal. Lemma 2.2 states that an erasure sequence e_1,...,e_k in K_n results in a chordal graph G exactly when the corresponding facets F_i = [n] \\ e_i form a shelling of X(G^C). Starting from a shelling of X and deleting the edges e_i = V \\ F_i from K_n, the final graph is K_n - {e_i}, which is G(X)^C, not G(X). The graph G(X) is the complement of the chordal graph produced by the erasure process and need not be chordal; for example, if the erasure process ends at the chordal graph K_3 ∪ K_2 on five vertices, then G(X) = K_{3,2}, which contains an induced 4-cycle. The same complement inversion occurs for the partial shelling: the graph obtained from K_{d+3} by erasing the edges corresponding to Y's facets is G(Y)^C, not G(Y). Consequently the inclusion H ⊂ G used to invoke Proposition 1.2 has the wrong direction: one has G(X)^C ⊂ G(Y)^C, not G(X) ⊂ G(Y). As written, the proof of Theorem 1.1 therefore does not go through. The error is localized and fixable: run Proposition 1.2 with H = G(X)^C and G = G(Y)^C. The central claim is not obviously false, but the printed argument requires this correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that every shellable d-dimensional simplicial complex on at most d+3 vertices is extendably shellable. The proof translates shellings into erasure sequences of edges in complete graphs via Lemma 2.2, reducing the problem to a chordal-graph statement (Proposition 1.2) about obtaining a chordal subgraph from a chordal supergraph by removing exposed edges. The graph-theoretic part is proved self-contained in Section 3, while the bridge between shellings and erasures is imported from the authors' earlier work.","tokens_in":7034,"tokens_out":7983,"duration_ms":75930,"significance":"If the proof is corrected, this is a strong result: it establishes the k = n-3 case of Simon's conjecture and generalizes Kleinschmidt's theorem on d-spheres with d+3 vertices. The exposed-edge framework is clean, and the graph-theoretic heart (Lemmas 3.1--3.4) is self-contained and appears sound. However, the present submission contains a load-bearing complement-direction error in the application of Lemma 2.2 that invalidates the proof as written; the fix is straightforward and localized.","major_comments":[{"comment":"The application of Lemma 2.2 inverts the complement. Lemma 2.2 states that an erasure sequence e_1,...,e_k in K_n resulting in a chordal graph G corresponds to shelling steps resulting in the complex X(G^C). Here the facets F_i = V \\ e_i form a shelling of X, so the final complex is X, and hence the chordal graph produced by the erasure sequence is G(X)^C, not G(X). The paper instead sets H = G(X) and asserts 'By Lemma 2.2 the graph H is chordal'; this is not justified, and G(X) need not be chordal (for example, if the erasure sequence ends at K_3 ∪ K_2 on five vertices, then G(X) is K_{3,2}, which has an induced 4-cycle). Similarly, a partial shelling of Y corresponds to an erasure sequence ending in G(Y)^C, not G(Y), so the inclusion 'H ⊂ G' used to apply Proposition 1.2 has the wrong direction: the correct inclusion is G(X)^C ⊆ G(Y)^C. As printed, the proof of Theorem 1.1 does not go through. The fix is to run Proposition 1.2 with H = G(X)^C and G = G(Y)^C; both are chordal on the vertex set V, H ⊆ G holds, and Lemma 2.2 then converts the exposed-edge removals back into an extension of the partial shelling to a shelling of X.","section":"Section 3, proof of Theorem 1.1 (second and third paragraphs)"}],"minor_comments":[{"comment":"The statement should specify that H and G have the same vertex set. As written, with the subgraph definition in Section 2 allowing V(H) ⊆ V(G), the statement is false: take G = K_3 and H = K_2 on two of its vertices. The proof of Proposition 1.2 uses Lemma 3.4, which assumes V(H) = V(G). Since the application in Theorem 1.1 does use the same vertex set, this is a statement-level fix rather than a fatal flaw.","section":"Section 1, Proposition 1.2"},{"comment":"The first sentence contains a typo: 'alld' should be 'all d'.","section":"Abstract"},{"comment":"The phrase 'if the subgraph induced on N(V) is a complete graph' should use N_G(v) rather than N(V).","section":"Section 2, paragraph on simplicial vertices"},{"comment":"The sentence 'We thank an anonymous referee who provided helpful comments on a earlier draft' should read 'on an earlier draft'.","section":"Acknowledgements"},{"comment":"Reusing the symbol C for both a k-clutter and the simplicial complex it generates is confusing; consider distinguishing the two objects.","section":"Section 2, k-clutter notation"},{"comment":"Lemma 2.2 is imported from [10] and is load-bearing for the main theorem. A brief proof sketch or a precise statement of the full theorem from [10] would improve the self-containedness of the paper.","section":"Section 2, Lemma 2.2"}],"recommendation":"major_revision","confidential_remarks":"The complement-direction error is localized and the corrected proof seems likely to work, so I recommend major revision rather than rejection. The graph-theoretic lemmas in Section 3 appear sound and are a genuinely useful contribution. The authors should also tighten the statement of Proposition 1.2 to include the same-vertex-set hypothesis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the theorem is a real extension and the threshold is sharp, but the proof as written has a complement-direction error in the reduction to the graph lemma. The graph theory is sound; the fix is straightforward.\n\nWhat's good: proving extendable shellability for every shellable d-complex on at most d+3 vertices is a genuine step forward—it implies the k=n-3 case of Simon's conjecture, generalizes Kleinschmidt's sphere result, and the d+4-vertex counterexamples show the bound is best possible. Proposition 1.2 is also a real strengthening of Rose–Tarjan–Lueker's Lemma 2: exposed edges give a clean characterization of the edges you can delete while staying chordal, and the proof in Section 3 (Lemmas 3.1–3.4) is self-contained and convincing. The erasure-to-shelling dictionary is imported from the authors' prior work, but it is parameter-free and not a fitted claim.\n\nWhere it slips: in the proof of Theorem 1.1, Lemma 2.2 is applied to the wrong graph. If the shelling facets are F_i, the erasure sequence in K_{d+3} consists of e_i = V \\ F_i, so after deleting them the resulting chordal graph is K_{d+3} - {e_i} = G(X)^C, not G(X). The paper sets H = G(X) and calls it chordal; the same inversion happens for the partial shelling and G(Y), and the inclusion used for Proposition 1.2 has the wrong direction. The fix is to run Proposition 1.2 with H = G(X)^C and G = G(Y)^C, where the inclusion does hold. This is localized, and the authors' conclusion is very likely correct, but the printed proof does not go through. The other soft spot is that Lemma 2.2 is load-bearing and only cited from the authors' own earlier work; a referee should ask for the relevant statement to be reproduced or at least stated precisely.\n\nBottom line: this is a solid paper with one significant but fixable error in the main proof. It deserves a serious referee, not a desk reject. If the complement issue is corrected, it is a clean contribution to the low-vertex shellability program. I would not cite the current arXiv version, but I would follow a revised version.","headline":"The theorem is genuinely new and the threshold is sharp, but the printed proof applies the erasure-to-shelling dictionary to the wrong graph; the fix is elementary and the result is likely correct.","tokens_in":7688,"tokens_out":5071,"would_cite":false,"duration_ms":48839,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E45","13F55","05C75"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every shellable $d$-dimensional complex on at most $d+3$ vertices is extendably shellable.","keywords":["shellable simplicial complex","extendably shellable","chordal graph","exposed edge","erasure sequence","linear quotients","quadratic monomial ideals","simplex skeleton conjecture"],"falsifier":"The cleanest test is graph-theoretic: take a chordal graph $G$ and a chordal subgraph $H$ on the same vertex set and try to reduce $G$ to $H$ by deleting only exposed edges; Proposition 1.2 predicts success for every such pair, so one unreachable pair would refute the method, and through the dictionary it would produce a shellable complex on $d+3$ vertices with an uncompletable partial shelling.","tokens_in":6488,"feed_emoji":"🔺","tokens_out":13461,"duration_ms":128629,"temperature":0.7,"pith_summary":"The paper proves a sharp no-getting-stuck statement for simplicial complexes: if $X$ is a shellable $d$-dimensional complex with at most $d+3$ vertices, then any shelling of any subcomplex of $X$ can be extended to a shelling of all of $X$. Such complexes are called extendably shellable. The result includes, as a special case, the conjecture that every $k$-skeleton of a simplex is extendably shellable in the range $k=n-3$, and it generalizes an earlier theorem for $d$-spheres on $d+3$ vertices. The paper also shows the bound is best possible, since on $d+4$ vertices there are already shellable complexes that can get stuck.","feed_headline":"Shellable complexes on d+3 vertices never block a shelling","feed_subtitle":"Any partial shelling of a subcomplex extends to the whole complex, settling a boundary case of the simplex-skeleton conjecture.","key_machinery":"The central object is the exposed edge: an edge that lies in exactly one maximal clique of a graph, so that deleting it preserves chordality. The load-bearing mechanism is the dictionary of Lemma 2.2, equating erasure sequences of exposed edges with shelling steps of the complementary complex. Proposition 1.2, proved through Lemmas 3.1 to 3.4, shows that every chordal subgraph of a chordal graph can be reached by exposed-edge erasures. This reachability at the graph level is what carries the complex-level extension argument.","core_discovery":"The central claim is Theorem 1.1: for every $d\\geq 1$, a shellable $d$-dimensional simplicial complex on at most $d+3$ vertices is extendably shellable. The proof reduces this to a purely graph-theoretic statement. On $d+3$ vertices, each facet is a $(d+1)$-set whose complement is a pair of vertices, so facets correspond to edges of a graph. The dictionary of Lemma 2.2 identifies a shelling order with an erasure sequence in the complete graph $K_{d+3}$: at each step the erased edge is exposed, and the final graph is chordal. Proposition 1.2 then says that from any chordal graph one can keep erasing exposed edges until any prescribed chordal subgraph remains. Translating this back, a partial shelling of a subcomplex can always be completed to a shelling of the whole complex.","pith_inferences":["The known non-extendably shellable examples in dimension 2 on six vertices sit exactly one vertex beyond the theorem's range, so they separate the boundary case from the next one; analyzing why the chordal-graph mechanism fails there could indicate what a $d+4$ proof would need.","Proposition 1.2 stands on its own as a constructive graph algorithm: any chordal graph can be reduced to any chordal subgraph by deleting exposed edges one at a time, a primitive that may be useful outside shelling.","A testable extension would be to replace edges by exposed circuits in higher-dimensional clutters and ask whether every shellable complex on $d+4$ vertices still has the extension property; the paper's conclusion notes the chordal-graph tools do not lift, so a positive answer would need a new invariant."],"forward_implications":["Any shelling of a subcomplex of such an $X$ can be extended, so building $X$ by facets never gets stuck after a correct start.","The $k=n-3$ case of the simplex-skeleton conjecture follows as a corollary.","The theorem extends the known extendable shellability of $d$-spheres on $d+3$ vertices to all shellable complexes on that many vertices, not just spheres.","The bound is tight: on $d+4$ vertices there are shellable complexes that are not extendably shellable.","Since the cases with $d+1$ and $d+2$ vertices are straightforward, the entire statement is really about the $d+3$-vertex case."],"supporting_citations":[{"why":"Supplies Lemma 2.2, the erasure-to-shelling dictionary that converts the complex question into a graph question; the proof of Theorem 1.1 depends directly on it.","marker":"[10]"},{"why":"Introduces exposed edges and erasure sequences and establishes that chordal graphs are exactly the graphs obtainable from complete graphs by erasures.","marker":"[7]"},{"why":"Contains the weaker version of Proposition 1.2 that the paper strengthens, providing the core comparison for the graph-theoretic argument.","marker":"[15]"},{"why":"Establishes the earlier result that every $d$-sphere on $d+3$ vertices is extendably shellable, the statement Theorem 1.1 generalizes.","marker":"[12]"},{"why":"Shows rank-3 matroid independence complexes are extendably shellable, an earlier broad class with the same property.","marker":"[4]"},{"why":"Provides a six-vertex 2-dimensional complex that fails extendable shellability, showing the vertex bound in Theorem 1.1 is best possible.","marker":"[13]"}],"fun_headline_variants":["Shellable on d+3 vertices? Always extendable","d+3 vertices guarantee shelling completion","Every partial shelling on d+3 vertices completes","Boundary case solved: extendable shelling on d+3","No stuck shellings on d+3 vertices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on an unproved dictionary lemma from earlier work identifying erasure sequences of edges with shelling steps of complementary facets; as written, the proof must apply it to the complementary graphs, so if that translation cannot be repaired the graph-theoretic proposition does not reach the complexes.","fun_headline_variants_meta":{"raw":{"variants":["Shellable on d+3 vertices? Always extendable","d+3 vertices guarantee shelling completion","Every partial shelling on d+3 vertices completes","Boundary case solved: extendable shelling on d+3","No stuck shellings on d+3 vertices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000379,"raw_usage":{"total_tokens":1935,"prompt_tokens":783,"completion_tokens":1152,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":399,"completion_tokens_details":{"reasoning_tokens":1075}},"tokens_in":399,"tokens_out":1152,"duration_ms":8917,"temperature":1.0,"reasoning_tokens":1075,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:25:09.393917+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The cleanest test is graph-theoretic: take a chordal graph $G$ and a chordal subgraph $H$ on the same vertex set and try to reduce $G$ to $H$ by deleting only exposed edges; Proposition 1.2 predicts success for every such pair, so one unreachable pair would refute the method, and through the dictionary it would produce a shellable complex on $d+3$ vertices with an uncompletable partial shelling.","supporting_citations":[{"cited_title":"Exposed circuits, linear quotients, and chordal clutters","cited_arxiv_id":"1812.08128","evidence_quote":"Supplies Lemma 2.2, the erasure-to-shelling dictionary that converts the complex question into a graph question; the proof of Theorem 1.1 depends directly on it."},{"cited_title":"Edge Erasures and Chordal Graphs","cited_arxiv_id":"1706.04537","evidence_quote":"Introduces exposed edges and erasure sequences and establishes that chordal graphs are exactly the graphs obtainable from complete graphs by erasures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the weaker version of Proposition 1.2 that the paper strengthens, providing the core comparison for the graph-theoretic argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the earlier result that every $d$-sphere on $d+3$ vertices is extendably shellable, the statement Theorem 1.1 generalizes."},{"cited_title":"Bj¨orner, K","cited_arxiv_id":null,"evidence_quote":"Shows rank-3 matroid independence complexes are extendably shellable, an earlier broad class with the same property."},{"cited_title":"Moriyama, F","cited_arxiv_id":null,"evidence_quote":"Provides a six-vertex 2-dimensional complex that fails extendable shellability, showing the vertex bound in Theorem 1.1 is best possible."}],"review_version":1}