{"id":"dfc69de4-6a0a-4f59-99f8-6e406314dc01","arxiv_id":"2607.24476","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A simplicial set is upper/lower d-Segal iff it is (d+1)-coskeletal and satisfies the d-Segal conditions in dimensions d+1 and d+2 (with the proved 'if' direction using (d+2)-coskeletality).","lead":"A simplicial set is d-Segal exactly when it is highly coskeletal and the Segal condition holds in just the two lowest dimensions. This gives a finite check for d-Segality on simplicial sets with finitely many nondegenerate simplices.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The reader's flagged abstract/body gap is illusory: (d+1)-coskeletal implies (d+2)-coskeletal, so Theorem 25 immediately yields the abstract's iff. No significant objection to the central claim.","rationale":"The reader gave CONDITIONAL primarily on the grounds that the abstract promises an iff at (d+1)-coskeletal while the body proves the reverse only at (d+2). This inverts the strength relationship: smaller coskeletal level is the stronger hypothesis, so the body's Theorem 25 is strictly stronger than the abstract's reverse direction, and the abstract is fully supported. The verdict should therefore be ACCEPT rather than CONDITIONAL. On internal correctness, which the reader flagged via the weakest_assumption field: the proofs are elementary, combinatorial, and complete; the Wiggle Lemma's cube-pasting argument is given in detail with the single-gap iteration made explicit; Proposition 20 and Lemma 21 handle the parity-dependent boundary cases (p = n, p = n+2) individually with the critical conditions supplying exactly the missing face compatibility (e.g., ue^k_{p-1} yielding epf_{p+1} = ep+1f_p in Prop. 20(2)); the induction in Theorem 15 closes at the base n = 2k+2 using precisely the two critical conditions. Edge cases d = 0, 1 are treated by hand in Lemma 19, and I verified the abstract's d = 0 case independently (constant sets are 1-coskeletal; 1-coskeletal ⟹ 2-coskeletal, so Lemma 19 applies). Independent support: the result recovers the classical d = 1 (nerve) and d = 2 ([1, Cor. 1.7]) characterizations at the same coskeletal levels, the simplicial-spaces counterexample (Remark 26) is correct and shows the hypotheses are calibrated, and the finite-check application rests on a published result [12, Thm 3.19]. Residual risks are minor and non-load-bearing: heavy reliance on the author's own prior work [10] for the cube-pasting lemma and Prop. 9, and no formal verification — acceptable for a short, fully explicit combinatorial paper.","tokens_in":13043,"tokens_out":12887,"duration_ms":414982,"concrete_test":"Two-line audit: (1) From Definition 3, confirm n-coskeletality quantifies over all m-spheres with m > n, hence (d+1)-coskeletal implies (d+2)-coskeletal; then check that for each parity (Thm 15, Thm 24(1)–(3)) the coskeletal level required by the body is exactly (d+2), so the abstract's reverse direction follows a fortiori, while Theorem 22 supplies the forward direction at (d+1). (2) Independently re-derive the base case of Theorem 15's induction for k=1, n=4 (the classical nerve case, lo^1_2 + lo^1_3 + 3-coskeletal ⟹ lo^1_4) directly from Lemmas 16–17; if this recovers the known result, the induction engine is sound.","verdict_should_be":"ACCEPT","load_bearing_attack":"I read the central claim (Theorem 25 and the abstract's iff) looking for the least secure point, and checked both the internal machinery and the reader's stated defect. On the internal machinery: the reduction chain (Wiggle Lemma 11, Prop. 9's décalage shift of conditions, Prop. 20's transfer of coskeletality under décalage with critical hypotheses, Lemma 21's converse with its two sharpened parity clauses) is fully written out; I traced each parity case of Theorems 22 and 24 against these lemmas and the bookkeeping is consistent (e.g., Thm 22 upper-odd uses Prop. 12 on dec⊤X then Lemma 21 clause (2) with uo^k_{2k+2}; Thm 24(2) uses Prop. 20(2) then Thm 15 on dec⊤X). Lemma 19 correctly quarantines k=0. Remark 26's counterexample for spaces is correct and properly delimits scope. On the reader's verdict-driving defect: the claim that the abstract's reverse direction ((d+1)-coskeletal + d-critical ⟹ d-Segal) is unproven is mistaken. Coskeletality is monotone in the trivial direction: by Definition 3, n-coskeletal quantifies over all m-spheres with m > n, so (d+1)-coskeletal a fortiori gives (d+2)-coskeletal, and Theorem 25's first statement applies verbatim. The forward direction is Theorem 22 exactly. The abstract's iff is thus an immediate corollary of the body; footnotes 2 and 4 (\"3-coskeletal may replace 2-coskeletal in the reverse direction\", etc.) describe precisely this phenomenon for the classical cases, and \"See Theorem 25 for a slightly stronger statement\" is accurate, since the reverse direction alone needs only (d+2). I find no load-bearing concern.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper proves a coskeletality criterion for the higher Segal conditions on simplicial sets: X is upper (resp. lower) d-Segal if and only if it is (d+1)-coskeletal and satisfies the upper (resp. lower) d-Segal conditions in the lowest two non-vacuous dimensions, d+1 and d+2 (\"d-critical\"). Theorem 25 states the slightly sharper body version: (d+2)-coskeletal plus d-critical implies d-Segal (Theorems 15 and 24), and d-Segal implies (d+1)-coskeletal (Proposition 12 and Theorem 22). The lower-odd case is proved directly by an inductive filler argument (Lemmas 16–17, Proposition 18) built on the Wiggle Lemma; the remaining three parities are reduced to it via Poguntke's path-space criterion and a careful comparison of coskeletality under décalage (Proposition 20, Lemma 21). Remark 26 shows the analogue for simplicial spaces fails, via a constant simplicial space on a space with infinitely many nontrivial homotopy groups.","tokens_in":13384,"tokens_out":3230,"duration_ms":104228,"significance":"The result resolves a question attributed to Walker Stern and unifies two classical recognition principles — the nerve criterion (2-coskeletal plus unique inner-horn fillers in dimensions 2 and 3) and the Bergner–Osorno–Ozornova–Rovelli–Scheimbauer / Stern criterion for 2-Segal sets (3-coskeletal plus square and pentagon conditions) — into a single statement valid for all d. The practical content is real: combined with the Kennett–Riehl–Roy–Zaks result that n-skeletal simplicial sets are (2n−1)-coskeletal, the criterion reduces d-Segality of a finite simplicial set (for d large enough) to a finite check in dimensions d+1 and d+2, and the author has already used this to exhibit the strictness of the higher Segal hierarchy on the examples Δn/∂Δn. The proofs are self-contained and combinatorial, with the four parity cases handled explicitly, the degenerate k = 0 cases quarantined by a correct hand computation (Lemma 19), and the failure for simplicial spaces delimited by a correct counterexample (Remark 26) that prevents overgeneralization. There are no fitted parameters or external black boxes beyond standard, cited results (Walde's theorem, Poguntke's path-space criterion).","major_comments":[{"comment":"I have no load-bearing objections. I specifically checked the one place where the abstract appears stronger than the body: Theorem 25's reverse direction assumes (d+2)-coskeletality, while the abstract asserts the iff with (d+1)-coskeletal. By Definition 3, n-coskeletality quantifies over all m-spheres with m > n, so (d+1)-coskeletal implies (d+2)-coskeletal and Theorem 25 applies verbatim; the forward direction is Theorem 22 exactly. Footnotes 2 and 4 describe precisely this slack in the classical cases, and the line 'See Theorem 25 for a slightly stronger statement' is accurate. The abstract is therefore supported by the body.","section":"Abstract / Theorem 25"},{"comment":"The induction in Theorem 15 rests on Lemma 16 and Lemma 17 producing a full compatible collection from the I-indexed data, with Proposition 18 then giving existence and uniqueness. The gap-filling for J = {1,3,...,2k+1} (Lemma 17(2)) is the most intricate step: the verification that L_{j,j'} is gapped in S_{j,j'} via the isomorphism to the even elements of [0,2k] in [0,n-2], and the induction on N = (j'-j)/2, are correct as written. I confirmed the N=1 base case is fully handled by the even-element argument and that Lemma 2 is applied with consistent parameter choices throughout. No correction needed, but this is the argument a reader must trust, and it holds up.","section":"§3, Lemmas 16-17, Proposition 18"},{"comment":"The parity reduction via décalage (Propositions 9 and 20, Lemma 21) is consistent in all four cases. In particular: Theorem 22 (upper odd) correctly chains Theorem 8 with Proposition 12 applied to dec^T X and Lemma 21; Theorem 24(3) invokes item (1) rather than Theorem 15, which is legitimate since (1) is established (via (2) and duality) earlier in the same proof — there is no circularity. Lemma 21's two sharpened clauses match exactly the hypotheses available in each parity (the 'in particular' sentence is correct: ue^k_{2k+1}/le^k_{2k+1} hold for 2k-critical, uo^k_{2k+2} for (2k+1)-critical). The k=0 exceptional cases are quarantined by Lemma 19 before the general argument, which is necessary (Remark 14 shows Proposition 12 fails at k=0).","section":"§4, Proposition 20, Lemma 21, Theorems 22 and 24"}],"minor_comments":[{"comment":"The proof is outsourced entirely to [10, Proposition 3.13] ('As in the proof of...'). Since this proposition is used in every parity reduction in §4, one or two sentences indicating the idea (or at least a more precise pointer to the relevant steps in [10]) would make the paper more self-contained.","section":"Proposition 9"},{"comment":"Only the upper 1-critical case is treated, with 'the other cases being similar.' The lower 0-critical case is not literally identical (it uses le^0_1 rather than uo^0_2), so a sentence indicating what changes would be welcome, especially since this lemma is the sole guard for the k=0 exceptions.","section":"Lemma 19"},{"comment":"Reference [14] is an nLab page (with a revision date); for the classical fact that nerves of categories are 2-coskeletal, a stable reference (e.g., a Kerodon tag, which is already cited as [13], or a textbook source) would be preferable.","section":"References, footnote 3"},{"comment":"The claim that the cyclic-polytope formulation of d-critical is equivalent to Definition 7 is supported only by 'it's not so arduous to prove this directly' plus a reading of Walde's Theorem 7.2.1. A slightly more detailed sketch, or an explicit statement of which steps of [18] are needed, would help readers who want to use the geometric form.","section":"Remark 10"},{"comment":"The effectiveness claim in the introduction ('allows for an effective, finite check of d-Segality... for d suitably large') would benefit from stating the explicit bound: an n-skeletal simplicial set is (2n-1)-coskeletal by [12, Theorem 3.19], so the criterion applies once d+1 >= 2n-1, i.e. d >= 2n-2. Giving this inequality explicitly would make the remark immediately usable.","section":"Introduction, paragraph on effectiveness"},{"comment":"The notation S_i = S \\ {i} and e_i is convenient but collides visually with simplicial face notation; the remark after the definition helps, but a one-line reminder at its first use in §2 (Definition 4 context) would prevent confusion. Also, in the abstract line of the arXiv text, spacing appears to be lost in 'isd-Segal'/'is(d+1)-coskeletal' — presumably a rendering artifact, but worth checking in the final version.","section":"§1, notation; Abstract"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"Punchline: this is a solid, elementary recognition theorem that finishes the pattern already known for nerves of categories and for 2-Segal sets. A simplicial set is d-Segal exactly when it is (d+1)-coskeletal and satisfies the two lowest nontrivial d-Segal conditions. That answers Stern’s question and gives a finite check once you already know high enough coskeletality (via the Kennett–Riehl–Roy–Zaks bound for skeletal sets).\n\nWhat is actually new is the uniform argument for arbitrary d. The lower-odd case is done by direct inductive filling (Prop. 12, Thm. 15); the remaining parities ride on Poguntke’s path-space criterion plus a careful comparison of coskeletality under décalage (Prop. 20, Lemma 21). The Wiggle Lemma and the compatibility lemmas are written out cleanly. Remark 26 correctly kills the analogous statement for simplicial spaces, so the scope is honest.\n\nThe reader’s main defect is illusory. Coskeletality is monotone: (d+1)-coskeletal already implies (d+2)-coskeletal, so Theorem 25 immediately yields the abstract’s iff. The footnotes about classical low-d cases describe exactly this phenomenon. I traced the parity bookkeeping in Theorems 22–24 against the décalage lemmas; it closes. No circularity, no free parameters, citations are the right ones (Dyckerhoff–Kapranov, Poguntke, Walde, Bergner et al.).\n\nSoft spots are minor and proportional. The paper is pure combinatorial set-level work; anyone wanting ∞-categorical or space-level strengthenings will have to do more. The inductive filling arguments are a bit long but not fragile. Impact is real inside the higher-Segal / decomposition-space corner and essentially zero outside it.\n\nThis is for people already working with higher Segal conditions or coskeletal simplicial sets. It deserves a serious referee. I would accept it for peer review and would cite it if I needed a recognition criterion or a finite check for d-Segality.","headline":"Clean uniform recognition theorem for d-Segal sets; the abstract/body gap the reader flagged is not real.","tokens_in":14151,"tokens_out":523,"would_cite":true,"duration_ms":13525,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N50","55U10","18G30"],"pacs":[],"model":"grok-4.5","headline":"A simplicial set is d-Segal exactly when it is (d+1)-coskeletal and satisfies the d-Segal conditions in the two lowest relevant dimensions.","keywords":["higher Segal conditions","coskeletal simplicial sets","décalage","path space criterion","gapped subsets","2-Segal sets","simplicial sets"],"falsifier":"Exhibit a simplicial set that is (d+1)-coskeletal and satisfies the two critical d-Segal conditions yet fails a higher-dimensional gapped-cube condition, or a d-Segal simplicial set whose (d+1)-spheres do not fill uniquely.","tokens_in":13849,"feed_emoji":"△","tokens_out":972,"duration_ms":20550,"temperature":0.7,"pith_summary":"Higher Segal conditions ask that certain cubes built from a simplicial set be cartesian; they generalize the classical Segal condition that models categories. Checking them in every dimension looks infinite. This paper proves that for ordinary simplicial sets the infinite list collapses: the set is upper or lower d-Segal if and only if it is (d+1)-coskeletal and the conditions hold in just the two lowest non-vacuous dimensions (called d-critical). The converse direction also shows every d-Segal set is automatically (d+1)-coskeletal. The reduction turns an infinite exactness check into a finite one, and it is sharp: the analogous statement fails for simplicial spaces.","feed_headline":"d-Segal sets reduce to two dimensions plus coskeletality","feed_subtitle":"An infinite exactness check on simplicial sets collapses to a finite one; the space case fails","key_machinery":"The path-space criterion relating Segality of X to Segality of its upper and lower décalages, together with the Wiggle Lemma that propagates cartesianness of one gapped cube to all others in the same dimension, and the comparison theorems that transfer coskeletality between X and its décalages. These close the inductive step from the two critical dimensions to all higher ones.","core_discovery":"For a simplicial set X and d ≥ 0, X is upper (respectively lower) d-Segal if and only if it is (d+1)-coskeletal and upper (respectively lower) d-critical—that is, it satisfies the corresponding d-Segal cube conditions in simplicial dimensions d+1 and d+2. Equivalently, (d+2)-coskeletal plus d-critical already forces full d-Segality, while every d-Segal set is at least (d+1)-coskeletal.","pith_inferences":["The same coskeletal-plus-critical pattern may supply practical recognition algorithms for higher Segal structures arising in Hall algebras or incidence coalgebras once those structures are presented by finite generators.","Because n-skeletal sets are automatically (2n−1)-coskeletal, the criterion immediately classifies the Segal height of many quotients and suspensions already studied in algebraic topology.","A computer-checkable test for d-criticality on truncated simplicial sets could turn the theorem into a decision procedure for small examples."],"forward_implications":["d-Segality of a finite simplicial set (finitely many nondegenerate simplices) becomes a finite, effective check once d is large enough relative to skeletal dimension.","The hierarchy of higher Segal conditions is strict: familiar examples such as Δ^n/∂Δ^n are 2n-Segal but not lower (2n−1)-Segal.","Every upper or lower d-Segal simplicial set is automatically (d+1)-coskeletal, recovering the classical facts for nerves of categories (d=1) and for 2-Segal sets.","The corresponding reduction fails for simplicial spaces, so the result is special to discrete simplicial sets."],"fun_headline_variants":["d-Segal sets = (d+1)-coskeletal + two low-dimension checks","Full d-Segality collapses to coskeletality plus d-critical cubes","(d+1)-coskeletal and d-critical already force d-Segal sets","Upper or lower d-Segal reduces to two cube conditions plus coskeleton","d-Segal exactness needs only dimensions d+1, d+2 and coskeletality"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument assumes that unique fillers for the two lowest gapped cubes, once coskeletality is known, can be propagated to every higher dimension by wiggling between adjacent gapped sets and by the décalage comparison; if that transfer failed for any parity, the induction would stop.","fun_headline_variants_meta":{"raw":{"variants":["d-Segal sets = (d+1)-coskeletal + two low-dimension checks","Full d-Segality collapses to coskeletality plus d-critical cubes","(d+1)-coskeletal and d-critical already force d-Segal sets","Upper or lower d-Segal reduces to two cube conditions plus coskeleton","d-Segal exactness needs only dimensions d+1, d+2 and coskeletality"]},"model":"grok-4.5","effort":"low","cost_usd":0.003903,"raw_usage":{"total_tokens":1123,"prompt_tokens":600,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":39028000,"prompt_tokens_details":{"text_tokens":600,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":425,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":600,"tokens_out":98,"duration_ms":6970,"temperature":1.0,"reasoning_tokens":425,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T13:46:15.336045+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a simplicial set that is (d+1)-coskeletal and satisfies the two critical d-Segal conditions yet fails a higher-dimensional gapped-cube condition, or a d-Segal simplicial set whose (d+1)-spheres do not fill uniquely.","supporting_citations":[],"review_version":1}