{"id":"7e4c155b-321d-4c8d-b952-c22d0e80ce60","arxiv_id":"2606.29449","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves local-to-global KKL-type theorem for simplicial complexes (if links satisfy KKL then global does) and weaker version for any non-trivially two-sided expanding complexes, with applications to combinatorial HDX and Ramanujan complexes.","lead":"This paper introduces a local-to-global method proving that if the links of a simplicial complex satisfy a KKL theorem then the complex does globally, plus a weaker dimension-dependent version for any non-trivially expanding complex. Smart generalists might read it to see how local expansion properties lift to global structure in high-dimensional combinatorics and theoretical computer science.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the link-satisfaction premise as the weakest assumption aligns exactly with the conditional form of the claim. No additional load-bearing risk (e.g., hidden assumptions or inconsistency with the weaker result) appears in the abstract-level description.","tokens_in":1796,"tokens_out":251,"duration_ms":24904,"concrete_test":"Extract the statement and proof of the main local-to-global theorem; verify that the argument derives the global KKL bound solely from the link hypothesis without inserting extra expansion or dimension assumptions not present in the link KKL statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is explicitly conditional: simplicial complexes inherit a global KKL-type result precisely when their links satisfy the property (or, in the weaker case, when the complex has non-trivial two-sided expansion). The abstract presents this as the output of a new local-to-global method, with no indication of an internal gap, circularity, or unstated quantitative requirement that would invalidate the transfer. The dimension dependence of the weaker result is stated openly. Absent a concrete flaw in the transfer argument itself, the conditional structure holds as described.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces a local-to-global method for low-influence functions on simplicial complexes. It proves that any simplicial complex whose links satisfy a KKL theorem also satisfies a global KKL-type result, and establishes a weaker dimension-dependent version for complexes with non-trivial two-sided expansion (building on Gotlib-Kaufman). Applications include the first characterization of non-expanding functions on combinatorial HDX such as dense clique complexes (with a corresponding Kruskal-Katona theorem) and a small-set expansion result for the Ramanujan complexes of Lubotzky-Samuels-Vishne.","tokens_in":1901,"tokens_out":355,"duration_ms":22966,"significance":"If the transfer argument holds, the work removes the strong quantitative expansion barriers of prior KKL analogs on HDX (Bafna et al., Gur et al.) and supplies a general framework applicable whenever links satisfy the local property. The weaker result for arbitrary non-trivial expansion and the concrete applications to combinatorial HDX and Ramanujan complexes are concrete strengths; the conditional structure of the main theorem is stated clearly.","major_comments":[],"minor_comments":[{"comment":"Abstract: the phrase 'combinatorial HDX such as dense clique complexes' is used without a precise definition or citation to the exact class of complexes; adding a one-sentence clarification would improve readability.","section":"Abstract"},{"comment":"The dimension dependence of the weaker theorem is noted in the abstract but should be stated quantitatively (e.g., the precise dependence on dimension d) in the introduction or theorem statement for immediate comparison with prior work.","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report correctly identifies the local-to-global transfer as the central contribution and notes the concrete applications to combinatorial HDX and Ramanujan complexes.","responses":[],"tokens_in":1342,"tokens_out":63,"duration_ms":20533,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper gives a local-to-global lift for KKL results on simplicial complexes, relaxing the strong expansion needed in the 2022 STOC papers and reaching clique complexes plus Ramanujan complexes.\n\nThe new piece is the lifting method itself: if every link satisfies a KKL theorem then the whole complex does. They also get a weaker, dimension-dependent version that needs only non-trivial two-sided expansion, using the Gotlib-Kaufman RANDOM 2023 result. The applications follow directly: first characterization of non-expanding functions on dense clique complexes, a Kruskal-Katona theorem there, and small-set expansion on the Lubotzky-Samuels-Vishne Ramanujan complexes.\n\nThe argument is presented as conditional on the links, which matches the abstract exactly, and the dimension dependence is stated openly. No internal contradiction or hidden quantitative blow-up appears in the description. The method looks like a straightforward way to move from local to global without the prior strong-expansion barrier.\n\nOne soft spot is that the weaker result loses strength as dimension grows, which limits its use in very high-dimensional settings even if the complex expands. Checking the link condition for new complexes may also require separate work, though the paper uses it to reach examples the STOC results could not.\n\nThis is for people working on high-dimensional expanders and boolean analysis on non-product domains. A reader who knows the 2022 papers will see the technical step and the new examples clearly. It has enough new technique and concrete reach to deserve a serious referee.","headline":"The paper gives a local-to-global lift for KKL results on simplicial complexes, relaxing the strong expansion needed in the 2022 STOC papers and reaching clique complexes plus Ramanujan complexes.","tokens_in":2390,"tokens_out":396,"would_cite":false,"duration_ms":20599,"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":"Any simplicial complex inherits a KKL theorem from its links via a local-to-global argument.","keywords":["KKL theorem","high-dimensional expanders","simplicial complexes","local-to-global","boolean function analysis","expansion","Kruskal-Katona theorem","Ramanujan complexes"],"falsifier":"A simplicial complex in which every link satisfies a KKL theorem but the global complex fails to exhibit the corresponding low-influence structure would disprove the local-to-global transfer.","tokens_in":2681,"feed_emoji":"","tokens_out":561,"duration_ms":23092,"temperature":0.7,"pith_summary":"The paper develops a method to lift KKL-type results from the links of a simplicial complex to the whole complex. This allows proving that if every link satisfies a KKL theorem, then the complex does too. It also gives a weaker result for any complex with two-sided expansion. This extends previous work that required strong global expansion. The approach yields new results for combinatorial HDX and Ramanujan complexes.","feed_headline":"KKL theorem transfers from links to any simplicial complex","feed_subtitle":"Local-to-global method proves the result whenever links satisfy it, plus a weaker version for any expanding complex.","key_machinery":"A simple local-to-global method for analyzing low-influence functions on simplicial complexes.","core_discovery":"The central discovery is a local-to-global KKL theorem: any simplicial complex whose links satisfy a KKL theorem also satisfies one globally. Building on prior work, a dimension-dependent version holds for any non-trivially expanding complex. This yields the first characterization of non-expanding functions on dense clique complexes together with a Kruskal-Katona theorem and a small-set expansion result for Ramanujan complexes.","pith_inferences":["The method suggests that verifying KKL on small links may suffice for global results across many complexes.","It could extend to other local-to-global phenomena in high-dimensional expanders beyond KKL.","Similar transfers might apply to additional theorems from boolean analysis on HDX."],"forward_implications":["First characterization of non-expanding functions on dense clique complexes.","Corresponding Kruskal-Katona theorem for those complexes.","Small-set expansion theorem for Ramanujan complexes.","KKL-type results now hold without the strong quantitative expansion requirements of earlier HDX theorems."],"fun_headline_variants":["KKL theorem from links to global on simplicial complexes","Local-to-global KKL on any simplicial complex","Complexes inherit KKL when links satisfy it","KKL holds globally if links hold locally","Dimension-dependent KKL for expanding complexes"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The links of the simplicial complex must themselves satisfy a KKL theorem, or the complex must have non-trivial two-sided expansion.","fun_headline_variants_meta":{"raw":{"variants":["KKL theorem from links to global on simplicial complexes","Local-to-global KKL on any simplicial complex","Complexes inherit KKL when links satisfy it","KKL holds globally if links hold locally","Dimension-dependent KKL for expanding complexes"]},"model":"grok-4.3","cost_usd":0.003192,"raw_usage":{"total_tokens":1756,"prompt_tokens":743,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":31924500,"prompt_tokens_details":{"text_tokens":743,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":944,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":743,"tokens_out":69,"duration_ms":13646,"temperature":1.0,"reasoning_tokens":944,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T02:03:40.729258+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A simplicial complex in which every link satisfies a KKL theorem but the global complex fails to exhibit the corresponding low-influence structure would disprove the local-to-global transfer.","supporting_citations":[],"review_version":1}