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The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincar\\'e polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is controlled by a Sturm--Rolle argument on the interval $0<y<1-t$. The original polynomial is recovered on the slice $y=1$, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\\overline{\\mathcal M}_{0,n}$ form an ultra-log-concave seque","title":"Real-rootedness of the Poincar\\'e polynomials of $\\overline{\\mathcal M}_{0,n}$: an AI-assisted proof","url":"https://arxiv.org/abs/2605.29151","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.29151v2 Announce Type: replace-cross \nAbstract: We prove real-rootedness for the Poincar\\'e polynomial \\[\n  P_n(t)=\\sum_{i=0}^{n-3} \\dim H^{2i}(\\overline{\\mathcal M}_{0,n};\\mathbb{Q})t^i \\] of the Deligne--Mumford moduli space $\\overline{\\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincar\\'e polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is controlled by a Sturm--Rolle argument on the interval $0<y<1-t$. The original polynomial is recovered on the slice $y=1$, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\\overline{\\mathcal M}_{0,n}$ form an ultra-log-concave seque","title":"Real-rootedness of the Poincar\\'e polynomials of $\\overline{\\mathcal M}_{0,n}$: an AI-assisted proof","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-12T04:43:44Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.29151"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:9ab1c9dd17d666b8218013cb5d9111221faeab74187dcf6c36b08f949a82f1f6d1f4114b7bd8bc762e849fa98a17cfc0d0671903030aba789b04ab128e79b90a","signer":"crovia.substrate","subject":{"observed_at":"2026-06-12T04:43:44Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.29151"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"b85071bbf4d684be10bc87f9fe0531e6653af54a9de7b9951e0b2a5b3e979281","leaf_index":229900,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"01a4b844667d8a92d28d4c7d943df70a09dbbe15c96c020badbf1cd60b0ea094","side":"right"},{"sibling":"a42a3e304c382e8cef3ea3fe5e6700f4fa694179bdf3357a1970351caa91386f","side":"right"},{"sibling":"17a7bb223f40ac8bcfb469208702c0c0c08f6e87446cc7d346647eb62b9954ad","side":"left"},{"sibling":"8ab03aca969463c34cc99cb0dd3f3af38a1525c3bc6244d32ef87ee111ca3cf6","side":"left"},{"sibling":"d47a2fa876204988c23786d19670280cb3dd8acbc098de7c6aad71b07fafb32f","side":"right"},{"sibling":"bf0e631fe6e0bc7e2f7b67b2bac99bc045b976d64492dd645f4d30340e7d6c02","side":"right"},{"sibling":"6bde248411f8ec3173e97ec3895054895c25f208bf85a22fd09c8fdf4911671b","side":"right"},{"sibling":"bc1002bb7e3da047b8a7c8f3990db61fb56edf499868605dc0c31aa0dee387b7","side":"right"},{"sibling":"14c50c43949e1ad41f149ffea691627d3f715c5861766c693b9fbac9d03b0d90","side":"right"},{"sibling":"74897e850164dddc689c3c65b33f9bae0268ab0bf429867a4e193d9b9b685040","side":"left"},{"sibling":"bde25d7e94e64717e426a97f6fcb4907e92b5c61fc89d92d7e0947a2249c3f6b","side":"right"},{"sibling":"d10d772a4984cae00e65ab24af21d1d260e475bbaae3f17871859eb705bd3999","side":"right"},{"sibling":"e79159853f2f35ddae8e3247d515e433c534277b287d65bbd77ae989aa4992fa","side":"right"},{"sibling":"3054319f1840cce0eaaf0bc4b1ae38e5bf8b6210927d924a750775cc7d77cca6","side":"right"},{"sibling":"0fd8b5059f279c4a4a6688de2472fdbc25543fee183df19b21dacca880354cff","side":"right"},{"sibling":"a04392fb9f2a3a620840e3b3fecd93d12e6d224e481c162389d8ae64e7194599","side":"left"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":232015,"merkle_root":"62bfb7809bb55667ad7eeebdb48267b9b2c1ee89bb808ea1e6d3a814a15aa402","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260617T133701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-17T13:39:59Z","sig_algorithm":"ed25519","signature":"930a3563c643cc7518d048b12a1f5392a96a5edce49533ba0a56f11b6cc69319bb1c800aacc46b52a6941c4d05dae255a45cac757d6093c971b96852c24f140e","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_4ff184cfbe721e9b5a04bf41d0b8335e59f393dc54fd2611ef4ed98f0499f75c"}}