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Let $\\delta_{1,\\mathcal{T}}$ be the signed edge-triangle incidence matrix, and $\\lambda(\\mathcal{T})$ the spectral gap of $\\delta_{1,\\mathcal{T}}^T\\delta_{1,\\mathcal{T}}.$\n  Our main results show that large $\\lambda(\\mathcal{T})$ forces strong overlap and a large minimum degree in the support graph. In particular, every support edge lies in at least $\\lceil \\lambda(\\mathcal{T})\\rceil-2$ triangles in $\\mathcal{T}$ and hence the graph"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.27307","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T17:18:11Z","cross_cats_sorted":[],"title_canon_sha256":"7ae17ff07891d0039f91434a7f3b5cb636b2153cb1fe815b7b66cc262d65a707","abstract_canon_sha256":"6c3cf1700285401f28bd1a054b17d6fa3f8115227b6d569aca1a621e33f3e96d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T02:06:17.367650Z","signature_b64":"Fbz2YDHdxNrJfAMHbD5/qeIFR8Y1SuPv5gBE9HXGJ+XhjOss8dCqEDicw9ofUGeq5f/eJTgEP5FQeVQvsi2TAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c784818b94110795f8f2d322178ecf7db639cb8b0ea7024c39312ca29cee59c","last_reissued_at":"2026-05-27T02:06:17.366774Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T02:06:17.366774Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Triangle Families with Large Edge Up-Laplacian Spectral Gap","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mutasim Mim","submitted_at":"2026-05-26T17:18:11Z","abstract_excerpt":"Let $\\mathcal{T}$ be a finite nonempty set of $3$-element subsets of a totally ordered set $V$. 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