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We prove a product lemma that combines gadgets of different dimensions while preserving the required independence conditions. Starting from the size-$367$ independent set in $C_7^5$ of S. C. Polak and A. Schrijver (Information Processing Letters 143 (2019), 37-40), the construction gives an explicitly specified independent set in $C_7^{200}$. Consequently, \\[ \\Theta(C_7)\\geq 3.2587891539086910161967650155"},"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":"2607.27869","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-30T08:44:15Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"fa2fe154d79396852fda29809446b48154870194c2d83df81001503cd8a9cf62","abstract_canon_sha256":"285c6680b4ce499f036a1f8f9f434f2569566ee1d17526901e94ded77e3b362a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dcba7db8d8fe2e6a47a52e36f45b9f8c13e999e17d4319965aed5df472d43144","last_reissued_at":"2026-07-31T01:34:31.565463Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-31T01:34:31.565463Z"},"graph_snapshot":{"paper":{"title":"A Recursive Construction Improving the Lower Bound on the Shannon Capacity of $C_7$","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Yu Gao","submitted_at":"2026-07-30T08:44:15Z","abstract_excerpt":"We give a recursive reformulation and extension of the independent set of size $134753$ in $C_7^{10}$ constructed by N. Itty, C. D. Rosin, C. Carstensen, and D. Reichman (arXiv:2607.21517v1). We prove a product lemma that combines gadgets of different dimensions while preserving the required independence conditions. Starting from the size-$367$ independent set in $C_7^5$ of S. C. Polak and A. Schrijver (Information Processing Letters 143 (2019), 37-40), the construction gives an explicitly specified independent set in $C_7^{200}$. 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