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Our framework gives depth-zero projections from colored subgraph isomorphism to the three targets at dimension, row count, or bit width $O(k\\log n)$, without increasing depth or size, and preserving gate orientation. For every fixed depth and every sufficiently large fixed $k$, we obtain unconditional bounds $n^{\\Omega(k)}$ for $k$-OV and $(n/k)^{\\Omega(k)}$ for $k$-XOR and $k$-SUM, with an absolute exponent-rate constant inde"},"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":"2608.08578","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-09T08:43:25Z","cross_cats_sorted":[],"title_canon_sha256":"dc52b2b1959565aac82017d9da079b77fb25900cdca95a689812d1e587872575","abstract_canon_sha256":"93ab44c860a25a4250d88e028964a4432ad3bc06f39554e42253eba8d3bf2d42"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T01:23:02.807221Z","signature_b64":"5CzWoQ3MZBnbF3YWT0PKBefVeoBnre39TwI3Z5gn8s1j/m7hh2pbKhJN/v3Vd59op0232fHWAaC/90Kbe4g8AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c25353c21b10b56278971880a799d6c128f0441847a9674ecb2a0dbc37f5cca","last_reissued_at":"2026-08-11T01:23:02.803032Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T01:23:02.803032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fine-Grained AC$^0$ Lower Bounds for $k$-OV, $k$-XOR, and $k$-SUM via Colored Subgraph Isomorphism","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Haoxing Lin","submitted_at":"2026-08-09T08:43:25Z","abstract_excerpt":"We prove lower bounds for $k$-OV, $k$-XOR, and $k$-SUM in nonuniform AC$^0$, tracking how the circuit-size exponent scales with $k$ and using no running-time hypothesis. Our framework gives depth-zero projections from colored subgraph isomorphism to the three targets at dimension, row count, or bit width $O(k\\log n)$, without increasing depth or size, and preserving gate orientation. 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