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We charge only $E_{\\dim}(\\rho_{LR})=\\log_2\\min\\{\\operatorname{rank}\\rho_L,\\operatorname{rank}\\rho_R\\}$, the logarithm of the smaller marginal support dimension of the shared state prepared before the inputs arrive. The state may be arbitrary and mixed. For the inner product modulo $2$, we pr"},"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":"2608.05775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2026-08-06T09:09:44Z","cross_cats_sorted":[],"title_canon_sha256":"0e97c7e5e95d950c9395d4bc7e476d2ca510fba0d57fbde1b2d855ab70c59cfd","abstract_canon_sha256":"640dc7fbedef91342d48954b80f4dde090c185b5b459d080e68340b06c47e7bc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-07T00:52:43.812730Z","signature_b64":"EnOvLelOQSnNtLpEQqVSfgkP7HnfJJDqIGcoOauOog2LvOotnZzhfCPwSjbs8gUlozDVGQ7iKowKnJA+q6kADw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3080c13bb59b1ae3fea6e80d830daade2f9e4ad03f78bcd499d162d08f64cb49","last_reissued_at":"2026-08-07T00:52:43.811329Z","signature_status":"signed_v1","first_computed_at":"2026-08-07T00:52:43.811329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Robust logarithmic lower bound on shared-resource cost for $f$-routing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Kevin Bogner","submitted_at":"2026-08-06T09:09:44Z","abstract_excerpt":"In one-round $f$-routing, Alice receives an $n$-bit string and an unknown qubit, while Bob receives another $n$-bit string. They exchange one simultaneous message each, and the value of $f$ determines which party must recover the qubit. Message lengths, local systems, and local operations are unrestricted. We charge only $E_{\\dim}(\\rho_{LR})=\\log_2\\min\\{\\operatorname{rank}\\rho_L,\\operatorname{rank}\\rho_R\\}$, the logarithm of the smaller marginal support dimension of the shared state prepared before the inputs arrive. The state may be arbitrary and mixed. 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