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E.g. for $\\pi^0 \\pi^0$ scattering we show that for c.m. energy $\\sqrt{s}\\rightarrow \\infty $, $\\bar{\\sigma}_{tot}(s,\\infty)\\equiv s\\int_{s} ^{\\infty} ds'\\sigma_{tot}(s')/s'^2 \\leq \\pi (m_{\\pi})^{-2} [\\ln (s/s_0)+(1/2)\\ln \\ln (s/s_0) +1]^2$ where $m_\\pi^2/s_0= 17\\pi \\sqrt{\\pi/2} $ ."},"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":"1306.5210","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-ph","submitted_at":"2013-06-21T18:29:40Z","cross_cats_sorted":[],"title_canon_sha256":"7f727664fa8a348254e6d8f68eb6f01b9fcf2e26b863b020a8d1c62ab557120d","abstract_canon_sha256":"47cbc3af1a6480dac8cc41a807dbcb623397336259245b40ecbb86435e92100b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:23.801087Z","signature_b64":"K6FGfKmzKeaEMvl4fpgUgJhRh1kRRCDp49WxqFVuUsHKsaC9/IyQI+F1y5mUrfC+XKLfmLj8JzKrzEq10CXCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5a16b4118c72392a91663023e6636a20c78b83d9f8ae21007bd50a574dfb9ce","last_reissued_at":"2026-05-18T02:57:23.800576Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:23.800576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Froissart Bound on Total Cross-section without Unknown Constants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Andr\\'e Martin (CERN, Geneva), Mumbai), S. 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