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We estimate $\\lambda = 1.744550 \\pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628 < \\lambda < 1.782.$ We give exact results for the number of SAW of length $2L + 2K$ for $K = 0, 1, 2$ and asymptotic results for $K = o(L^{1/3})$.\n  We also consider the model in which a weight or {\\em fugacity}"},"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":"cond-mat/0506341","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"cond-mat.stat-mech","submitted_at":"2005-06-14T23:25:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"80bf8e1a1093ee8683ca4d169416081c85b2969aca9f38435baa813f9c6285b6","abstract_canon_sha256":"e3e09d9505be1f09ced91b15e10fbc34020602b1e6449ddf2ccd2dd6aca11c7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:52.586400Z","signature_b64":"J3KqZ6IQx4O1P1Mg9N25WOk9KYpDCX19xg1SOBssHuSrZje20f7vjOx3tuGoAqb5X3AaSFh9GS5+cyt82ANnCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a2494532e0026abafcd2ac2c502c4d0e910022e45cfa2caf705117691a2c833","last_reissued_at":"2026-05-18T01:06:52.585962Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:52.585962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-avoiding walks crossing a square","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"cond-mat.stat-mech","authors_text":"A. 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