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The set of solutions of a random formula exhibits various geometrical transitions as the ratio $\\frac{m}{n}$ varies.\n  We consider a {\\em coupled} $K$-XORSAT ensemble, consisting of a chain of random XORSA"},"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":"1303.0540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2013-03-03T18:23:58Z","cross_cats_sorted":["cs.DM","cs.IT","math.IT"],"title_canon_sha256":"7bad9c17ab001d85094bb3aae16442590d243f86c3c223993a3274f192303dff","abstract_canon_sha256":"951ad30392b38bb65eedcff141d0ae5071ad4ca39b2ddeb2478e793cdfb50b1c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:55.623227Z","signature_b64":"pvR5EZ6aWCV0j0WaZoZva0QWx2T7vzVHAtisjVf+ZC2S2kkmgEwXionEgmx2NSYWhzWKATqH7R2MErOfC2x/Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1fbd3d99dbde7a6a3716b7ee2ecc42086891131c5cc65717ac047f560e25fc01","last_reissued_at":"2026-05-18T03:31:55.622688Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:55.622688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Space of Solutions of Coupled XORSAT Formulae","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.IT","math.IT"],"primary_cat":"cond-mat.dis-nn","authors_text":"Nicolas Macris, Rudiger Urbanke, S. 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