{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:GMHUUCCX3VB7H6P5FSVWIVNOBN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6b4b5f2c38b14511d38baddb543e5964bf5b66155ae811bdac016eff2f9fbe0e","cross_cats_sorted":[],"license":"","primary_cat":"math.MG","submitted_at":"1994-05-13T00:00:00Z","title_canon_sha256":"315cc90b5701d34fcff35e99680c7732959601b8fc526b627e65bb42e8b622c1"},"schema_version":"1.0","source":{"id":"math/9405218","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9405218","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9405218v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9405218","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"GMHUUCCX3VB7","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"GMHUUCCX3VB7H6P5","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"GMHUUCCX","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:5b0963be733444fce2f8f7d949e2dc9fdececf28da5ff61511c590eb806a55da","target":"graph","created_at":"2026-05-18T01:05:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies\n 12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928\n  We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contrad","authors_text":"Greg Kuperberg, Oded Schramm","cross_cats":[],"headline":"","license":"","primary_cat":"math.MG","submitted_at":"1994-05-13T00:00:00Z","title":"Average kissing numbers for non-congruent sphere packings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9405218","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:814c4caf9dd18dd13a1eb13edbf8da4de6211c9ac663219ef6e08cf24a7e7e08","target":"record","created_at":"2026-05-18T01:05:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6b4b5f2c38b14511d38baddb543e5964bf5b66155ae811bdac016eff2f9fbe0e","cross_cats_sorted":[],"license":"","primary_cat":"math.MG","submitted_at":"1994-05-13T00:00:00Z","title_canon_sha256":"315cc90b5701d34fcff35e99680c7732959601b8fc526b627e65bb42e8b622c1"},"schema_version":"1.0","source":{"id":"math/9405218","kind":"arxiv","version":1}},"canonical_sha256":"330f4a0857dd43f3f9fd2cab6455ae0b4fbe968f825918fa2009375175478f3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"330f4a0857dd43f3f9fd2cab6455ae0b4fbe968f825918fa2009375175478f3f","first_computed_at":"2026-05-18T01:05:51.353963Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.353963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UzwOpFGk/aI46Gxhx0Zp+bKwldnYEgMs9yL1/AI/cC5jZCG4BToPvOCgRXB/uSH8hnXoLasYPQ6rsOtFOLK0Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.354451Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9405218","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:814c4caf9dd18dd13a1eb13edbf8da4de6211c9ac663219ef6e08cf24a7e7e08","sha256:5b0963be733444fce2f8f7d949e2dc9fdececf28da5ff61511c590eb806a55da"],"state_sha256":"2ac4c4a0c755746a386da8250fd1ca8c4b1d26efe0a4a6d848cbc29cabda8ab9"}