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This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $\\alpha = \\Theta(1)$ and $\\alpha = \\Theta(\\log n)$, but nowhere in between.\n  More specifically, we exhibit $n$-point spaces with doubling constant $\\lambda$ requiring Eucli"},"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":"0910.0277","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-10-01T21:45:27Z","cross_cats_sorted":[],"title_canon_sha256":"14ae5d6fbbbf2bf2c6e9c8a8ff79fb569212523c4023398f5997c8feef8b27b9","abstract_canon_sha256":"b6df7a90b7ff16a65083b739643b473a8188122b58e05a161b43a21fc6bac859"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:35.141639Z","signature_b64":"0evKjDvClxJEvKePSZF/V9tDXIyCmJ1BF0KibGrxE6FJtbocHDD2rzBJrRGLV4GH7MavlE/Emv7UbnHspyvmDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5624ce05e10a8b6886526329472fb74b3b42c2c52b1a02c0c68d9d74c022322d","last_reissued_at":"2026-05-18T02:11:35.140938Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:35.140938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the optimality of gluing over scales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Jaffe, James R. 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