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We denote by $c:=2\\sum_{j\\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \\[ \\log\\frac{1}{\\lambda(t)} = \\left(\\frac{9c}{4}\\right)^{1/3}t^{2/3}+O(t^{1/3}) \\qquad \\text{as } t\\to+\\infty . \\] This concentration rate comes fr"},"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":"2606.04449","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-03T04:51:52Z","cross_cats_sorted":[],"title_canon_sha256":"4d6a042355f621f852536f9b850845d9a99a58c99c33c18a41ccf1988e18b5ff","abstract_canon_sha256":"b84eab7d6eef39bb9e43b75bf4308adb3705c40102b145d8c72f3e186f6d4196"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:09:08.582078Z","signature_b64":"LbC+ycrx8WzbMusIUnn20J4CABJBOUYfKw3dEc6LHPRoZq4BbvIrz1vSnF3iVMyH5TPyRs1YEWf9cRcV91DKAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15f5f0d310bf7f05f2b2c92c23929d6e161ee51070d8b3898f37dadb3219e403","last_reissued_at":"2026-06-04T01:09:08.581630Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:09:08.581630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Construction of multi-bubble solutions for the energy-critical wave equation in dimension four","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jacek Jendrej, Jingyuan Gu, Lifeng Zhao","submitted_at":"2026-06-03T04:51:52Z","abstract_excerpt":"For any $N\\geq 2$, we construct a global solution of the energy-critical focusing wave equation in dimension four which blows up in infinite time at $N$ prescribed points $z_1,\\ldots,z_N\\in \\mathbb R^4$, provided that the points form one orbit under a finite group of orthogonal symmetries. We denote by $c:=2\\sum_{j\\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \\[ \\log\\frac{1}{\\lambda(t)} = \\left(\\frac{9c}{4}\\right)^{1/3}t^{2/3}+O(t^{1/3}) \\qquad \\text{as } t\\to+\\infty . \\] This concentration rate comes fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04449","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.04449/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.04449","created_at":"2026-06-04T01:09:08.581693+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.04449v1","created_at":"2026-06-04T01:09:08.581693+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04449","created_at":"2026-06-04T01:09:08.581693+00:00"},{"alias_kind":"pith_short_12","alias_value":"CX27BUYQX57Q","created_at":"2026-06-04T01:09:08.581693+00:00"},{"alias_kind":"pith_short_16","alias_value":"CX27BUYQX57QL4VS","created_at":"2026-06-04T01:09:08.581693+00:00"},{"alias_kind":"pith_short_8","alias_value":"CX27BUYQ","created_at":"2026-06-04T01:09:08.581693+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY","json":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY.json","graph_json":"https://pith.science/api/pith-number/CX27BUYQX57QL4VSZEWCHEU5NY/graph.json","events_json":"https://pith.science/api/pith-number/CX27BUYQX57QL4VSZEWCHEU5NY/events.json","paper":"https://pith.science/paper/CX27BUYQ"},"agent_actions":{"view_html":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY","download_json":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY.json","view_paper":"https://pith.science/paper/CX27BUYQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.04449&json=true","fetch_graph":"https://pith.science/api/pith-number/CX27BUYQX57QL4VSZEWCHEU5NY/graph.json","fetch_events":"https://pith.science/api/pith-number/CX27BUYQX57QL4VSZEWCHEU5NY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY/action/storage_attestation","attest_author":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY/action/author_attestation","sign_citation":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY/action/citation_signature","submit_replication":"https://pith.science/pith/CX27BUYQX57QL4VSZEWCHEU5NY/action/replication_record"}},"created_at":"2026-06-04T01:09:08.581693+00:00","updated_at":"2026-06-04T01:09:08.581693+00:00"}