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In more details, let \\[ \\mathbf X_t=\\exp(tl) \\,\\text{ with }\\, l=l_1+\\cdots+l_m\\,\\text{ and }\\, l_r\\in\\mathcal L_r(V), \\] be a pure $m$-rough path over a finite dimensional real or complex Banach space, and equip the tensor powers of $V$ with arbitrary reasonable tensor algebra norms. We prove that \\[ \\limsup_{n\\to\\infty}\\left(\\left(\\frac{n}{m}\\right)!\\left\\|\\pi_n(\\exp l)\\right\\|_n\\right)^{m/n}=\\|l_m\\|_m . \\] In particular, this identifies the signatu"},"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":"2606.30055","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-29T09:45:41Z","cross_cats_sorted":[],"title_canon_sha256":"0e35c148afc391573d471dca540c120b00f3c89f23a6ce5ca96d5779784f7c24","abstract_canon_sha256":"b211cd0687ce5092a149ea219f0302323fe567a08bbcdd7c647d41b5f88e4ea4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T02:17:47.688162Z","signature_b64":"ur1bgvSPEEZempUIk/7r74/XxTgN1mUJx+19gNsF3i8UGuRw9bs3IlZaOGa+Y7xta2FGvKFeB8V+9IsDc80uAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"472ed84d1853f1c1badba444c2b061563785982dc0a1a91aa6f863368112e3a5","last_reissued_at":"2026-06-30T02:17:47.687633Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T02:17:47.687633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact Signature Tail Asymptotics for Pure Rough Paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nannan Li, Xing Gao","submitted_at":"2026-06-29T09:45:41Z","abstract_excerpt":"We prove~\\cite[Conjecture 2.12]{BGS20} on the signature tail asymptotics of pure rough paths and extend it to arbitrary reasonable tensor norms. In more details, let \\[ \\mathbf X_t=\\exp(tl) \\,\\text{ with }\\, l=l_1+\\cdots+l_m\\,\\text{ and }\\, l_r\\in\\mathcal L_r(V), \\] be a pure $m$-rough path over a finite dimensional real or complex Banach space, and equip the tensor powers of $V$ with arbitrary reasonable tensor algebra norms. We prove that \\[ \\limsup_{n\\to\\infty}\\left(\\left(\\frac{n}{m}\\right)!\\left\\|\\pi_n(\\exp l)\\right\\|_n\\right)^{m/n}=\\|l_m\\|_m . \\] In particular, this identifies the signatu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30055","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/2606.30055/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":"2606.30055","created_at":"2026-06-30T02:17:47.687713+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.30055v1","created_at":"2026-06-30T02:17:47.687713+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.30055","created_at":"2026-06-30T02:17:47.687713+00:00"},{"alias_kind":"pith_short_12","alias_value":"I4XNQTIYKPY4","created_at":"2026-06-30T02:17:47.687713+00:00"},{"alias_kind":"pith_short_16","alias_value":"I4XNQTIYKPY4DOW3","created_at":"2026-06-30T02:17:47.687713+00:00"},{"alias_kind":"pith_short_8","alias_value":"I4XNQTIY","created_at":"2026-06-30T02:17:47.687713+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/I4XNQTIYKPY4DOW3URCMFMDBKY","json":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY.json","graph_json":"https://pith.science/api/pith-number/I4XNQTIYKPY4DOW3URCMFMDBKY/graph.json","events_json":"https://pith.science/api/pith-number/I4XNQTIYKPY4DOW3URCMFMDBKY/events.json","paper":"https://pith.science/paper/I4XNQTIY"},"agent_actions":{"view_html":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY","download_json":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY.json","view_paper":"https://pith.science/paper/I4XNQTIY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.30055&json=true","fetch_graph":"https://pith.science/api/pith-number/I4XNQTIYKPY4DOW3URCMFMDBKY/graph.json","fetch_events":"https://pith.science/api/pith-number/I4XNQTIYKPY4DOW3URCMFMDBKY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY/action/storage_attestation","attest_author":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY/action/author_attestation","sign_citation":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY/action/citation_signature","submit_replication":"https://pith.science/pith/I4XNQTIYKPY4DOW3URCMFMDBKY/action/replication_record"}},"created_at":"2026-06-30T02:17:47.687713+00:00","updated_at":"2026-06-30T02:17:47.687713+00:00"}