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More generally, put $A=\\log_3 x+\\log_4 x-\\log 2$, $G=\\sqrt{\\log x,A}$, and $V=\\log x/G$. For every fixed integer $J\\ge 1$, uniformly for $1\\le h\\le \\exp{G/\\sqrt{J}}$, we obtain $S_h^\\varphi(x)=D_{h,>Y_J}^\\varphi(x)+O_J(x\\exp{-\\sqrt{J},G+o_J(V)})$, where $Y_J=\\exp{\\sqrt{J},G}$. Here $D_{h,>Y_J}^\\varphi(x)$ is the above-cutoff part of the classical Graham--Holt--Pomerance same-support family; it is empty for od"},"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":"2606.23681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-22T17:59:25Z","cross_cats_sorted":[],"title_canon_sha256":"7d6b05089ea801f43c905f63c1ef1b3bf245bea0eb7c6769fd215cea8beba464","abstract_canon_sha256":"554250a8cd80955ab88f060b8e7a0db7b15fa71592a07a4d7c22d853bfe9c3c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:14:34.546884Z","signature_b64":"PmiBkVPtVDPSfkp36cv31iKTwCDXHEWPO3xJwfei/sgESMB3ix3ogzSqJH6A+1n88qyT4/j95vmRStPvi/B3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00afb010589db176f133d667c5f924d6e326d1dabafd70f3e6688c1a85d71100","last_reissued_at":"2026-06-23T03:14:34.546538Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:14:34.546538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rank Amplification for Shifted Equal Values of Euler's Totient Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-22T17:59:25Z","abstract_excerpt":"Let $S_h^\\varphi(x)$ denote the number of integers $n\\le x$ for which $\\varphi(n)=\\varphi(n+h)$. For the unit shift, we prove $S_1^\\varphi(x)\\ll x\\exp{-(1/2-o(1))\\sqrt{\\log x,\\log_2 x}}$. More generally, put $A=\\log_3 x+\\log_4 x-\\log 2$, $G=\\sqrt{\\log x,A}$, and $V=\\log x/G$. For every fixed integer $J\\ge 1$, uniformly for $1\\le h\\le \\exp{G/\\sqrt{J}}$, we obtain $S_h^\\varphi(x)=D_{h,>Y_J}^\\varphi(x)+O_J(x\\exp{-\\sqrt{J},G+o_J(V)})$, where $Y_J=\\exp{\\sqrt{J},G}$. Here $D_{h,>Y_J}^\\varphi(x)$ is the above-cutoff part of the classical Graham--Holt--Pomerance same-support family; it is empty for od"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23681","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/2606.23681/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":"2606.23681","created_at":"2026-06-23T03:14:34.546598+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.23681v1","created_at":"2026-06-23T03:14:34.546598+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.23681","created_at":"2026-06-23T03:14:34.546598+00:00"},{"alias_kind":"pith_short_12","alias_value":"ACX3AECYTWYX","created_at":"2026-06-23T03:14:34.546598+00:00"},{"alias_kind":"pith_short_16","alias_value":"ACX3AECYTWYXN4JT","created_at":"2026-06-23T03:14:34.546598+00:00"},{"alias_kind":"pith_short_8","alias_value":"ACX3AECY","created_at":"2026-06-23T03:14:34.546598+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/ACX3AECYTWYXN4JT2ZT4L6JE23","json":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23.json","graph_json":"https://pith.science/api/pith-number/ACX3AECYTWYXN4JT2ZT4L6JE23/graph.json","events_json":"https://pith.science/api/pith-number/ACX3AECYTWYXN4JT2ZT4L6JE23/events.json","paper":"https://pith.science/paper/ACX3AECY"},"agent_actions":{"view_html":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23","download_json":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23.json","view_paper":"https://pith.science/paper/ACX3AECY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.23681&json=true","fetch_graph":"https://pith.science/api/pith-number/ACX3AECYTWYXN4JT2ZT4L6JE23/graph.json","fetch_events":"https://pith.science/api/pith-number/ACX3AECYTWYXN4JT2ZT4L6JE23/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23/action/storage_attestation","attest_author":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23/action/author_attestation","sign_citation":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23/action/citation_signature","submit_replication":"https://pith.science/pith/ACX3AECYTWYXN4JT2ZT4L6JE23/action/replication_record"}},"created_at":"2026-06-23T03:14:34.546598+00:00","updated_at":"2026-06-23T03:14:34.546598+00:00"}