{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:6ZAENYDDDJW5YGI5J3765NT7NN","short_pith_number":"pith:6ZAENYDD","canonical_record":{"source":{"id":"2607.25442","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-28T08:33:30Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8f3117127c76f01ad3b9b34c4bf23627561c1ab892cd32c3d63ea0526b74440b","abstract_canon_sha256":"e051542aa99e66a801ee5a7e0803e5140addf7b53fcd606e1c22114a20722e82"},"schema_version":"1.0"},"canonical_sha256":"f64046e0631a6ddc191d4effeeb67f6b65e2b160ccad23c1dd2d4b24631f6b7d","source":{"kind":"arxiv","id":"2607.25442","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25442","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25442v1","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25442","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"pith_short_12","alias_value":"6ZAENYDDDJW5","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"pith_short_16","alias_value":"6ZAENYDDDJW5YGI5","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"pith_short_8","alias_value":"6ZAENYDD","created_at":"2026-07-29T01:25:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:6ZAENYDDDJW5YGI5J3765NT7NN","target":"record","payload":{"canonical_record":{"source":{"id":"2607.25442","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-28T08:33:30Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8f3117127c76f01ad3b9b34c4bf23627561c1ab892cd32c3d63ea0526b74440b","abstract_canon_sha256":"e051542aa99e66a801ee5a7e0803e5140addf7b53fcd606e1c22114a20722e82"},"schema_version":"1.0"},"canonical_sha256":"f64046e0631a6ddc191d4effeeb67f6b65e2b160ccad23c1dd2d4b24631f6b7d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T01:25:15.793137Z","signature_b64":"2Ho4AHKuGVbm8s/dyGURkb1nRfA4/ykGmJh+bCFBMaayoIhTfv/Hu0/LB1vpbzNJDgQz6TPrgrgfN9ugLiTzDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f64046e0631a6ddc191d4effeeb67f6b65e2b160ccad23c1dd2d4b24631f6b7d","last_reissued_at":"2026-07-29T01:25:15.792379Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T01:25:15.792379Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.25442","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-29T01:25:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Is0kaYfc/c+VECXeFPB6TY1URlQ0hl9nMLiaQnaty4SpOPsyPRSqxJ2T0b5wElbxi1FgMI4NpPxJTEIVCVoCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T16:46:19.335967Z"},"content_sha256":"f957a478ed718fbebaeea18a7cd7d6e9ee1029640253aa4714a6e2bf1b84413b","schema_version":"1.0","event_id":"sha256:f957a478ed718fbebaeea18a7cd7d6e9ee1029640253aa4714a6e2bf1b84413b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:6ZAENYDDDJW5YGI5J3765NT7NN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On an asymmetric additive energy inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Akshat Mudgal","submitted_at":"2026-07-28T08:33:30Z","abstract_excerpt":"Let $d \\geq 1$ be an integer, $G$ be an abelian group and $\\nu, w_1, \\dots, w_{2d}: G \\to [0, \\infty)$ be functions with finite, non-empty supports. Define the generalised additive energy \\[ E_{2d, \\nu}(w_1, \\dots, w_{2d}) = \\sum_{y,y' \\in G}\\sum_{a_1, \\dots, a_{2d} \\in G } w_1(a_1) \\dots w_{2d}(a_{2d}) \\nu(y) \\nu(y') 1_{\\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .\\] Moreover, for every $1 \\leq i \\leq 2d$, let $E_{2d, \\nu}(w_i) = E_{2d, \\nu}(w_i, \\dots, w_i)$. A standard Fourier analytic argument delivers the estimate \\[ E_{2d,\\nu}(w_1, \\dots, w_{2d}) \\leq \\prod_{1 \\leq i \\leq 2d} E_{2d, \\nu}(w_i)^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25442","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/2607.25442/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-29T01:25:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WVY8TDjk2nRn+8j2zARLrHEiCno+/GyvahMcKnp+rmkMZ8NeeR2vyhuf5rvhn1vzFPAcUKbPkiNSfw/Ni3r2Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T16:46:19.337103Z"},"content_sha256":"9b22f7ee10944f383302e85c07e80da5db81832c48d45bcf2f56c1dcde0a2dc5","schema_version":"1.0","event_id":"sha256:9b22f7ee10944f383302e85c07e80da5db81832c48d45bcf2f56c1dcde0a2dc5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6ZAENYDDDJW5YGI5J3765NT7NN/bundle.json","state_url":"https://pith.science/pith/6ZAENYDDDJW5YGI5J3765NT7NN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6ZAENYDDDJW5YGI5J3765NT7NN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-18T16:46:19Z","links":{"resolver":"https://pith.science/pith/6ZAENYDDDJW5YGI5J3765NT7NN","bundle":"https://pith.science/pith/6ZAENYDDDJW5YGI5J3765NT7NN/bundle.json","state":"https://pith.science/pith/6ZAENYDDDJW5YGI5J3765NT7NN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6ZAENYDDDJW5YGI5J3765NT7NN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:6ZAENYDDDJW5YGI5J3765NT7NN","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":"e051542aa99e66a801ee5a7e0803e5140addf7b53fcd606e1c22114a20722e82","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-28T08:33:30Z","title_canon_sha256":"8f3117127c76f01ad3b9b34c4bf23627561c1ab892cd32c3d63ea0526b74440b"},"schema_version":"1.0","source":{"id":"2607.25442","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25442","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25442v1","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25442","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"pith_short_12","alias_value":"6ZAENYDDDJW5","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"pith_short_16","alias_value":"6ZAENYDDDJW5YGI5","created_at":"2026-07-29T01:25:15Z"},{"alias_kind":"pith_short_8","alias_value":"6ZAENYDD","created_at":"2026-07-29T01:25:15Z"}],"graph_snapshots":[{"event_id":"sha256:9b22f7ee10944f383302e85c07e80da5db81832c48d45bcf2f56c1dcde0a2dc5","target":"graph","created_at":"2026-07-29T01:25:15Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.25442/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $d \\geq 1$ be an integer, $G$ be an abelian group and $\\nu, w_1, \\dots, w_{2d}: G \\to [0, \\infty)$ be functions with finite, non-empty supports. Define the generalised additive energy \\[ E_{2d, \\nu}(w_1, \\dots, w_{2d}) = \\sum_{y,y' \\in G}\\sum_{a_1, \\dots, a_{2d} \\in G } w_1(a_1) \\dots w_{2d}(a_{2d}) \\nu(y) \\nu(y') 1_{\\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .\\] Moreover, for every $1 \\leq i \\leq 2d$, let $E_{2d, \\nu}(w_i) = E_{2d, \\nu}(w_i, \\dots, w_i)$. A standard Fourier analytic argument delivers the estimate \\[ E_{2d,\\nu}(w_1, \\dots, w_{2d}) \\leq \\prod_{1 \\leq i \\leq 2d} E_{2d, \\nu}(w_i)^{","authors_text":"Akshat Mudgal","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-28T08:33:30Z","title":"On an asymmetric additive energy inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25442","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:f957a478ed718fbebaeea18a7cd7d6e9ee1029640253aa4714a6e2bf1b84413b","target":"record","created_at":"2026-07-29T01:25:15Z","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":"e051542aa99e66a801ee5a7e0803e5140addf7b53fcd606e1c22114a20722e82","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-28T08:33:30Z","title_canon_sha256":"8f3117127c76f01ad3b9b34c4bf23627561c1ab892cd32c3d63ea0526b74440b"},"schema_version":"1.0","source":{"id":"2607.25442","kind":"arxiv","version":1}},"canonical_sha256":"f64046e0631a6ddc191d4effeeb67f6b65e2b160ccad23c1dd2d4b24631f6b7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f64046e0631a6ddc191d4effeeb67f6b65e2b160ccad23c1dd2d4b24631f6b7d","first_computed_at":"2026-07-29T01:25:15.792379Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T01:25:15.792379Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2Ho4AHKuGVbm8s/dyGURkb1nRfA4/ykGmJh+bCFBMaayoIhTfv/Hu0/LB1vpbzNJDgQz6TPrgrgfN9ugLiTzDw==","signature_status":"signed_v1","signed_at":"2026-07-29T01:25:15.793137Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.25442","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f957a478ed718fbebaeea18a7cd7d6e9ee1029640253aa4714a6e2bf1b84413b","sha256:9b22f7ee10944f383302e85c07e80da5db81832c48d45bcf2f56c1dcde0a2dc5"],"state_sha256":"2a2fbc6e5005262f5e1903e8959ccd3628ff175d61025e05ffee89be6d9983e9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9hOBAQMhuz0iH2ddNTCvDGfrofvi3fNzLAOVitgjQqMO6baw5mO4fxOj1OYMKwunDIvphRW0FH8y3m/rFNY+Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T16:46:19.364433Z","bundle_sha256":"7aadbea11762a715a5f40112123b03dff0942b4db8e2f0f81ab71cadc9ff993d"}}