{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TP7MQ3P5CUTYA6IW7H2QP2PERR","short_pith_number":"pith:TP7MQ3P5","canonical_record":{"source":{"id":"2607.28467","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-30T16:25:10Z","cross_cats_sorted":[],"title_canon_sha256":"9672c4b7653dec932dca5f9a5300ad6468482f8161bd84f72190951bb8bd4870","abstract_canon_sha256":"f7c0097f6510795f108bd94610759ea3221bbc934ef21fd13bba4e6c01e27a17"},"schema_version":"1.0"},"canonical_sha256":"9bfec86dfd1527807916f9f507e9e48c69d8086079524727f25b8bbe4f52c768","source":{"kind":"arxiv","id":"2607.28467","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.28467","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"arxiv_version","alias_value":"2607.28467v1","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28467","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"pith_short_12","alias_value":"TP7MQ3P5CUTY","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"pith_short_16","alias_value":"TP7MQ3P5CUTYA6IW","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"pith_short_8","alias_value":"TP7MQ3P5","created_at":"2026-07-31T01:37:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TP7MQ3P5CUTYA6IW7H2QP2PERR","target":"record","payload":{"canonical_record":{"source":{"id":"2607.28467","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-30T16:25:10Z","cross_cats_sorted":[],"title_canon_sha256":"9672c4b7653dec932dca5f9a5300ad6468482f8161bd84f72190951bb8bd4870","abstract_canon_sha256":"f7c0097f6510795f108bd94610759ea3221bbc934ef21fd13bba4e6c01e27a17"},"schema_version":"1.0"},"canonical_sha256":"9bfec86dfd1527807916f9f507e9e48c69d8086079524727f25b8bbe4f52c768","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bfec86dfd1527807916f9f507e9e48c69d8086079524727f25b8bbe4f52c768","last_reissued_at":"2026-07-31T01:37:50.229933Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-31T01:37:50.229933Z"},"source_kind":"arxiv","source_id":"2607.28467","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":"2026-07-31T01:37:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UmhN0WyftyF+ofmJE4wpRdCUBW5CVmmiPS1S+uqn6rXROdT0a0XhCLD4tGhHkS2qXZQWe7jeZz4qJOKGLQwyDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T10:24:46.319699Z"},"content_sha256":"7056825691302e3aeba3e6cc467f13009c2468fe7e2edcd7e4a595beed0e39b2","schema_version":"1.0","event_id":"sha256:7056825691302e3aeba3e6cc467f13009c2468fe7e2edcd7e4a595beed0e39b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TP7MQ3P5CUTYA6IW7H2QP2PERR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The space of metrics with positive generalized conformal Laplacian","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bernhard Hanke, Georg Frenck, Gioacchino Antonelli","submitted_at":"2026-07-30T16:25:10Z","abstract_excerpt":"Let $n\\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\\gamma\\Delta_g+\\mathrm{R}_g$ is strictly positive.\n  We prove that if $n=2$ and $\\gamma\\ge0$, or if $n\\ge3$ and $0\\leq \\gamma \\leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\\hookrightarrow R^\\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\\ge3$ and $\\gamma>4(n-1)/(n-2)$, the spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28467","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/2607.28467/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":"2026-07-31T01:37:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ppis+l1P47rJtok0RSLHp/P/8F8FOl1o3SFsCxwTQs0dTyiXDeO7Q/X46LZx2iF5/8VN9f2WtJ7yTQd18wF/Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T10:24:46.320606Z"},"content_sha256":"ecbef3cec22f2bda426b6bd09c6e41a1c2303703442deda66c00d5694cf70e7d","schema_version":"1.0","event_id":"sha256:ecbef3cec22f2bda426b6bd09c6e41a1c2303703442deda66c00d5694cf70e7d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR/bundle.json","state_url":"https://pith.science/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR/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":"2026-08-07T10:24:46Z","links":{"resolver":"https://pith.science/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR","bundle":"https://pith.science/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR/bundle.json","state":"https://pith.science/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TP7MQ3P5CUTYA6IW7H2QP2PERR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TP7MQ3P5CUTYA6IW7H2QP2PERR","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":"f7c0097f6510795f108bd94610759ea3221bbc934ef21fd13bba4e6c01e27a17","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-30T16:25:10Z","title_canon_sha256":"9672c4b7653dec932dca5f9a5300ad6468482f8161bd84f72190951bb8bd4870"},"schema_version":"1.0","source":{"id":"2607.28467","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.28467","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"arxiv_version","alias_value":"2607.28467v1","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28467","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"pith_short_12","alias_value":"TP7MQ3P5CUTY","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"pith_short_16","alias_value":"TP7MQ3P5CUTYA6IW","created_at":"2026-07-31T01:37:50Z"},{"alias_kind":"pith_short_8","alias_value":"TP7MQ3P5","created_at":"2026-07-31T01:37:50Z"}],"graph_snapshots":[{"event_id":"sha256:ecbef3cec22f2bda426b6bd09c6e41a1c2303703442deda66c00d5694cf70e7d","target":"graph","created_at":"2026-07-31T01:37:50Z","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/2607.28467/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $n\\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\\gamma\\Delta_g+\\mathrm{R}_g$ is strictly positive.\n  We prove that if $n=2$ and $\\gamma\\ge0$, or if $n\\ge3$ and $0\\leq \\gamma \\leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\\hookrightarrow R^\\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\\ge3$ and $\\gamma>4(n-1)/(n-2)$, the spa","authors_text":"Bernhard Hanke, Georg Frenck, Gioacchino Antonelli","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-30T16:25:10Z","title":"The space of metrics with positive generalized conformal Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28467","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":"sha256:7056825691302e3aeba3e6cc467f13009c2468fe7e2edcd7e4a595beed0e39b2","target":"record","created_at":"2026-07-31T01:37:50Z","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":"f7c0097f6510795f108bd94610759ea3221bbc934ef21fd13bba4e6c01e27a17","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-30T16:25:10Z","title_canon_sha256":"9672c4b7653dec932dca5f9a5300ad6468482f8161bd84f72190951bb8bd4870"},"schema_version":"1.0","source":{"id":"2607.28467","kind":"arxiv","version":1}},"canonical_sha256":"9bfec86dfd1527807916f9f507e9e48c69d8086079524727f25b8bbe4f52c768","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9bfec86dfd1527807916f9f507e9e48c69d8086079524727f25b8bbe4f52c768","first_computed_at":"2026-07-31T01:37:50.229933Z","kind":"pith_receipt","last_reissued_at":"2026-07-31T01:37:50.229933Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.28467","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7056825691302e3aeba3e6cc467f13009c2468fe7e2edcd7e4a595beed0e39b2","sha256:ecbef3cec22f2bda426b6bd09c6e41a1c2303703442deda66c00d5694cf70e7d"],"state_sha256":"bfa03fabfff0681fe3a73800ca035ded09752c8d3db4840c72a385edbc7966b5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CDoSynDwU7BBV6jsoh5UTb0Uyd/WQHpsBD1fxFIlPu05iFLTlNuNXfpVtUFkEmte/EXEDS2BW6NroAJozvx4BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T10:24:46.325615Z","bundle_sha256":"84647a5bbd4207375e2ecbffabf417ad3da30e9d72ee9594570b72fa9123a4d8"}}