{"as_of":"2026-08-21T12:53:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:e42b6aa5655fe54d975908369dc5fb12052a3ac5ea1ba7f7d5e81a692768f775","coverage":[{"denominator":25,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":25,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-02T10:07:08.010259Z","state":"measured"},{"denominator":25,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":25,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-21T06:32:19.484+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2606.27869/citation-record","integrity":"/paper/2606.27869/integrity","json":"/paper/2606.27869/citation-record.json","paper":"/paper/2606.27869"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.310034Z","title":"Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.310034Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:7af526aa85b4bc377de16050100ac8b2a0c57f1fa86b9fc16566d42fc0522dda","observation_id":"f2a0d5f6-0ea7-44c1-bcca-347ce2160e5d","resolution":{"observed_at":"2026-08-02T10:07:06.310034Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.329444Z","title":null,"venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":2,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.329444Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:85766edb370c257383284f3625101bfc7690c9cb71566b4fba450cec99994e69","observation_id":"714b5180-4813-4596-8cfc-89424514b13c","resolution":{"observed_at":"2026-08-02T10:07:06.329444Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.333183Z","title":"A course in metric geometry , volume 33 of Graduate Studies in Mathematics","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":3,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.333183Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:c1c27454adc732940eb82231340a7f39851f3477e351d995658375237c2b3a6e","observation_id":"daf24277-82eb-4881-81c8-93d82266cb55","resolution":{"observed_at":"2026-08-02T10:07:06.333183Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.337147Z","title":"Functional analysis, S obolev spaces and partial differential equations","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":4,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.337147Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:865f169f56bc30e27202c3058ea3569a6f4110507168499af06777f3c41062bd","observation_id":"31d2a0f7-b594-4ad2-bfee-5215c06f8e97","resolution":{"observed_at":"2026-08-02T10:07:06.337147Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.341145Z","title":"Stability of heat kernel bounds under pointed G romov- H ausdorff convergence","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":5,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.341145Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:3babebc56e3b50d90b0a0c968d956fd388f4d911747fb0dd338b7777c3bdd770","observation_id":"8da1b1af-fa92-42a1-8181-d84eca1ffdb0","resolution":{"observed_at":"2026-08-02T10:07:06.341145Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.344773Z","title":"Elliptic Harnack inequalities for mixed local and nonlocal p -energy form on metric measure spaces, 2026","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":6,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.344773Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:2424c695f924c02ca9e9cbb7551decb02ad56098de650a2b682b83a2666237ad","observation_id":"40a4977e-a10c-4c37-b295-858993f3ffbe","resolution":{"observed_at":"2026-08-02T10:07:06.344773Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2201.02059","last_updated":"2022-01-06T13:50:40Z","snapshot_observed_at":"2026-08-16T17:29:56.046536Z","submitted_at":"2022-01-06T13:50:40Z","title":"On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2201.02059","snapshot_observed_at":"2026-08-02T10:07:06.349068Z","title":"On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity, 2022","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":7,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.349068Z"},"links":{"cited_paper":"/paper/2201.02059","citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:4c8a8a24cca9ca937356c343892c6fd9ab8216436cf9880930b96fa8d187a0cb","observation_id":"15db47a2-87ca-466a-b89c-fe019877fde0","resolution":{"observed_at":"2026-08-02T10:07:06.349068Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.353227Z","title":"On the Resistance Conjecture, 2026","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":8,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.353227Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:9e1d54ac99660edc69dda5391346b1c8dffa2702267d099e82f78f826a018cac","observation_id":"ed4c657b-a394-458c-bdcf-81b6d1679abb","resolution":{"observed_at":"2026-08-02T10:07:06.353227Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.357026Z","title":"On the energy image density conjecture of Bouleau and Hirsch, 2025","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":9,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.357026Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:e29d42626b06796094a6969574740529fbbf16de3d52a40e7479d1721ea4899e","observation_id":"f385876a-acc9-4788-9f87-1c0b0ad8704e","resolution":{"observed_at":"2026-08-02T10:07:06.357026Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.362110Z","title":"Density of continuous functions in S obolev spaces with applications to capacity","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":10,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.362110Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:6f5e93adfde4d42cf225d2afbaebe298031fc6e6f607f5520c7f31d91d411338","observation_id":"d74ef6f8-1645-400b-9cb4-7f81b49a8e20","resolution":{"observed_at":"2026-08-02T10:07:06.362110Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.388290Z","title":"Dirichlet forms and symmetric M arkov processes , volume 19 of De Gruyter Studies in Mathematics","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":11,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.388290Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:1bb7ba93be8c9f0fe230411829f6ebebe7fffd4ffb0e98bc8fd265a640af6afc","observation_id":"87aae151-4892-4a4e-8029-27877e6f66e0","resolution":{"observed_at":"2026-08-02T10:07:06.388290Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.516738Z","title":"Generalized capacity, H arnack inequality and heat kernels of D irichlet forms on metric measure spaces","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":12,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.516738Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:0beb6e6728b51658d421eb9007accaa7cc4addb7b7614c7d70af635e467b907d","observation_id":"6e31e0c5-172f-44b3-96a9-3750de2ef5f9","resolution":{"observed_at":"2026-08-02T10:07:06.516738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.648909Z","title":"Lectures on analysis on metric spaces","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":13,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.648909Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:44f594732ac712640a8cc8780247f2b1bc90889087ab0e3f69a86595ae2819aa","observation_id":"ea32131b-f49d-451d-bf96-7d98bfb92bc8","resolution":{"observed_at":"2026-08-02T10:07:06.648909Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.758485Z","title":"Quasiconformal maps in metric spaces with controlled geometry","venue":null,"work_id":null,"year":1998},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":14,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.758485Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:0da759b188ac52789f0f74f036c725b26eec6271b60bb0bea8b0b2ea1c9ecfd4","observation_id":"c83fbf45-c0e3-486d-82cc-62474be9c9c3","resolution":{"observed_at":"2026-08-02T10:07:06.758485Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:06.900132Z","title":null,"venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":15,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:06.900132Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:f9d6ce8bcebb47ba3375cbb6e00b11bb5f55599c2439812a9d47a4fd4026f965","observation_id":"cdbdbfc0-fe42-4dce-9441-56012fc468f8","resolution":{"observed_at":"2026-08-02T10:07:06.900132Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.016148Z","title":"An elementary proof that walk dimension is greater than two for B rownian motion on S ierpi\\'nski carpets","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":16,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.016148Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:aaa1afc8761781498c4fc159f13fe3aafc24cac44aaa9b60d8942a8f687337cd","observation_id":"ecf8e678-3294-4b00-a6e4-a2c06d0439a9","resolution":{"observed_at":"2026-08-02T10:07:07.016148Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.220516Z","title":"Conductive homogeneity of compact metric spaces and construction of p -energy , volume 5 of Memoirs of the European Mathematical Society","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":17,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.220516Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:30bd7530d52c2d0a0ea92e29782fa7d546ca46ecb6584a98bced4d28ddebab62","observation_id":"1850b48e-f281-4dd2-9720-5939aaeacbf9","resolution":{"observed_at":"2026-08-02T10:07:07.220516Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.381917Z","title":"Keith and T","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":18,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.381917Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:a37e1cf2f7f07d4008cafde97ec8749bfc044f3f6c348d116dc72600b2bce9cd","observation_id":"5965aff0-8cb6-44a5-b341-1a547252f1dc","resolution":{"observed_at":"2026-08-02T10:07:07.381917Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.466840Z","title":"On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":19,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.466840Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:9f1aa54e173caa067528bb8656af0ef1035f952c35e86a490635ce88365c5f9c","observation_id":"b5a195ef-4a2d-4491-8fe5-d612f4e7d715","resolution":{"observed_at":"2026-08-02T10:07:07.466840Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.536793Z","title":"On the conformal walk dimension: quasisymmetric uniformization for symmetric diffusions","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":20,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.536793Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:f77b501041c9bdbb5ac3c0c19060d00126b6b92307de4508689d18fe40a8a63d","observation_id":"98159dd7-6d21-42b1-ad27-d4ab19964e15","resolution":{"observed_at":"2026-08-02T10:07:07.536793Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2404.13668","last_updated":"2026-08-13T06:15:46Z","snapshot_observed_at":"2026-08-16T23:08:56.950832Z","submitted_at":"2024-04-21T14:17:56Z","title":"Contraction properties and differentiability of $p$-energy forms with applications to nonlinear potential theory on self-similar sets","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2404.13668","snapshot_observed_at":"2026-08-02T10:07:07.628190Z","title":"Contraction properties and differentiability of p -energy forms with applications to nonlinear potential theory on self-similar sets, 2026","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":21,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.628190Z"},"links":{"cited_paper":"/paper/2404.13668","citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:773a3589f052f554e3993a5b3bebbe16bec4191427b2ad3e3cf7912fcf6eb312","observation_id":"40425610-61a6-4f2a-8f6e-d083c557d87b","resolution":{"observed_at":"2026-08-02T10:07:07.628190Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.728142Z","title":"Scale-invariant boundary H arnack principle on inner uniform domains in fractal-type spaces","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":22,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.728142Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:a4b62d444b62e4cc405330a29b90b42ce42b14234213b0c4451b47834edcc904","observation_id":"fd9da3a9-dc87-49c9-9546-a4ed0d3de12b","resolution":{"observed_at":"2026-08-02T10:07:07.728142Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:07.827196Z","title":"Gromov-Hausdorff distances in Euclidean spaces","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":23,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.827196Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:2d9fa7621be7d0eae53838f647a375cace8da4881c615d192eb73fab5d4f545e","observation_id":"c3160a88-6c7a-4570-8274-b3cddaa46e29","resolution":{"observed_at":"2026-08-02T10:07:07.827196Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2505.12468","last_updated":"2026-06-05T13:50:02Z","snapshot_observed_at":"2026-08-15T20:31:03.151682Z","submitted_at":"2025-05-18T15:46:23Z","title":"On singularity of $p$-energy measures on metric measure spaces","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2505.12468","snapshot_observed_at":"2026-08-02T10:07:07.864478Z","title":"On singularity of p -energy measures on metric measure spaces, 2025","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":24,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:07.864478Z"},"links":{"cited_paper":"/paper/2505.12468","citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:72064b35a88e0c0cc35dd20599099446addd6848b266f8cb46ac01233a564859","observation_id":"1759ceb2-b51b-40a3-b163-c8ee7b1c7392","resolution":{"observed_at":"2026-08-02T10:07:07.864478Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T10:07:08.010259Z","title":"On the dichotomy of p -walk dimensions on metric measure spaces, 2025","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces","version":2},"reference_index":25,"source":"arxiv_source","source_observed_at":"2026-08-02T10:07:08.010259Z"},"links":{"citing_paper":"/paper/2606.27869"},"observation_digest":"sha256:a0ba46ebbe88cdb2198b2fd5ca3df9bb66e8c453f1aceefbcf26b32c38340236","observation_id":"daee4694-290c-45c5-b394-d95d096c18fd","resolution":{"observed_at":"2026-08-02T10:07:08.010259Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2606.27869","last_updated":"2026-07-22T07:35:40Z","latest_version":2,"primary_category":"math.MG","snapshot_observed_at":"2026-08-17T01:23:59.086442Z","submitted_at":"2026-06-26T09:10:15Z","title":"Walk dimension and vanishing curve modulus in metric measure spaces"},"reference_resolution":{"displayed":25,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":25,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":25},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"thesis":"As of 21 August 2026, this Paper Citation Record lists 25 of 25 outbound references and 0 inbound Pith citation observations for arXiv:2606.27869."}