{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:LRFAPEGQV6U24HD2D3HER73LTT","short_pith_number":"pith:LRFAPEGQ","canonical_record":{"source":{"id":"2605.20905","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T08:49:13Z","cross_cats_sorted":[],"title_canon_sha256":"4174386ea694116cadc93318df44ff3ea53b4980cdf68ff8cb34d273d8173d3f","abstract_canon_sha256":"606b37bf6822fa1c15b89f48b1a69f5ad310196659bcdce168da0865d4b2b1e8"},"schema_version":"1.0"},"canonical_sha256":"5c4a0790d0afa9ae1c7a1ece48ff6b9cff37147ccdd2f30b82085fc0c5a427b1","source":{"kind":"arxiv","id":"2605.20905","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20905","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20905v1","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20905","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"pith_short_12","alias_value":"LRFAPEGQV6U2","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"pith_short_16","alias_value":"LRFAPEGQV6U24HD2","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"pith_short_8","alias_value":"LRFAPEGQ","created_at":"2026-05-21T01:05:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:LRFAPEGQV6U24HD2D3HER73LTT","target":"record","payload":{"canonical_record":{"source":{"id":"2605.20905","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T08:49:13Z","cross_cats_sorted":[],"title_canon_sha256":"4174386ea694116cadc93318df44ff3ea53b4980cdf68ff8cb34d273d8173d3f","abstract_canon_sha256":"606b37bf6822fa1c15b89f48b1a69f5ad310196659bcdce168da0865d4b2b1e8"},"schema_version":"1.0"},"canonical_sha256":"5c4a0790d0afa9ae1c7a1ece48ff6b9cff37147ccdd2f30b82085fc0c5a427b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:05:27.325649Z","signature_b64":"14DYOe2w4+HaqNq8W9sPsPjgsKyfR63fTIOqfGbH7MlQYCUyRCuDIwb4nNIQDrjwCLpbUlNSXVjrm9i8REQoDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c4a0790d0afa9ae1c7a1ece48ff6b9cff37147ccdd2f30b82085fc0c5a427b1","last_reissued_at":"2026-05-21T01:05:27.325051Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:05:27.325051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.20905","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":"2026-05-21T01:05:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QxhIpT5OGn1IKp7brnWNCyW3+/VJuLV2W4JWCkSHroTDn/MXMRZHka05COFULfaxG9u537gY8Gfo8QGhUnY3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T21:01:10.107246Z"},"content_sha256":"3e00d7b0561541d9bacdd65b720e7e4d01755b6e7200ca22be4c273b3087186d","schema_version":"1.0","event_id":"sha256:3e00d7b0561541d9bacdd65b720e7e4d01755b6e7200ca22be4c273b3087186d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:LRFAPEGQV6U24HD2D3HER73LTT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Horizontal miniatures and normal-sized miniatures of convex lattice polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Takashi Hirotsu","submitted_at":"2026-05-20T08:49:13Z","abstract_excerpt":"Let $d$ be a nonnegative integer, and let $P \\subset \\mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\\binom{2d+1}{d},$ which generalizes the known results for the unit hypercube and lattice simplices provided by the author. This theorem is proven by establishing that the number of horizontal miniatures of $P$ with resolution $t$ is a polynomial of degree $d+1$ in $t$ whose leading coefficient is $\\mathrm{vol}\\,(P)/(d+1),$ which is derived from Ehrhart theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20905","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/2605.20905/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":"2026-05-21T01:05:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lOCvJQYuj4gGx6PvmZ6PW0J9DX1C3TZzvUzYnx0XHARMJJt3vA9fHrUaQtqmqF0hQcRTAvkwghOD+/Vok92CBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T21:01:10.107967Z"},"content_sha256":"289e22ef5a862c2e4c4089cfb3ac9987d2eaeb441afaeb17ce357076cfe67d1b","schema_version":"1.0","event_id":"sha256:289e22ef5a862c2e4c4089cfb3ac9987d2eaeb441afaeb17ce357076cfe67d1b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LRFAPEGQV6U24HD2D3HER73LTT/bundle.json","state_url":"https://pith.science/pith/LRFAPEGQV6U24HD2D3HER73LTT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LRFAPEGQV6U24HD2D3HER73LTT/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":"2026-05-22T21:01:10Z","links":{"resolver":"https://pith.science/pith/LRFAPEGQV6U24HD2D3HER73LTT","bundle":"https://pith.science/pith/LRFAPEGQV6U24HD2D3HER73LTT/bundle.json","state":"https://pith.science/pith/LRFAPEGQV6U24HD2D3HER73LTT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LRFAPEGQV6U24HD2D3HER73LTT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LRFAPEGQV6U24HD2D3HER73LTT","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":"606b37bf6822fa1c15b89f48b1a69f5ad310196659bcdce168da0865d4b2b1e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T08:49:13Z","title_canon_sha256":"4174386ea694116cadc93318df44ff3ea53b4980cdf68ff8cb34d273d8173d3f"},"schema_version":"1.0","source":{"id":"2605.20905","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20905","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20905v1","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20905","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"pith_short_12","alias_value":"LRFAPEGQV6U2","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"pith_short_16","alias_value":"LRFAPEGQV6U24HD2","created_at":"2026-05-21T01:05:27Z"},{"alias_kind":"pith_short_8","alias_value":"LRFAPEGQ","created_at":"2026-05-21T01:05:27Z"}],"graph_snapshots":[{"event_id":"sha256:289e22ef5a862c2e4c4089cfb3ac9987d2eaeb441afaeb17ce357076cfe67d1b","target":"graph","created_at":"2026-05-21T01:05:27Z","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/2605.20905/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $d$ be a nonnegative integer, and let $P \\subset \\mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\\binom{2d+1}{d},$ which generalizes the known results for the unit hypercube and lattice simplices provided by the author. This theorem is proven by establishing that the number of horizontal miniatures of $P$ with resolution $t$ is a polynomial of degree $d+1$ in $t$ whose leading coefficient is $\\mathrm{vol}\\,(P)/(d+1),$ which is derived from Ehrhart theory.","authors_text":"Takashi Hirotsu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T08:49:13Z","title":"Horizontal miniatures and normal-sized miniatures of convex lattice polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20905","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":"sha256:3e00d7b0561541d9bacdd65b720e7e4d01755b6e7200ca22be4c273b3087186d","target":"record","created_at":"2026-05-21T01:05:27Z","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":"606b37bf6822fa1c15b89f48b1a69f5ad310196659bcdce168da0865d4b2b1e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T08:49:13Z","title_canon_sha256":"4174386ea694116cadc93318df44ff3ea53b4980cdf68ff8cb34d273d8173d3f"},"schema_version":"1.0","source":{"id":"2605.20905","kind":"arxiv","version":1}},"canonical_sha256":"5c4a0790d0afa9ae1c7a1ece48ff6b9cff37147ccdd2f30b82085fc0c5a427b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c4a0790d0afa9ae1c7a1ece48ff6b9cff37147ccdd2f30b82085fc0c5a427b1","first_computed_at":"2026-05-21T01:05:27.325051Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:05:27.325051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"14DYOe2w4+HaqNq8W9sPsPjgsKyfR63fTIOqfGbH7MlQYCUyRCuDIwb4nNIQDrjwCLpbUlNSXVjrm9i8REQoDQ==","signature_status":"signed_v1","signed_at":"2026-05-21T01:05:27.325649Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.20905","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e00d7b0561541d9bacdd65b720e7e4d01755b6e7200ca22be4c273b3087186d","sha256:289e22ef5a862c2e4c4089cfb3ac9987d2eaeb441afaeb17ce357076cfe67d1b"],"state_sha256":"1ba6fc45037895929229a153f13596cd19d27e6d5040828b3306102145a50387"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AXft5Ka7vKU1d4w2QMz90ldlIrhHKPKXfqXKA5LJ/LnB8OBUIuMIahhCvOronPBpzE/FhsmPTbGhtdHmuh9eDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T21:01:10.111588Z","bundle_sha256":"7d72aa6b2fc7d0e71e04075e60488a713b3c9fb8580bf7c2eab1ba775e9c8bd7"}}