{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"labels": {
"zh-CN": "边缘",
"en": "boundary"
},
"alternative_labels": [
{
"lang": "en",
"value": "p-boundary"
},
{
"lang": "en",
"value": "r-boundary"
},
{
"lang": "zh-CN",
"value": "$p$-边界"
},
{
"lang": "zh-CN",
"value": "$p$-边缘"
},
{
"lang": "zh-CN",
"value": "$r$-边界"
},
{
"lang": "zh-CN",
"value": "$r$-边缘"
},
{
"lang": "zh-CN",
"value": "边界"
}
],
"scope_note": "同调边缘：p-链 C 若存在 (p+1)-链 C′ 使 C=∂C′，则 C 为 p-boundary，即 C∈im ∂_{p+1}。来源 Szekeres §17.4；不是集合的 closure-minus-interior 边界，也不是边缘算子或边缘群本身。证据 reports/coordination-20261020/boundary-sense-review/evidence.json#szekeres-homological-boundary。",
"identity_status": "active",
"identity_revision": "698472308669796680bfc0e50009540c5c78b3184e4d94d14a685b03c8205d3b",
"occurrences": [
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"lang": "zh-CN",
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"anchor": "bound-first-d95600b3e2cb680080386f74",
"chapter": "3",
"section": "3.3",
"offset": 24100,
"range": [
54643,
55190
],
"source_revision": "3a655ef421c1b2cd7cfb292d1fce3192aafb66fee9db0926c5d8b817c663acf2",
"source_category": "machine_draft",
"review_status": "unreviewed",
"role": "first-mention",
"role_basis": {
"kind": "explicit-source-uuid-marker",
"original_concept_id": "urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179",
"identity_resolution_basis": "explicit-approved-merge",
"source_bound_role": "first-mention",
"first_mention": true,
"source_node": "a-9e2da34264795702adf3f727",
"source_sidecar": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/provenance/chapter-03/concept-annotations.json",
"sha256": "e9b83f3a6bb889fdb2fa61ca3e5d151e2e4946f2172ab31e9e59245b49b7649c",
"page": "3.3",
"annotation_count": 83,
"policy": "explicit-source-bound-UUIDs; no string-based identity allocation",
"rendered_annotation_count": 40,
"plain_mention_count": 43,
"display_policy": "clickable-only"
}
},
"label": "边缘",
"context": {
"range": [
52769,
55303
],
"quote": null,
"quote_sha256": "0a00984afb987c6879e24a8912386fa4961f1de1582b69dcc09bbca00b5d4804",
"environment": null,
"excerpt_omitted": true
},
"parallel_readings": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-03/03-section-3-3",
"lang": "en",
"source_category": "machine_draft",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-88344bd705205e684822e0c7"
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"lang": "zh-CN",
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"anchor": "bound-mention-c5761648b6197ceacbe79141",
"chapter": "4",
"section": "4",
"offset": 1234,
"range": [
8248,
8881
],
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_category": "machine_draft",
"review_status": "unreviewed",
"role": "discussion",
"role_basis": {
"kind": "explicit-source-uuid-marker",
"source_bound_role": "discussion",
"first_mention": false,
"source_node": "a-e89612c8019eac959ef72ae0",
"source_sidecar": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/provenance/chapter-04/concept-annotations.json",
"sha256": "d5e62dfb62385102900f839c1162f627e780f8e168759526961482268ef49ac4",
"page": "4",
"annotation_count": 10,
"policy": "explicit-source-bound-UUIDs; no string-based identity allocation",
"rendered_annotation_count": 8,
"plain_mention_count": 2,
"display_policy": "clickable-only"
}
},
"label": "边缘",
"context": {
"range": [
7209,
15079
],
"quote": null,
"quote_sha256": "740c139ab53b2a81f09a5d0217c62874dfdb915fa96d721978fbf24eb859fb22",
"environment": null,
"excerpt_omitted": true
},
"parallel_readings": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-04/index",
"lang": "en",
"source_category": "machine_draft",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-2ee2492cd15dc230abc55711"
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"lang": "zh-CN",
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/01-section-6-1",
"anchor": "bound-mention-87599ba23745b8b83b46cda7",
"chapter": "6",
"section": "6.1",
"offset": 11287,
"range": [
24511,
25156
],
"source_revision": "e4b5540a1170224114cc2ab29049010d355eaf7b3257a6c2f94afce3ab5037fb",
"source_category": "machine_draft",
"review_status": "unreviewed",
"role": "discussion",
"role_basis": {
"kind": "explicit-source-uuid-marker",
"source_bound_role": "discussion",
"first_mention": false,
"source_node": "a-6507ec7a16008fdecc630cd9",
"source_sidecar": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/provenance/chapter-06/concept-annotations.json",
"sha256": "8e24dc92eba8bc34f780867e45c26d2a5c8c45b757e7b0baaa2fb1e5a283fd69",
"page": "6.1",
"annotation_count": 24,
"policy": "explicit-source-bound-UUIDs; no string-based identity allocation",
"rendered_annotation_count": 17,
"plain_mention_count": 7,
"display_policy": "clickable-only"
}
},
"label": "边缘(boundary)",
"context": {
"range": [
21751,
25678
],
"quote": null,
"quote_sha256": "b93ce5639963785a8f0452438d7d8e361cc9c738b7d475ddea55d6dd9a72ecd6",
"environment": null,
"excerpt_omitted": true
},
"parallel_readings": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-06/01-section-6-1",
"lang": "en",
"source_category": "machine_draft",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-9aeb67c5ee1c10bf3d114463"
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"lang": "zh-CN",
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/02-section-6-2",
"anchor": "bound-mention-15cb8a24f0b9b8c3d51908d7",
"chapter": "6",
"section": "6.2",
"offset": 37408,
"range": [
52873,
53506
],
"source_revision": "2adf7861492ef32e8411451cbaaa1ac2e305e6d8da633adb4624cf35a45d937d",
"source_category": "machine_draft",
"review_status": "unreviewed",
"role": "discussion",
"role_basis": {
"kind": "explicit-source-uuid-marker",
"source_bound_role": "discussion",
"first_mention": false,
"source_node": "a-0c1466a855a241b4bc1191ea",
"source_sidecar": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/provenance/chapter-06/concept-annotations.json",
"sha256": "8e24dc92eba8bc34f780867e45c26d2a5c8c45b757e7b0baaa2fb1e5a283fd69",
"page": "6.2",
"annotation_count": 48,
"policy": "explicit-source-bound-UUIDs; no string-based identity allocation",
"rendered_annotation_count": 33,
"plain_mention_count": 15,
"display_policy": "clickable-only"
}
},
"label": "边缘",
"context": {
"range": [
50955,
54110
],
"quote": null,
"quote_sha256": "d070dd00413ad856559bb46193fa929894f7974d92bf16a56c1d94edf0711d25",
"environment": null,
"excerpt_omitted": true
},
"parallel_readings": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-06/02-section-6-2",
"lang": "en",
"source_category": "machine_draft",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-679e1f6b7854685345b2b84a"
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-17/04-homology-and-cohomology",
"anchor": "first-ccd810e4a97cf523defc-homology-zh",
"chapter": "17",
"section": "17.4",
"offset": 7404,
"range": [
11701,
11941
],
"source_revision": "90f459d7613ac36fccd50d5e31fd52b3e28884c63a94670a7dc59abff13007f2",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "边界(boundary)",
"context": {
"range": [
11688,
12346
],
"quote": "一个$p$-链$C$若没有边界(boundary)，即$\\partial C = 0 .$，则称其为循环(cycle)。若存在一个$(p+1)$-链$C^{\\prime}$使得$C = \\partial C ^ { \\prime }$，则称其为边界。显然，每个边界都是$p$-循环，因为$\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$，但反之未必成立。",
"quote_sha256": "fcea4805800933623c7b8ee9d45e0610b1f2fb47089c205cb0632306cdff4d54",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "en/chapter-17/04-homology-and-cohomology",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-08e4f297979474c60d397dcd"
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "en",
"route": "en/chapter-17/04-homology-and-cohomology",
"anchor": "first-ccd810e4a97cf523defc-homology-en",
"chapter": "17",
"section": "17.4",
"offset": 9557,
"range": [
13381,
13617
],
"source_revision": "f56b6ea9d7772435362632085cbb020a2064f36ef90fde89e6dfc5d109825a80",
"source_category": "original_transcription",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "boundary",
"context": {
"range": [
12964,
13886
],
"quote": "A $p$-chain $C$ is said to be a cycle if it has no boundary, $\\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\\prime}$ such that $C = \\partial C ^ { \\prime }$ . Clearly every boundary is a $p$-cycle since $\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$ , but the converse need not be true.",
"quote_sha256": "7f220240b60d8aa197969f73e22e5999e055449d4a01300e06aa869cf2448eec",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"lang": "zh-CN",
"source_category": "translation",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-507ade36b5f5d4108a7dc797"
}
],
"registered_positions": [
{
"route": "en/chapter-17/04-homology-and-cohomology",
"lang": "en",
"section": "17.4",
"signal": "term-mark",
"label": "boundary",
"english": "boundary",
"offset": 8264
},
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"lang": "zh-CN",
"section": "17.4",
"signal": "term-mark",
"label": "边界",
"english": "boundary",
"offset": 6484
}
],
"statements": [],
"resolved_references": [
{
"raw": "href=\"03-section-3-3.md#bound-definition-1c3bccaa36fa6f6567f60365\"",
"href": "03-section-3-3.md#bound-definition-1c3bccaa36fa6f6567f60365",
"range": [
52872,
52938
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"target_anchor": "bound-definition-1c3bccaa36fa6f6567f60365",
"resolved": true,
"reason": null,
"source_revision": "3a655ef421c1b2cd7cfb292d1fce3192aafb66fee9db0926c5d8b817c663acf2",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"occurrence_id": "occ-88344bd705205e684822e0c7"
},
{
"raw": "href=\"../chapter-03/03-section-3-3.md#bound-definition-1ccd5413f50b8ab807558a49\"",
"href": "../chapter-03/03-section-3-3.md#bound-definition-1ccd5413f50b8ab807558a49",
"range": [
7314,
7394
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"target_anchor": "bound-definition-1ccd5413f50b8ab807558a49",
"resolved": true,
"reason": null,
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"occurrence_id": "occ-2ee2492cd15dc230abc55711"
},
{
"raw": "href=\"../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c\"",
"href": "../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c",
"range": [
8915,
8995
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"target_anchor": "bound-definition-b6c4b4cd58ce97d21892883c",
"resolved": true,
"reason": null,
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"occurrence_id": "occ-2ee2492cd15dc230abc55711"
},
{
"raw": "href=\"05-section-4-5.md#bound-definition-aa4fa93d3f466df668189f39\"",
"href": "05-section-4-5.md#bound-definition-aa4fa93d3f466df668189f39",
"range": [
9785,
9851
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/05-section-4-5",
"target_anchor": "bound-definition-aa4fa93d3f466df668189f39",
"resolved": true,
"reason": null,
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"occurrence_id": "occ-2ee2492cd15dc230abc55711"
},
{
"raw": "href=\"../chapter-02/03-section-2-3.md#bound-definition-bea63120260758e422dc7e27\"",
"href": "../chapter-02/03-section-2-3.md#bound-definition-bea63120260758e422dc7e27",
"range": [
11059,
11139
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-02/03-section-2-3",
"target_anchor": "bound-definition-bea63120260758e422dc7e27",
"resolved": true,
"reason": null,
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"occurrence_id": "occ-2ee2492cd15dc230abc55711"
},
{
"raw": "href=\"01-section-4-1.md#bound-definition-4a2489d02e04924c4017048c\"",
"href": "01-section-4-1.md#bound-definition-4a2489d02e04924c4017048c",
"range": [
12722,
12788
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/01-section-4-1",
"target_anchor": "bound-definition-4a2489d02e04924c4017048c",
"resolved": true,
"reason": null,
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"occurrence_id": "occ-2ee2492cd15dc230abc55711"
},
{
"raw": "href=\"../chapter-02/01-section-2-1.md#bound-definition-c624ec2b869cd03c185835db\"",
"href": "../chapter-02/01-section-2-1.md#bound-definition-c624ec2b869cd03c185835db",
"range": [
13488,
13568
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-02/01-section-2-1",
"target_anchor": "bound-definition-c624ec2b869cd03c185835db",
"resolved": true,
"reason": null,
"source_revision": "b7d42262569f737fdcc782c96c9642398e52705f3527709f121edb29646591c1",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-04/index",
"occurrence_id": "occ-2ee2492cd15dc230abc55711"
},
{
"raw": "href=\"../chapter-05/index.md#bound-definition-2d833ac0bb979a1a384519d5\"",
"href": "../chapter-05/index.md#bound-definition-2d833ac0bb979a1a384519d5",
"range": [
21865,
21936
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-05/index",
"target_anchor": "bound-definition-2d833ac0bb979a1a384519d5",
"resolved": true,
"reason": null,
"source_revision": "e4b5540a1170224114cc2ab29049010d355eaf7b3257a6c2f94afce3ab5037fb",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/01-section-6-1",
"occurrence_id": "occ-9aeb67c5ee1c10bf3d114463"
},
{
"raw": "href=\"../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c\"",
"href": "../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c",
"range": [
23559,
23639
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"target_anchor": "bound-definition-b6c4b4cd58ce97d21892883c",
"resolved": true,
"reason": null,
"source_revision": "e4b5540a1170224114cc2ab29049010d355eaf7b3257a6c2f94afce3ab5037fb",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/01-section-6-1",
"occurrence_id": "occ-9aeb67c5ee1c10bf3d114463"
},
{
"raw": "href=\"../chapter-03/02-section-3-2.md#section-3-2\"",
"href": "../chapter-03/02-section-3-2.md#section-3-2",
"range": [
25523,
25573
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/02-section-3-2",
"target_anchor": "section-3-2",
"resolved": true,
"reason": null,
"source_revision": "e4b5540a1170224114cc2ab29049010d355eaf7b3257a6c2f94afce3ab5037fb",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/01-section-6-1",
"occurrence_id": "occ-9aeb67c5ee1c10bf3d114463"
},
{
"raw": "href=\"../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c\"",
"href": "../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c",
"range": [
51074,
51154
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"target_anchor": "bound-definition-b6c4b4cd58ce97d21892883c",
"resolved": true,
"reason": null,
"source_revision": "2adf7861492ef32e8411451cbaaa1ac2e305e6d8da633adb4624cf35a45d937d",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/02-section-6-2",
"occurrence_id": "occ-679e1f6b7854685345b2b84a"
},
{
"raw": "[练习 6.4](02-section-6-2.md#exercise-6-4)",
"href": "02-section-6-2.md#exercise-6-4",
"range": [
52700,
52740
],
"relation": "references",
"target_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/02-section-6-2",
"target_anchor": "exercise-6-4",
"resolved": true,
"reason": null,
"source_revision": "2adf7861492ef32e8411451cbaaa1ac2e305e6d8da633adb4624cf35a45d937d",
"source_route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/02-section-6-2",
"occurrence_id": "occ-679e1f6b7854685345b2b84a"
}
],
"reading_order": [
"occ-88344bd705205e684822e0c7",
"occ-2ee2492cd15dc230abc55711",
"occ-9aeb67c5ee1c10bf3d114463",
"occ-679e1f6b7854685345b2b84a",
"occ-08e4f297979474c60d397dcd",
"occ-507ade36b5f5d4108a7dc797"
],
"recorded_dependencies": [],
"reading_paths": [],
"coverage": {
"registered_positions": 2,
"located_occurrences": 6,
"whole_book_search_complete": false,
"registered_positions_truncated": false,
"independent_human_review_implied": false
},
"unresolved": [],
"reviewed_definitions": [
{
"id": "szekeres-simplex-boundary",
"role": "definition",
"sense": "homology",
"path": "library/books/szekeres/source-snapshot-bc3404156f6e/sources/chapter-17/en/04-homology-and-cohomology.md",
"document_sha256": "f20e33e4c5a6277f4b4f3a4fe4e24383dfcb988a565b3d676d50911708cb2d74",
"range": [
4259,
4789
],
"range_unit": "Unicode code points; half-open; full decoded file",
"line": 42,
"quote": "and the boundary of a $p$-simplex $\\Delta$ is defined as the $( p - 1 )$ -chain\n\n$$\n\\partial \\Delta = \\sum_ {i = 0} ^ {p} \\Delta_ {i} = \\sum_ {i = 0} ^ {p} (- 1) ^ {i} \\langle \\mathbf {x} _ {0}, \\ldots , \\widehat {\\mathbf {x}} _ {i}, \\ldots , \\mathbf {x} _ {p} \\rangle ,\n$$\n\n\\\n*Figure 17.5 Standard low dimensional $p$-simplexes*\n\nand extends to all $p$-chains by linearity.",
"quote_sha256": "13d62541bbedd6bd985b3383d9cd43af202c98a5890e6c941ab10a8e5886d0a8",
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"section": "17.4",
"source_label": "boundary",
"source_zh": "边缘",
"review_status": "librarian-original-source-reviewed",
"construction": "oriented simplex alternating boundary chain; linear extension",
"conditions": "链群语境的边缘；各构造按原书保留，不与点集边界合并。",
"historical_evidence_pack": "reports/coordination-20261020/boundary-sense-review/evidence.json",
"historical_evidence_pack_sha256": "7321767e7468fa6495ed8b074f3069312a0735430e90b218e759df8e4ecf4c79",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"reading_links": [
{
"route": "en/chapter-17/04-homology-and-cohomology",
"anchor": "section-17-4",
"lang": "en",
"basis": "registered-section-context"
},
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"anchor": "section-17-4",
"lang": "zh-CN",
"basis": "registered-section-context"
}
]
},
{
"id": "szekeres-homological-boundary",
"role": "definition",
"sense": "homology",
"path": "library/books/szekeres/source-snapshot-bc3404156f6e/sources/chapter-17/en/04-homology-and-cohomology.md",
"document_sha256": "f20e33e4c5a6277f4b4f3a4fe4e24383dfcb988a565b3d676d50911708cb2d74",
"range": [
7638,
8022
],
"range_unit": "Unicode code points; half-open; full decoded file",
"line": 94,
"quote": "A $p$-chain $C$ is said to be a cycle if it has no boundary, $\\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\\prime}$ such that $C = \\partial C ^ { \\prime }$ . Clearly every boundary is a $p$-cycle since $\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$ , but the converse need not be true.",
"quote_sha256": "0339bf00eaa16a8b626f1f96c4d33d81aa60e3007934d43a453aaafb009b7b14",
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"section": "17.4",
"source_label": "boundary",
"source_zh": "边缘",
"review_status": "librarian-original-source-reviewed",
"construction": "p-chain in the image of the next boundary map",
"conditions": "链群语境的边缘；各构造按原书保留，不与点集边界合并。",
"historical_evidence_pack": "reports/coordination-20261020/boundary-sense-review/evidence.json",
"historical_evidence_pack_sha256": "7321767e7468fa6495ed8b074f3069312a0735430e90b218e759df8e4ecf4c79",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"reading_links": [
{
"route": "en/chapter-17/04-homology-and-cohomology",
"anchor": "section-17-4",
"lang": "en",
"basis": "registered-section-context"
},
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"anchor": "section-17-4",
"lang": "zh-CN",
"basis": "registered-section-context"
}
]
},
{
"id": "szekeres-singular-boundary",
"role": "definition",
"sense": "homology",
"path": "library/books/szekeres/source-snapshot-bc3404156f6e/sources/chapter-17/en/04-homology-and-cohomology.md",
"document_sha256": "f20e33e4c5a6277f4b4f3a4fe4e24383dfcb988a565b3d676d50911708cb2d74",
"range": [
9933,
10248
],
"range_unit": "Unicode code points; half-open; full decoded file",
"line": 142,
"quote": "A $p$-boundary $B$ is a singular $p$-cycle on $M$ that is the boundary of a $(p+1)$-chain, $B = \\partial _ { p + 1 } C . \\mathrm { ~ A ~ }$ $p$-cycle $C$ is a singular $p$-chain on $M$ whose boundary vanishes, $\\partial _ { p } C = 0$ Since $\\partial ^ { 2 } = 0$ it is clear that every $p$-boundary is a $p$-cycle.",
"quote_sha256": "3c58cc7a019df937cc1038cd6128dd7d261317c3b22beedaf2e9d6ae61bc93f7",
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"section": "17.4",
"source_label": "boundary",
"source_zh": "边缘",
"review_status": "librarian-original-source-reviewed",
"construction": "singular p-chain in the image of the next boundary map",
"conditions": "链群语境的边缘；各构造按原书保留，不与点集边界合并。",
"historical_evidence_pack": "reports/coordination-20261020/boundary-sense-review/evidence.json",
"historical_evidence_pack_sha256": "7321767e7468fa6495ed8b074f3069312a0735430e90b218e759df8e4ecf4c79",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"reading_links": [
{
"route": "en/chapter-17/04-homology-and-cohomology",
"anchor": "section-17-4",
"lang": "en",
"basis": "registered-section-context"
},
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"anchor": "section-17-4",
"lang": "zh-CN",
"basis": "registered-section-context"
}
]
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"definition_id": "urn:uuid:b85ec1b0-0e2a-5236-b0e3-0aff073ad1cb",
"definition_group": "definition-3-4",
"path": "library/concepts/source-evidence/mcp/3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c.txt",
"range": [
0,
154
],
"quote": "Definition 3.4. Let $K$ be an $n$ -dimensional simplicial complex and let $c \\in C_r(K)$ . If there exists an element $d \\in C_{r + 1}(K)$ such that",
"document_sha256": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"quote_sha256": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"section": "3.3.2",
"pdf_page_index": 123,
"node": "a-075fd90cce3c868365101fbd",
"source_node_sha256": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"upstream": {
"path": "library/concepts/source-evidence/mcp/d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5.json",
"document_sha256": "d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5",
"node_id": "a-075fd90cce3c868365101fbd",
"node_source_sha256": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c"
},
"conditions": "Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。",
"review_status": "librarian-original-source-reviewed",
"source_locator": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml",
"sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"nodes": [
"a-075fd90cce3c868365101fbd",
"a-1e001c6774fe781a79ebe340",
"a-a640d82e944756100d483ec3",
"a-19671c7e478fcddd158d5517"
],
"node_hashes": {
"a-075fd90cce3c868365101fbd": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"a-1e001c6774fe781a79ebe340": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"a-a640d82e944756100d483ec3": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"a-19671c7e478fcddd158d5517": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e"
},
"environment": "definition-3-4"
},
"environment_basis": {
"path": "library/concepts/source-evidence/environment-basis/bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde.yaml.txt",
"document_sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"source_path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml"
},
"role": "definition",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"definition_part": "definition-environment",
"definition_part_basis": "original-environment-range",
"reading_links": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-03/03-section-3-3",
"anchor": "node-a-075fd90cce3c868365101fbd",
"lang": "en",
"basis": "explicit-source-anchor"
},
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"anchor": "node-a-075fd90cce3c868365101fbd",
"lang": "zh-CN",
"basis": "explicit-source-anchor"
}
]
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"definition_id": "urn:uuid:b85ec1b0-0e2a-5236-b0e3-0aff073ad1cb",
"definition_group": "definition-3-4",
"path": "library/concepts/source-evidence/mcp/1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6.txt",
"range": [
0,
43
],
"quote": "\n$$\nc = \\partial_ {r + 1} d \\tag {3.25}\n$$\n",
"document_sha256": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"quote_sha256": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"section": "3.3.2",
"pdf_page_index": 123,
"node": "a-1e001c6774fe781a79ebe340",
"source_node_sha256": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"upstream": {
"path": "library/concepts/source-evidence/mcp/d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5.json",
"document_sha256": "d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5",
"node_id": "a-1e001c6774fe781a79ebe340",
"node_source_sha256": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6"
},
"conditions": "Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。",
"review_status": "librarian-original-source-reviewed",
"source_locator": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml",
"sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"nodes": [
"a-075fd90cce3c868365101fbd",
"a-1e001c6774fe781a79ebe340",
"a-a640d82e944756100d483ec3",
"a-19671c7e478fcddd158d5517"
],
"node_hashes": {
"a-075fd90cce3c868365101fbd": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"a-1e001c6774fe781a79ebe340": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"a-a640d82e944756100d483ec3": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"a-19671c7e478fcddd158d5517": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e"
},
"environment": "definition-3-4"
},
"environment_basis": {
"path": "library/concepts/source-evidence/environment-basis/bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde.yaml.txt",
"document_sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"source_path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml"
},
"role": "definition",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"definition_part": "definition-environment",
"definition_part_basis": "original-environment-range",
"reading_links": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-03/03-section-3-3",
"anchor": "node-a-1e001c6774fe781a79ebe340",
"lang": "en",
"basis": "explicit-source-anchor"
},
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"anchor": "node-a-1e001c6774fe781a79ebe340",
"lang": "zh-CN",
"basis": "explicit-source-anchor"
}
]
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"definition_id": "urn:uuid:b85ec1b0-0e2a-5236-b0e3-0aff073ad1cb",
"definition_group": "definition-3-4",
"path": "library/concepts/source-evidence/mcp/469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3.txt",
"range": [
0,
244
],
"quote": "then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \\operatorname{im} \\partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]",
"document_sha256": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"quote_sha256": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"section": "3.3.2",
"pdf_page_index": 123,
"node": "a-a640d82e944756100d483ec3",
"source_node_sha256": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"upstream": {
"path": "library/concepts/source-evidence/mcp/d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5.json",
"document_sha256": "d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5",
"node_id": "a-a640d82e944756100d483ec3",
"node_source_sha256": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3"
},
"conditions": "Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。",
"review_status": "librarian-original-source-reviewed",
"source_locator": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml",
"sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"nodes": [
"a-075fd90cce3c868365101fbd",
"a-1e001c6774fe781a79ebe340",
"a-a640d82e944756100d483ec3",
"a-19671c7e478fcddd158d5517"
],
"node_hashes": {
"a-075fd90cce3c868365101fbd": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"a-1e001c6774fe781a79ebe340": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"a-a640d82e944756100d483ec3": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"a-19671c7e478fcddd158d5517": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e"
},
"environment": "definition-3-4"
},
"environment_basis": {
"path": "library/concepts/source-evidence/environment-basis/bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde.yaml.txt",
"document_sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"source_path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml"
},
"role": "definition",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"definition_part": "definition-environment",
"definition_part_basis": "original-environment-range",
"reading_links": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-03/03-section-3-3",
"anchor": "node-a-a640d82e944756100d483ec3",
"lang": "en",
"basis": "explicit-source-anchor"
},
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"anchor": "node-a-a640d82e944756100d483ec3",
"lang": "zh-CN",
"basis": "explicit-source-anchor"
}
]
},
{
"concept_id": "urn:uuid:464db988-1584-4cad-b2c3-a93d61fae5ad",
"definition_id": "urn:uuid:b85ec1b0-0e2a-5236-b0e3-0aff073ad1cb",
"definition_group": "definition-3-4",
"path": "library/concepts/source-evidence/mcp/199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e.txt",
"range": [
0,
221
],
"quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.",
"document_sha256": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e",
"quote_sha256": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"section": "3.3.2",
"pdf_page_index": 123,
"node": "a-19671c7e478fcddd158d5517",
"source_node_sha256": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e",
"upstream": {
"path": "library/concepts/source-evidence/mcp/d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5.json",
"document_sha256": "d302f693145ffcdbf903716892cc04dbeeb8f3677ef76a47eb2a635be10c99b5",
"node_id": "a-19671c7e478fcddd158d5517",
"node_source_sha256": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e"
},
"conditions": "Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。",
"review_status": "librarian-original-source-reviewed",
"source_locator": {
"path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml",
"sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"nodes": [
"a-075fd90cce3c868365101fbd",
"a-1e001c6774fe781a79ebe340",
"a-a640d82e944756100d483ec3",
"a-19671c7e478fcddd158d5517"
],
"node_hashes": {
"a-075fd90cce3c868365101fbd": "3f42a112eeb27ec3dffb363fd73d6cf6dd643eb6657527d02199662cb35d6b8c",
"a-1e001c6774fe781a79ebe340": "1c6103f6fc9eafcc9a9a280f3a4f0cbccc30dd97020b4507540dccf083a6b2b6",
"a-a640d82e944756100d483ec3": "469c177f129631e9ac63dc63f18db8f8ac7571eb6b54a48be7432c1877ed4ba3",
"a-19671c7e478fcddd158d5517": "199589346fa789422c82ee9115fe4650e0869fd29968a40c2eb2802c86791c1e"
},
"environment": "definition-3-4"
},
"environment_basis": {
"path": "library/concepts/source-evidence/environment-basis/bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde.yaml.txt",
"document_sha256": "bac03a89e2212ddb99847b73c6aa1a10e5e7c4fbff75c8a8bcd05336d8780bde",
"source_path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/nakahara-3.yaml"
},
"role": "definition",
"first_in_book": "not-established",
"evidence_origin": "explicit-reviewed-source-manifest",
"definition_part": "definition-environment",
"definition_part_basis": "original-environment-range",
"reading_links": [
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-03/03-section-3-3",
"anchor": "node-a-19671c7e478fcddd158d5517",
"lang": "en",
"basis": "explicit-source-anchor"
},
{
"route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/03-section-3-3",
"anchor": "node-a-19671c7e478fcddd158d5517",
"lang": "zh-CN",
"basis": "explicit-source-anchor"
}
]
}
],
"base_revision": "646d4dc3bca535d988f22b139303d8a5e822a77c7795ebbcbc4869fda9262e2a",
"optional_enrichments": []
}