{ "schema": "kt-cp-versioned-result-source-use/v1", "course": "KT-CP", "recorded_at_utc": "2026-10-03T19:12:58.732234+00:00", "results": [ { "id": "KT-CP-07/Theorem-7B.1", "lesson": "KT-CP-07", "kind": "theorem", "title": "Exactness for closed and cocompact subgroups", "label": "Theorem 7B.1", "source": "src/KT-CP-07.md", "full_conditions": "All locally compact Hausdorff groups, closed subgroups; converse requires compact G/H.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "echterhoff-2017-v4", "role": "proof construction", "correspondence": "exact closed/cocompact subgroup statement", "locator": "Definition 4.9; Theorem 6.11, PDF pp.13,32–33" } ], "source_sha256": "5572D654E9A38CC3FB3C1FCBF0B8A071829075FA947EE6FDBCF1E2555F5E8E58", "reader": "KT-CP-07.html", "proof_location": { "source": "src/KT-CP-07.md", "start_line": 598, "end_line": 637, "sha256_utf8_lf": "5C26ABCBD23CEAFA8A6C00935EC4AEDA2EB875C608E7A76A36FA2171E8736EDC" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-07.md", "source_sha256": "5572D654E9A38CC3FB3C1FCBF0B8A071829075FA947EE6FDBCF1E2555F5E8E58", "locator": "Theorem 7.4", "state": "written", "reader": "KT-CP-07.html" }, { "course": "KT-CP", "source": "src/KT-CP-07.md", "source_sha256": "5572D654E9A38CC3FB3C1FCBF0B8A071829075FA947EE6FDBCF1E2555F5E8E58", "locator": "## The reduced form", "state": "written", "reader": "KT-CP-07.html" }, { "course": "KT-CP", "source": "src/KT-CP-02.md", "source_sha256": "660177BE6291AB788BD43F946CB74805E477F1080AEA722A285587063D8BFC72", "locator": "Theorem 2.5", "state": "written", "reader": "KT-CP-02.html" }, { "course": "hilbert-c-star-modules-and-morita-equivalence", "source": "prerequisites/src/exact/hilbert-c-star-modules-and-morita-equivalence/81E3988691F5/the-rieffel-correspondence-and-induced-representations.md", "source_sha256": "1A5C275C3B04C0554DBC654BAB60A9B2ADBFB4D69F0050322E445147DED404F6", "locators": [ "Theorem 2.1", "Proposition 2.2", "Theorem 3.1" ], "state": "written", "reader": "prerequisites/exact/hilbert-c-star-modules-and-morita-equivalence/81E3988691F5/the-rieffel-correspondence-and-induced-representations.html" } ] }, { "id": "KT-CP-07/Theorem-7B.2", "lesson": "KT-CP-07", "kind": "theorem", "title": "Covariant localization dilation", "label": "Theorem 7B.2", "source": "src/KT-CP-07.md", "full_conditions": "Nondegenerate completely positive C0(G/H) map and strongly continuous covariant G representation; arbitrary closed H and Hilbert space.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "landsman-1998-v1", "role": "proof construction and application", "correspondence": "generalized from the localization treatment to arbitrary closed H using the existing Green module", "locator": "Theorems 4.4.2 and 4.5.2, PDF pp.87–89" } ], "source_sha256": "5572D654E9A38CC3FB3C1FCBF0B8A071829075FA947EE6FDBCF1E2555F5E8E58", "reader": "KT-CP-07.html", "proof_location": { "source": "src/KT-CP-07.md", "start_line": 658, "end_line": 697, "sha256_utf8_lf": "4A9B9FD33EF0B46AE6D35F8E21F8339E8F01288CFEFE0B7831E16D43A5E7F997" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-07.md", "source_sha256": "5572D654E9A38CC3FB3C1FCBF0B8A071829075FA947EE6FDBCF1E2555F5E8E58", "locator": "Theorem 7.5", "state": "written", "reader": "KT-CP-07.html" }, { "course": "hilbert-c-star-modules-and-morita-equivalence", "source": "prerequisites/src/exact/hilbert-c-star-modules-and-morita-equivalence/81E3988691F5/the-rieffel-correspondence-and-induced-representations.md", "source_sha256": "1A5C275C3B04C0554DBC654BAB60A9B2ADBFB4D69F0050322E445147DED404F6", "locators": [ "Theorem 2.1", "Proposition 2.2", "Theorem 3.1" ], "state": "written", "reader": "prerequisites/exact/hilbert-c-star-modules-and-morita-equivalence/81E3988691F5/the-rieffel-correspondence-and-induced-representations.html" }, { "course": "foundations-of-von-neumann-algebras", "source": "prerequisites/src/exact/foundations-of-von-neumann-algebras/DD4372862B2D/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md", "source_sha256": "EE1759A464C684B18E0CAE413D41C04604A339037E83F133199B61B8B8F50674", "locators": [ "Theorem 4.7" ], "state": "written", "reader": "prerequisites/exact/foundations-of-von-neumann-algebras/DD4372862B2D/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html" } ] }, { "id": "KT-CP-10/Theorem-10C.1", "lesson": "KT-CP-10", "kind": "theorem", "title": "Cooper theorem and continuous Wold decomposition", "label": "Theorem 10C.1", "source": "src/KT-CP-10.md", "full_conditions": "Arbitrary Hilbert space and strongly continuous semigroup of isometries, arbitrary multiplicity cardinal.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "sundar-2025-v1", "role": "permitted adaptation and correction", "correspondence": "full semigroup classification, extended to arbitrary multiplicity with completed bridges", "locator": "Native TeX Cooper section; PDF pp.118–129", "component_notice": "sundar-cooper-2025-v1" } ], "source_sha256": "C4BB6E6785C8EEEEE67AFD20899296CC965B58144B0FB3AD50589A39D1FC80B8", "reader": "KT-CP-10.html", "proof_location": { "source": "src/KT-CP-10.md", "start_line": 304, "end_line": 387, "sha256_utf8_lf": "9DC9210135D59400CE2E8AD39069E21ECF73A0FDE5D81C49E964C2C718B88E88" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-07.md", "source_sha256": "5572D654E9A38CC3FB3C1FCBF0B8A071829075FA947EE6FDBCF1E2555F5E8E58", "locator": "## Translation gives compact operators for every group", "state": "written", "reader": "KT-CP-07.html" }, { "course": "foundations-of-von-neumann-algebras", "source": "prerequisites/src/exact/foundations-of-von-neumann-algebras/50F1FFA37C83/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md", "source_sha256": "CD5D5E40B3FDCB9A64DFE3387A3C83477350C03C28F62C7A573465EBEA7CA21F", "locators": [ "Theorem 2.1" ], "state": "written", "reader": "prerequisites/exact/foundations-of-von-neumann-algebras/50F1FFA37C83/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html" }, { "course": "function-algebras-and-approximation", "source": "prerequisites/src/exact/function-algebras-and-approximation/CD62124F940F/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md", "source_sha256": "4A1061EE4B1E37E024E19FC6FAF233E505D744134871ACFFC8428BC75ABC5344", "locators": [ "Theorem 10.1" ], "state": "written", "reader": "prerequisites/exact/function-algebras-and-approximation/CD62124F940F/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.html" }, { "course": "harmonic-analysis-on-locally-compact-groups", "source": "prerequisites/src/exact/harmonic-analysis-on-locally-compact-groups/A6D03E284155/measure-and-hilbert-space-tools.md", "source_sha256": "94712BF576D0EC0AF9122C381FA7998FDD042F2D9B155F809827885F89CA9C75", "locators": [ "Theorem 4.2", "Theorem 3.2" ], "state": "written", "reader": "prerequisites/exact/harmonic-analysis-on-locally-compact-groups/A6D03E284155/measure-and-hilbert-space-tools.html" } ] }, { "id": "KT-CP-12/Theorem-12B.1", "lesson": "KT-CP-12", "kind": "theorem", "title": "Normalized continuous twisted-action stabilization", "label": "Theorem 12B.1", "source": "src/KT-CP-12.md", "full_conditions": "Locally compact Hausdorff G; arbitrary A; point-norm automorphisms and normalized jointly strict continuous cocycle.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "buss-meyer-zhu-2009-v1", "role": "proof construction", "correspondence": "group case of stabilization, with explicit full/reduced norm comparisons and translated cocycle convention", "locator": "Proposition 5.2 and proof, PDF pp.30–31" }, { "source_key": "clay-topics-2012", "role": "context and proof construction", "correspondence": "Busby–Smith coherence and cochain change", "locator": "Meyer chapter, PDF pp.95–96" } ], "source_sha256": "FA38622020E82CA5129260239A0AFE3402CE48750224425C15F87049F05A0965", "reader": "KT-CP-12.html", "proof_location": { "source": "src/KT-CP-12.md", "start_line": 181, "end_line": 215, "sha256_utf8_lf": "7CFD5ED053B87E2CD193DCC1DD4D6F1E9DC16FC25949C92846B331989D36CA5A" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-02.md", "source_sha256": "660177BE6291AB788BD43F946CB74805E477F1080AEA722A285587063D8BFC72", "locator": "## A regular model with no chosen Hilbert space", "state": "written", "reader": "KT-CP-02.html" }, { "course": "hilbert-c-star-modules-and-morita-equivalence", "source": "prerequisites/src/hilbert-c-star-modules-and-morita-equivalence/tensor-products-and-c-star-correspondences.md", "source_sha256": "9DABA13E015B1CC27EA0519246C0F7D01654DA68142A1FA687B2419B009F4CFF", "locators": [ "Theorem 4.1", "Theorem 5.1" ], "state": "written", "reader": "prerequisites/hilbert-c-star-modules-and-morita-equivalence/tensor-products-and-c-star-correspondences.html" } ] }, { "id": "KT-CP-12/Example-12B.14", "lesson": "KT-CP-12", "kind": "example", "title": "Pointwise inner projective obstruction", "label": "Example 12B.14", "source": "src/KT-CP-12.md", "full_conditions": "Irrational rotation scalar c and compact algebra on l2(Z).", "status": "proved; author self-checked", "source_uses": [ { "source_key": "clay-topics-2012", "role": "example", "correspondence": "explicit rotation example of the pointwise-inner obstruction, with sign and universal inverse maps proved", "locator": "Meyer chapter, pointwise inner versus inner discussion" } ], "source_sha256": "FA38622020E82CA5129260239A0AFE3402CE48750224425C15F87049F05A0965", "reader": "KT-CP-12.html", "proof_location": { "source": "src/KT-CP-12.md", "start_line": 219, "end_line": 239, "sha256_utf8_lf": "D92B4ECD148DDE4E4ACEDBC02C8C0641485DA102F665EC49C9616ABD579712C3" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-03.md", "source_sha256": "A7CE0F91F306B229DE55C28095E93B7CA9DAC9A460305DC128FE5E12DDD9ECB7", "locator": "Corollary 3.6", "state": "written", "reader": "KT-CP-03.html" }, { "course": "KT-CP", "source": "src/KT-CP-03.md", "source_sha256": "A7CE0F91F306B229DE55C28095E93B7CA9DAC9A460305DC128FE5E12DDD9ECB7", "locator": "**Irrational rotation.**", "state": "written", "reader": "KT-CP-03.html" }, { "course": "KT-CP", "source": "src/KT-CP-04.md", "source_sha256": "F8732CC750C96004D13C7D8F49EB26067AF118BC4139604D9AA053E02C0F19BF", "locator": "## Compression proves the norm equality", "state": "written", "reader": "KT-CP-04.html" }, { "course": "hilbert-c-star-modules-and-morita-equivalence", "source": "prerequisites/src/hilbert-c-star-modules-and-morita-equivalence/tensor-products-and-c-star-correspondences.md", "source_sha256": "9DABA13E015B1CC27EA0519246C0F7D01654DA68142A1FA687B2419B009F4CFF", "locators": [ "Theorem 4.1", "Theorem 5.1" ], "state": "written", "reader": "prerequisites/hilbert-c-star-modules-and-morita-equivalence/tensor-products-and-c-star-correspondences.html" } ] }, { "id": "KT-CP-14/Theorem-14I.1", "lesson": "KT-CP-14", "kind": "theorem", "title": "Effective etale diagonal uniqueness", "label": "Theorem 14I.1", "source": "src/KT-CP-14.md", "full_conditions": "Any locally compact Hausdorff effective etale groupoid; no second countability needed.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "sims-2017", "role": "proof construction and generalization", "correspondence": "countable density argument replaced by finite compact-support compression for arbitrary Hausdorff etale G", "locator": "PDF pp.35–37" } ], "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "reader": "KT-CP-14.html", "proof_location": { "source": "src/KT-CP-14.md", "start_line": 584, "end_line": 600, "sha256_utf8_lf": "64A199E5911B4817A236224B648F6F7B90743992A06FCC9DBE2D57F41904BE28" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14.4", "state": "written", "reader": "KT-CP-14.html" } ] }, { "id": "KT-CP-14/Theorem-14I.2", "lesson": "KT-CP-14", "kind": "theorem", "title": "Canonical Cartan diagonal criterion", "label": "Theorem 14I.2", "source": "src/KT-CP-14.md", "full_conditions": "Any locally compact Hausdorff etale groupoid; Cartan iff effective.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "sims-2017", "role": "proof construction", "correspondence": "canonical diagonal Cartan criterion; direct coefficient proof in the current Hausdorff scope", "locator": "Chapter 5, especially Lemma 5.1.14; PDF pp.49–51" } ], "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "reader": "KT-CP-14.html", "proof_location": { "source": "src/KT-CP-14.md", "start_line": 610, "end_line": 636, "sha256_utf8_lf": "EAD075DD727ED607333D381F288A7415F639F2B1D3D3B208269AE933BB7DDFD2" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14.4", "state": "written", "reader": "KT-CP-14.html" } ] }, { "id": "KT-CP-14/Theorem-14I.3", "lesson": "KT-CP-14", "kind": "theorem", "title": "Dynamical ideal correspondence and simplicity", "label": "Theorem 14I.3", "source": "src/KT-CP-14.md", "full_conditions": "Inner exact plus strongly effective for reduced ideals; minimal effective suffices for reduced simplicity; amenable converse uses standing second countability.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "sims-2017", "role": "proof construction and generalization", "correspondence": "reduced ideal theorem and amenable simplicity converse with all hypotheses explicit", "locator": "PDF pp.38–42" }, { "source_key": "exel-2017-v2", "role": "comparison", "correspondence": "related partial/discrete-action uniqueness and ideal results; does not substitute for the groupoid proof", "locator": "Theorems 29.5 and 29.9; PDF pp.250–254" } ], "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "reader": "KT-CP-14.html", "proof_location": { "source": "src/KT-CP-14.md", "start_line": 650, "end_line": 673, "sha256_utf8_lf": "8FB10D11FA86E2F73F1D06959743A11DEAAC6245D66185DC3FF998D095959144" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14I.1", "state": "written", "reader": "KT-CP-14.html" }, { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14.5", "state": "written", "reader": "KT-CP-14.html" }, { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14.7", "state": "written", "reader": "KT-CP-14.html" } ] }, { "id": "KT-CP-14/Theorem-14I.4", "lesson": "KT-CP-14", "kind": "theorem", "title": "Ideal sandwich and full residual triples", "label": "Theorem 14I.4", "source": "src/KT-CP-14.md", "full_conditions": "Any locally compact Hausdorff inner exact etale groupoid, including empty residual groupoid.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "brix-carlsen-sims-2024-v2", "role": "proof construction and correction", "correspondence": "exact full-support residual triple bijection, including empty residual", "locator": "Lemmas 3.4/3.6, Theorem 3.7 and Remark 3.8, PDF pp.6–8" } ], "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "reader": "KT-CP-14.html", "proof_location": { "source": "src/KT-CP-14.md", "start_line": 683, "end_line": 724, "sha256_utf8_lf": "D0EF87151518A4691C5158C85959099DF7A5B5EC31F26D3156B53C47EC89C3A8" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14.4", "state": "written", "reader": "KT-CP-14.html" }, { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14I.3", "state": "written", "reader": "KT-CP-14.html" } ] }, { "id": "KT-CP-15/Theorem-15I.1", "lesson": "KT-CP-15", "kind": "theorem", "title": "Morita invariance of inner exactness", "label": "Theorem 15I.1", "source": "src/KT-CP-15.md", "full_conditions": "Equivalent second countable locally compact Hausdorff groupoids with full Haar systems and Hausdorff equivalence space.", "status": "proved; author self-checked", "source_uses": [ { "source_key": "lalonde-2018-v3", "role": "proof construction", "correspondence": "exact Hausdorff linking proof for inner exactness; not coefficient functor exactness", "locator": "Theorem 6.1, PDF pp.25–26" }, { "source_key": "anantharaman-delaroche-2023", "role": "context", "correspondence": "same inner-exactness distinction", "locator": "Definition 5.9, PDF pp.17–18" } ], "source_sha256": "ABBC08B6EEAC42278AB5AE7588545215DAA918A28F49534B0D195AF06BCB6302", "reader": "KT-CP-15.html", "proof_location": { "source": "src/KT-CP-15.md", "start_line": 282, "end_line": 320, "sha256_utf8_lf": "20B6BC17B9A5FBD21D011A8EDBA710BBB30C7E2BA5E20F721B1F578B1E20DD8C" }, "checking": { "author": "GPT-6.1 Sol (OpenAI)", "reasoning_effort": "Ultra", "mathematical_self_check": "complete", "independent_review": "Not claimed for this edition." }, "exact_programme_dependencies": [ { "course": "KT-CP", "source": "src/KT-CP-15.md", "source_sha256": "ABBC08B6EEAC42278AB5AE7588545215DAA918A28F49534B0D195AF06BCB6302", "locator": "Theorem 15.3", "state": "written", "reader": "KT-CP-15.html" }, { "course": "KT-CP", "source": "src/KT-CP-14.md", "source_sha256": "430CCBAC9AFC70ED3C85F013C8607EB6C4B3FAE1F4B1F10E0BEFCA9AD543889E", "locator": "Theorem 14I.4", "state": "written", "reader": "KT-CP-14.html" }, { "course": "hilbert-c-star-modules-and-morita-equivalence", "source": "prerequisites/src/exact/hilbert-c-star-modules-and-morita-equivalence/81E3988691F5/the-rieffel-correspondence-and-induced-representations.md", "source_sha256": "1A5C275C3B04C0554DBC654BAB60A9B2ADBFB4D69F0050322E445147DED404F6", "locators": [ "Theorem 2.1", "Proposition 2.2", "Theorem 3.1" ], "state": "written", "reader": "prerequisites/exact/hilbert-c-star-modules-and-morita-equivalence/81E3988691F5/the-rieffel-correspondence-and-induced-representations.html" } ] } ], "source_to_results": [ { "source_key": "anantharaman-delaroche-2023", "result_ids": [ "KT-CP-15/Theorem-15I.1" ] }, { "source_key": "brix-carlsen-sims-2024-v2", "result_ids": [ 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