{ "schema": "oa-mod-spectral-exact-free-proof-bindings/v1", "lesson": { "id": "OA-MOD-SK", "title": "Spectral calculus with its domains retained", "source": "src/spectral-calculus-kernel.md", "reader": "spectral-calculus-kernel.html", "mathematical_item_ids": [ "OA-MOD-SK-01", "OA-MOD-SK-02", "OA-MOD-SK-03", "OA-MOD-SK-04", "OA-MOD-SK-05", "OA-MOD-SK-06", "OA-MOD-SK-07", "OA-MOD-SK-08", "OA-MOD-SK-09", "OA-MOD-SK-10", "OA-MOD-SK-11", "OA-MOD-SK-12" ] }, "original_reviewed_source_sha256": "b0e3221bb390aa0b11c6a7d7f5aa6a49e816cde8bac8dfcc47b5e757cc428d98", "public_projected_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0", "earlier_public_proofs": [ { "id": "OA-FND-HB-01", "source": "docs/courses/foundations-of-von-neumann-algebras/src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "f09b763b00c87141faac24be5c8d4a4760c3439085288bcdb56263731281e850", "proof_locus": { "line": 30, "through_line": 46 }, "proof_anchor": 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"OA-MOD-BK-08", "source": "docs/courses/OA-MOD/src/bounded-operator-kernel.md", "source_sha256": "04e0495ab7ec51fdc6b6f07a5dc835dfa623680103fc9f5bfc0cf30c9204fd3b", "proof_anchor": "OA-MOD-BK-08", "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/bounded-operator-kernel.html#OA-MOD-BK-08", "publication_commit": "8085ac1fff17797aca921090f70d49adafe6269b" } ], "scalar_source_proofs": [ { "id": "OPEN-SCALAR-RMK", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Real.lean", "source_sha256": "3B5649EF2AC22EE09C1DF19221E4C2797209DCEF34BAAE1D0C66A6D32B78A535", "source_bytes": 25466, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 65, "end_line": 65, "range_sha256": "B2F962527F3244923D99C2C2C30A79B2BE90391F61592A082F84B77F35086C26" }, { "start_line": 70, "end_line": 104, "range_sha256": "3C65042F1B7794FEDA7DB25E82EE0A244BC280E61B4286D54CF2C118E1A7BC63" }, { "start_line": 109, "end_line": 153, "range_sha256": "69D8513313BF78C0AB21CC3BF5717EA2E015ECC41F76B811983CCC4C814E8B01" }, { "start_line": 158, "end_line": 173, "range_sha256": "EDF8E645FFC55E281D8C0416362D260251EF6E7F8525D5A128618FF5752FDBC4" }, { "start_line": 176, "end_line": 185, "range_sha256": "6220249B083CA33E90657F1F8222656AEAC53D84AD2A6B7E57E10E85FC109BEF" }, { "start_line": 190, "end_line": 340, "range_sha256": "1BCBFD1997BF99609E86AD430D04344DFF6B72BFBEE880CE890850333E9A99A2" }, { "start_line": 345, "end_line": 354, "range_sha256": "5204D3E16C4165BAEE367C37FC1DECB739C5331C056AF85FBFDAF9F92E983F1D" }, { "start_line": 357, "end_line": 358, "range_sha256": "34F65479DAA9ADA8245B72787DA0F5FA99F2952D546B865BB79D3FFF375CB835" }, { "start_line": 371, "end_line": 408, "range_sha256": "D3D4A0A950569B6C3ADC573FDE337B086F630237F10FB97037275D893F53162F" }, { "start_line": 412, "end_line": 421, "range_sha256": "487D210FFC0CC5F65FB57889A30676FDE619AC6D51A06C91C0ED8350C31CFC03" }, { "start_line": 445, "end_line": 451, "range_sha256": "4A440C6F4C017C78A846AD9D6CA0D0B8CB8DA590F1A0D300DD3A7C3F9E4B0973" } ], "hypothesis_ranges": [ [ 52, 60 ], [ 362, 364 ], [ 443, 445 ] ], "contract": "For a locally compact Hausdorff X with its Borel sigma algebra and a positive real-linear functional on C_c(X,R), constructs a regular Borel measure realizing the functional. The measure is unique among regular measures. On compact X it is finite. Specialize C_c=C on a compact spectrum, apply integral_rieszMeasure to the constant one for total mass, and apply separately to real and imaginary parts for a complex positive functional.", "limitations": "The source theorem is real-valued. The compact C_c=C identification and complexification bridge must be supplied in original course prose. No separability or countability assumption on X. The long proof uses a compact-range partition and partition of unity; this is a primary formal proof, not a proofless theorem declaration. Bind the anonymous finite-measure instance by exact source range; no generated Lean identifier is asserted.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-RMK-CONTENT", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean", "source_sha256": "9D3DCE54E54BE48C202185AB2C1A121503700078254CE18C5D769EFC0B4240B2", "source_bytes": 16000, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 275, "end_line": 279, "range_sha256": "8D63107CDEDEC2FE61C2136F1FE814DFCECF5919865EEA45688C2C1101AD9805" }, { "start_line": 281, "end_line": 282, "range_sha256": "71AAA7F28085A5D813B1574ECB0946A589E2745025C548F346860325FEE990C8" }, { "start_line": 285, "end_line": 313, "range_sha256": "C025D7060D6ADEF343EB7AB078297E85CCD224FACF97C9BD3961EB18AF78D963" }, { "start_line": 324, "end_line": 324, "range_sha256": "3374A4CACA764438649B842D992FECA93D5E420B905D7FA6384528B6311F23A3" } ], "hypothesis_ranges": [ [ 42, 43 ], [ 66, 66 ], [ 319, 319 ] ], "contract": "The nonnegative functional gives a finite content on compact subsets; the regular-content proof yields its regularity; the Riesz measure is the measure induced by this content.", "limitations": "Supporting source bridge; not an independent substitute for the integral representation in Real.lean. Earlier content-additivity proofs are stored but not claimed reaudited here.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-CONTENT-REGULAR", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Measure/Content.lean", "source_sha256": "62CD59DCC7E403AEC035037D0A05730D0D83EBFD7E251E3A581B011FC6662705", "source_bytes": 20014, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 355, "end_line": 356, "range_sha256": "BA3339B6E260C71F318B356B2830F45C1392611420FB926102AA94B76044990A" }, { "start_line": 358, "end_line": 359, "range_sha256": "3A3AEDE73F321F99A731F259890FEAF9DB7A13390E4651F3F0A76659407E8152" }, { "start_line": 361, "end_line": 367, "range_sha256": "880C122FFEF231B806833323BB9F3A3F0607C2EE18E7F5AFEF829EC3FB7BAFDB" }, { "start_line": 370, "end_line": 381, "range_sha256": "782C3F832B3ADCAB862FDC0EBF259D75686F1893FE33A5916BEA64FB2BA3B3B1" } ], "hypothesis_ranges": [ [ 66, 66 ], [ 85, 92 ], [ 233, 233 ], [ 311, 311 ] ], "contract": "For a content on an R1 topological space with its Borel sigma algebra, the constructed Borel measure is outer regular and, on a weakly locally compact space, regular (finite on compacts and inner regular on open sets).", "limitations": "Use with the locally compact Hausdorff compact spectrum. Inherited hypotheses should be checked in the preserved file; no general arbitrary-topological-space regularity claim.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-REGULAR-BRIDGES", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Measure/Regular.lean", "source_sha256": "E2DA9C2A5D8874CF2397CC98C114D678FC648E74D62D96C58ED546BCACDAA6BE", "source_bytes": 64436, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 341, "end_line": 346, "range_sha256": "01CAC47B79F0EE77229290B6BD9881273F6D0001DBE5BC65A8909B33D60DE183" }, { "start_line": 582, "end_line": 603, "range_sha256": "892B1D14FA824A75D50BB6F4FBD20840B7B58CFAB4CC0FECD4F75780299C796A" }, { "start_line": 892, "end_line": 896, "range_sha256": "6E9B18B2A58F08039A9F515C52A0FC4CD8820A9490AFC1BF4B8B437324966F31" }, { "start_line": 1122, "end_line": 1123, "range_sha256": "DEDD1B411055D27918CD0AA3B4302045AB731263478D42D8B5893DAE05EE2223" } ], "hypothesis_ranges": [ [ 308, 337 ] ], "contract": "Regular measure on an R1 space is weakly regular; outer regularity plus inner compact approximation on open sets gives inner compact approximation on finite measurable sets; for finite total measure this gives inner regularity on every measurable set.", "limitations": "Regular in mathlib initially requires compact inner regularity only for open sets. Do not silently identify it with all-measurable-set inner regularity until these bridges or the finite-measure specialization are invoked.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-LP-CONTINUOUS-DENSITY", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "source_sha256": "CFC609ACEC4DDFB845C07C6628DDB6B67D6114899EF6080A560144608DDE245A", "source_bytes": 20163, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 79, "end_line": 131, "range_sha256": "5CA4002B1220754596D6336C978F8AE9C1A6D95E17F4037F102163B833F234B6" }, { "start_line": 233, "end_line": 280, "range_sha256": "26CFEE8CCCAE3CB215B19EB44427BDA6B9ED073562D66F2D36FCBAC64430B565" }, { "start_line": 321, "end_line": 330, "range_sha256": "26A1E805B003BA2E3351A24ED0751685C2AF079E9F8DD72D662B8B3EF083A502" }, { "start_line": 349, "end_line": 352, "range_sha256": "F7B9256B02BC25DA1CEF52127091EFC0AB19DAA91AB519F9C5C2AC89B9F8595D" }, { "start_line": 361, "end_line": 365, "range_sha256": "355070BE04AFDEDCD27D289A0FF02CC24B2F1B89BCD01BFC5E6961867415B3A1" } ], "hypothesis_ranges": [ [ 66, 72 ], [ 342, 345 ] ], "contract": "For a compact normal topological X with Borel sigma algebra, finite weakly regular measure, real normed vector codomain E, 1<=p=1, Lp(E,p,mu) is complete; in particular L2(C,mu) is complete. Proof takes a controlled Cauchy sequence, proves summable a.e. norm increments, constructs the pointwise limit and proves convergence in Lp.", "limitations": "No finite, sigma-finite, or separability hypothesis on mu or the domain. This is source inspection; no Lean run or full dependency-axiom audit.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-L2-INNER", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Function/L2Space.lean", "source_sha256": "AAED72005E80CDB9159E390BFB3B0B86EE6148964DC74EFA3CFCC7E8E508CCDF", "source_bytes": 13889, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 107, "end_line": 128, "range_sha256": "448B81F93119DDBB8CE31676A6FDA8C01AC6B18F9B61DAC2247F4B66B4A3A4C9" }, { "start_line": 134, "end_line": 138, "range_sha256": "6A8D73949D50DE92DBFFB2823E0336FE5D6461980284C09F7347844DD412F52B" }, { "start_line": 140, "end_line": 164, "range_sha256": "D1B58AE4A001AA18657F188C82E953FEC996178BA9FE4C117476AA88ADEECE0A" }, { "start_line": 166, "end_line": 176, "range_sha256": "3C00180E874D997B21C8E294E0287DC8F1C53D7B165C9AF2251E4E7CD4BE6093" }, { "start_line": 178, "end_line": 194, "range_sha256": "284292CFF5814F1474CEE81B729028D37111D16E50796A66F02B1FEAB32D4131" } ], "hypothesis_ranges": [ [ 100, 105 ] ], "contract": "For real or complex scalar field and any inner-product codomain E, L2 carries the integrated inner product, with its usual L2 norm. Combined with OPEN-SCALAR-LP-COMPLETE this makes scalar L2 a Hilbert space.", "limitations": "Mathlib inner product is conjugate-linear in the first variable. If the course uses linear-first notation, translate by swapping argument convention; do not identify textual formulas without that convention check.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-MCT", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Integral/Lebesgue/Add.lean", "source_sha256": "0C26250535B5B1F627534E54DF1F4B540099B82040CE1DB3B2C9BBB659D42E8E", "source_bytes": 24590, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 36, "end_line": 95, "range_sha256": "F0D311A1A24857B8E01358EE6869682C088FBDABCFFAA74431B981DDD92323A3" }, { "start_line": 98, "end_line": 112, "range_sha256": "A58136A77B0EF575096DD8E6BB31845C60B19C7EBAF679702E2AB07765EA35CE" }, { "start_line": 115, "end_line": 127, "range_sha256": "1DC354C73BD9E68F7C8B6AE68A12AA8EC4B58F75973CC1B2AA781094DDA6EF71" } ], "hypothesis_ranges": [ [ 23, 29 ] ], "contract": "For any measure and a pointwise monotone sequence of nonnegative extended-real measurable functions, integral of supremum equals supremum of integrals; also AE measurable/AE monotone version and convergence formulation.", "limitations": "Sequential result; no arbitrary-net interchange follows from this statement. Suitable for countable spectral cutoffs and pointwise truncations even when the Hilbert space has arbitrary dimension.", "previous_independent_proof_inspection_retained": true, "current_revalidation": "Exact complete source bytes and selected proof-range hashes match the existing independently inspected pinned proofs; no new Lean run or transitive axiom-cone audit claimed." }, { "id": "OPEN-SCALAR-DCT-NONNEGATIVE", "source_url": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/MeasureTheory/Integral/Lebesgue/DominatedConvergence.lean", "source_sha256": "DBE6F885DE76B2D25A6F9C398DECE3446066CBE8A422C6D6C113EF80EB7DE9FC", "source_bytes": 12969, "status": "PASS", "anonymous": true, "complete_selected_proof_ranges": [ { "start_line": 29, "end_line": 45, "range_sha256": "6743974783805C3C86C890FD2C5C833E093634FBACCD01C90F29FEA54720F156" }, { "start_line": 48, "end_line": 60, "range_sha256": "BF3AAFCDA266C52FC6B7649254126077D4DC6AE85AC8A04FD9C194703672F70C" }, { "start_line": 63, "end_line": 79, "range_sha256": "342C64F5A3E286A854E32970EE295C562AF1F2788B701028C7A8BC28503F19EF" } ], "hypothesis_ranges": [ [ 25, 27 ] ], "contract": "For any measure, a sequence of nonnegative measurable (or AE measurable) functions dominated AE by one function of finite integral and converging AE has converging integrals. Apply to squared L2 error with integrable domination to prove norm convergence of cutoffs.", "limitations": "Sequential result. Filter versions later require a countably generated filter. 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