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      "name": "Lemma 0.3 (Complex square roots).",
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      "name": "Lemma 0.2 (Quotient algebras and factorization).",
      "conditions": "Unital associative algebra, two-sided ideal, unital algebra maps. Zero quotient allowed.",
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      "name": "Lemma 0.1 (Rank–nullity).",
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      "name": "Theorem 6.2 (Poincaré–Birkhoff–Witt).",
      "conditions": "Alternating Lie algebra over any field; given totally ordered basis of arbitrary cardinality. Ordered monomials form a basis; associated graded is the symmetric algebra.",
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      "name": "Theorem 5.1.",
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      "name": "Theorem 3.3 (Weyl, algebraically closed case).",
      "conditions": "Finite-dimensional semisimple complex Lie algebra and finite-dimensional complex module.",
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      "name": "Theorem 4.1 (Weyl over an arbitrary field).",
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      "name": "Theorem 2.1.",
      "conditions": "Finite-dimensional complex sl2 modules; irreducible highest weights n are nonnegative integers.",
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      "name": "Proposition 2.2.",
      "conditions": "Polynomial Sym^n action of SL2 over every commutative ring; differential irreducibility asserted only over characteristic-zero fields.",
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      "name": "Theorem 4.1 (Whitehead).",
      "conditions": "Finite-dimensional semisimple Lie algebra and finite-dimensional module over any characteristic-zero field; H1 and H2 vanish.",
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      "name": "Proposition 2.2.",
      "conditions": "H1 identifies derivations modulo inner derivations for adjoint coefficients; combined with Theorem4.1 gives all derivations inner.",
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      "name": "Theorem 7.1 (Levi conjugacy).",
      "conditions": "Any two Levi subalgebras of a finite-dimensional Lie algebra over any characteristic-zero field are conjugate by one exp(ad a), with a in [g,rad g], hence in the nilradical; all constructions occur over the original field.",
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      "name": "Theorem 7.2 (Ado).",
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      "name": "Theorem 6.1 (Serre).",
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      "name": "Corollary 6.3.",
      "conditions": "Same finite-type presentation over Q and characteristic-zero extension fields; split semisimplicity, r+|Phi| dimensions and exact scalar extension.",
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      "name": "Theorem 9.1.",
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      "name": "Theorem 5.1 (integral root basis).",
      "conditions": "Complex semisimple Lie algebra with a chosen Cartan and base; simultaneous integral root basis with opposite normalization, all ±(p+1) constants and N(-alpha,-beta)=-N(alpha,beta). Includes the exhaustive exact rank-at-most-three certificate.",
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      "name": "Theorem 3.1.",
      "conditions": "Any complex highest weight; unique simple highest-weight quotient of the Verma module.",
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      "name": "Theorem 4.1 (highest-weight classification).",
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      "name": "Corollary 4.4.",
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      "name": "Highest-weight rational forms, Section4.5",
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    },
    {
      "lesson": "RT-LIE-15",
      "name": "Theorem 9.1.",
      "conditions": "Finite-dimensional complex semisimple Lie algebra of rank r, including products and zero: its Cartan Weyl invariant ring has r homogeneous algebraically independent polynomial generators. Formal completion proof; no general complex reflection-group CST theorem asserted.",
      "proof_locator": "theorem-9-1",
      "source": "src/RT-LIE-15.md",
      "source_sha256": "252271B819FAC2E0A9CD8A5468CFEA12B3D3593587570412098D3F81D4C1BCEE",
      "owner_source_sha256": "252271B819FAC2E0A9CD8A5468CFEA12B3D3593587570412098D3F81D4C1BCEE",
      "reader": "RT-LIE-15.html",
      "reader_sha256": "618DED0516013366802E7504CC09A85969FAE3374942BE73A844AC7B326A1DD8",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-15.html#theorem-9-1",
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    },
    {
      "lesson": "RT-LIE-17",
      "name": "Theorem 2.3.",
      "conditions": "Any two negative-Killing real forms of a finite-dimensional complex semisimple Lie algebra, any rank and components, are inner-conjugate.",
      "proof_locator": "theorem-2-3",
      "source": "src/RT-LIE-17.md",
      "source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "owner_source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "reader": "RT-LIE-17.html",
      "reader_sha256": "CCC042D7858D76427395874758339F11DEE43F6DCEF85376FE94060BB4427449",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-17.html#theorem-2-3",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-17",
      "name": "Theorem 3.4.",
      "conditions": "Compact connected finite-dimensional Lie group with semisimple real Lie algebra: finite fundamental group and compact connected simply connected cover.",
      "proof_locator": "theorem-3-4",
      "source": "src/RT-LIE-17.md",
      "source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "owner_source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "reader": "RT-LIE-17.html",
      "reader_sha256": "CCC042D7858D76427395874758339F11DEE43F6DCEF85376FE94060BB4427449",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-17.html#theorem-3-4",
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    },
    {
      "lesson": "RT-LIE-17",
      "name": "Theorem 8.1 (Cartan-involution existence).",
      "conditions": "Every real form of a finite-dimensional complex semisimple algebra has a Cartan involution; inner conjugacy reduces its conjugation to theta sigma for an involution theta of a fixed compact form.",
      "proof_locator": "theorem-8-1",
      "source": "src/RT-LIE-17.md",
      "source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "owner_source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "reader": "RT-LIE-17.html",
      "reader_sha256": "CCC042D7858D76427395874758339F11DEE43F6DCEF85376FE94060BB4427449",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-17.html#theorem-8-1",
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      "status": "proved_in_named_lesson_at_stated_scope",
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    },
    {
      "lesson": "RT-LIE-17",
      "name": "Theorem 8.2 (real forms and compact involutions).",
      "conditions": "Real forms of a fixed complex semisimple algebra, including component-exchange factors, are real-isomorphic exactly when their compact involutions are conjugate under the full compact-form automorphism group.",
      "proof_locator": "theorem-8-2",
      "source": "src/RT-LIE-17.md",
      "source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "owner_source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "reader": "RT-LIE-17.html",
      "reader_sha256": "CCC042D7858D76427395874758339F11DEE43F6DCEF85376FE94060BB4427449",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-17.html#theorem-8-2",
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    },
    {
      "lesson": "RT-LIE-17",
      "name": "Proposition 8.3.",
      "conditions": "Maximal abelian subspaces of the Cartan minus eigenspace are conjugate under the compact connected group with Lie algebra ad k; each extends by a maximal toral subalgebra of its compact centralizer to a complex Cartan.",
      "proof_locator": "proposition-8-3",
      "source": "src/RT-LIE-17.md",
      "source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "owner_source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "reader": "RT-LIE-17.html",
      "reader_sha256": "CCC042D7858D76427395874758339F11DEE43F6DCEF85376FE94060BB4427449",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-17.html#proposition-8-3",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-17",
      "name": "Theorem 8.4.",
      "conditions": "All finite-dimensional real semisimple Lie algebras, including zero, arbitrary products and complex simple algebras viewed as real: real-isomorphism classification by explicit admissible directed Satake diagrams. Full parity necessity, principal sl2 reconstruction, compact normalization and diagram equivalence are proved.",
      "proof_locator": "theorem-8-4",
      "source": "src/RT-LIE-17.md",
      "source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "owner_source_sha256": "E3B897CBFC079A4E2AE987D725311105853B92A7F9AF46F09E36255FCCBC1A39",
      "reader": "RT-LIE-17.html",
      "reader_sha256": "CCC042D7858D76427395874758339F11DEE43F6DCEF85376FE94060BB4427449",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-17.html#theorem-8-4",
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    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.1 (standard basis).",
      "conditions": "Finite Weyl group; standard Hecke basis over Z[v,v^-1] with q=v^2.",
      "proof_locator": "proposition-8-1",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-1",
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      "status": "proved_in_named_lesson_at_stated_scope",
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    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.1 (the canonical Hecke basis).",
      "conditions": "Finite Weyl group; unique bar-invariant canonical basis, integral KL polynomials and exact degree bounds. Positivity and Verma multiplicities are separate statements.",
      "proof_locator": "theorem-8-1",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-1",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.3 (finite KL inversion).",
      "conditions": "Every finite Weyl group, arbitrary rank and components: the signed unitriangular KL polynomial matrix over Z[q] has inverse Q_xy=P_(w0*y,w0*x)=P_(y*w0,x*w0). Complete trace, duality and Bruhat reversal argument; no geometric multiplicity or positivity conclusion asserted.",
      "proof_locator": "theorem-8-3",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-3",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.3 (dual-standard character, conditional on MP0–MP1).",
      "conditions": "Conditional on constructed mixed six operations, exact weight rules, ordinary finite-flag Bruhat geometry and the minimal-parabolic two-cell/affine-line package MP0–MP1: full dual-standard weighted character equals Hecke bar, with every boundary coefficient. No IC-purity assumption enters this deduction.",
      "proof_locator": "proposition-8-3",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-3",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.4 (conditional IC–KL identification).",
      "conditions": "Conditional on the explicit mixed-operation/minimal-parabolic package, normalized mixed IC, strict boundary bounds and every-stalk weight=cohomological-degree purity: canonical-basis recognition forces nonnegative even stalk degrees and identifies KL coefficients with their dimensions. The general pointwise-purity input is not proved.",
      "proof_locator": "proposition-8-4",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-4",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.5 (conditional simple-in-standard character expansion).",
      "conditions": "Conditional on the preceding actual IC stalk identity and an ordinary bounded finite-length perverse heart with constant IC simples and perverse cell standards: signed simple-in-standard Grothendieck relation and inverse-matrix standard composition coefficients, with exact transpose.",
      "proof_locator": "proposition-8-5",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-5",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.6 (conditional transfer to Verma multiplicities).",
      "conditions": "Conditional on the preceding perverse IC relation and a proved exact nonzero-simple localization dictionary Delta_x -> M((x*w0) dot lambda), I_y -> L((y*w0) dot lambda), at dominant integral lambda: full algebraic conversion to P_(w,z)(1), using the unconditional finite inversion Theorem8.3. Exact localization itself remains an unsupplied input.",
      "proof_locator": "proposition-8-6",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-6",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.6.",
      "conditions": "Finite graded modules over a finitely generated connected nonnegative real graded algebra: finite indecomposable decomposition and local degree-zero endomorphism algebra; every endomorphism of an indecomposable is a unit or nilpotent.",
      "proof_locator": "lemma-8-6",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-6",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.7.",
      "conditions": "At the same graded finite scope: uniqueness of indecomposable decompositions and actual direct-sum cancellation.",
      "proof_locator": "lemma-8-7",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-7",
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      "status": "proved_in_named_lesson_at_stated_scope",
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    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.8.",
      "conditions": "For the finite-Weyl polynomial ring with linear forms in degree two: every finite graded projective summand of a graded free module is graded free; complete graded Nakayama proof.",
      "proof_locator": "lemma-8-8",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-8",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.9.",
      "conditions": "Every finite Weyl group, any rank and components: homogeneous reflection-hyperplane localization of each actual Bott–Samelson object has only the specified lower graph and two-branch summands; both translation cases expanded.",
      "proof_locator": "lemma-8-9",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-9",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.4.",
      "conditions": "Every finite Weyl group: whole graded Hom from an actual descending graph-flag bimodule to a shifted Bott–Samelson sum is right graded free with the exact support-character pairing formula.",
      "proof_locator": "theorem-8-4",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-4",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.5.",
      "conditions": "Every object of the actual finite-Weyl Bott–Samelson idempotent closure: both primitive support maps have exact image multiplication by the product of right inversion roots; local divisibility and degreewise global equality are proved.",
      "proof_locator": "theorem-8-5",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-5",
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      "status": "proved_in_named_lesson_at_stated_scope",
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    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.6.",
      "conditions": "Every finite Weyl group: the exact right-free graded Hom formula holds for every Bott–Samelson summand, with actual extension lifting proved degree by degree.",
      "proof_locator": "theorem-8-6",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-6",
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    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.7.",
      "conditions": "Every finite Weyl group, including rank zero and products: one uniquely normalized indecomposable B_x for each x, all indecomposables its shifts, and B_x self-dual. Actual summands are constructed without a canonical-character assumption. Canonical characters and KL positivity are subsequently proved in Theorem8.10; the finite complex regular integral category-O transfer is proved in Theorem8.13.",
      "proof_locator": "theorem-8-7",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-7",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.10 (primitive decomposition and orthogonality).",
      "conditions": "Finite-dimensional graded real space with a degree-two hard-Lefschetz operator: primitive decomposition and orthogonality of the exact Lefschetz forms.",
      "proof_locator": "lemma-8-10",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-10",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.11 (continuous deformation).",
      "conditions": "Fixed nondegenerate symmetric graded form, continuous self-adjoint operators with hard Lefschetz throughout a connected interval: primitive Hodge signatures persist.",
      "proof_locator": "lemma-8-11",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-11",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.12 (balanced invariant subspaces).",
      "conditions": "Balanced graded stable subspace of a hard-Lefschetz Hodge space: hard Lefschetz, restricted-form nondegeneracy and inherited signs.",
      "proof_locator": "lemma-8-12",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-12",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.13 (an injective-map substitute for weak Lefschetz).",
      "conditions": "Degree-one intertwining injection in negative degrees with the exact form identity and Hodge target: Lefschetz injectivity on the source, even when its form is degenerate.",
      "proof_locator": "lemma-8-13",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-13",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.14 (vanishing for a displaced centre).",
      "conditions": "Graded self-adjoint symmetric form on a hard-Lefschetz space centered strictly below zero: the entire form vanishes, including pairings between unequal chains.",
      "proof_locator": "lemma-8-14",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-14",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.15 (a large-parameter rank-one extension).",
      "conditions": "Actual rank-one extension basis, square-zero middle operator and the stated mixed form identities: hard Lefschetz and standard Hodge signs for all sufficiently large positive parameters.",
      "proof_locator": "lemma-8-15",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-15",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.16 (singular classification).",
      "conditions": "Finite Weyl group and one right simple invariant ring: classification of the actual summand category of restricted regular bimodules, at the specified top-coset normalization.",
      "proof_locator": "lemma-8-16",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-16",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.8 (support-layer homotopy splitting).",
      "conditions": "Every chosen reduced expression in a finite Weyl group: actual standard and costandard support-layer complexes are homotopy equivalent to their single normalized graph in degree zero or a contractible complex. No braid-coherence theorem assumed.",
      "proof_locator": "theorem-8-8",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-8",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.9 (actual right-singular descent).",
      "conditions": "Every finite Weyl group and right-simple coset: induction of its normalized singular indecomposable is the regular indecomposable of the maximal member, with no extra shift. The controlled middle-action basis is retained.",
      "proof_locator": "theorem-8-9",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-9",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.10 (finite-Weyl Hodge theory and KL positivity).",
      "conditions": "Every finite Weyl group, including products and rank zero, over its real reflection realization: indecomposable canonical characters, hard Lefschetz, bottom-positive Hodge signs, nonnegative integer KL coefficients and nonnegative Laurent structure constants. The finite complex regular integral category-O Verma multiplicity transfer is proved in Theorem8.13.",
      "proof_locator": "theorem-8-10",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-10",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.17 (dominant orbit inequality).",
      "conditions": "Finite Weyl group, strictly dominant a and weakly dominant b: orbit pairing inequality and exact equality condition, including closed-chamber uniqueness and the lowest dot-orbit comparison.",
      "proof_locator": "lemma-8-17",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-17",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.18 (coatom containing a subword).",
      "conditions": "Finite Weyl reduced-subword order: every proper interval below w contains a coatom of w above its lower endpoint, proved by lifting and induction.",
      "proof_locator": "lemma-8-18",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-18",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.19 (extremal norm).",
      "conditions": "Finite simple highest-weight module: every weight has norm at most that of an extremal weight, with equality exactly on its Weyl orbit and multiplicity one.",
      "proof_locator": "lemma-8-19",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-19",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Lemma 8.20 (simple-root integrability).",
      "conditions": "Finite complex regular integral category O: each non-anti-dominant simple is integrable for some simple-root sl2; negative simple-root operators are injective on actual Verma-flag objects.",
      "proof_locator": "lemma-8-20",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#lemma-8-20",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.11 (actual simple-wall translations).",
      "conditions": "Finite-dimensional complex semisimple Lie algebra, dominant integral lambda and one simple wall: actual exact translations have the specified one- and two-factor Verma flags and both adjunctions, proved in Section8.12.",
      "proof_locator": "theorem-8-11",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-11",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.12 (big-projective wall effects).",
      "conditions": "At that finite ordinary simple-wall scope: translation out sends the singular anti-dominant big projective to the regular one; translation on gives two copies, so wall crossing gives two copies and preserves the structure-functor kernel.",
      "proof_locator": "theorem-8-12",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-12",
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      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.7 (free-orbit invariant jets).",
      "conditions": "Characteristic-zero field, finite linear group and a rational point with trivial stabilizer: invariant polynomials realize every finite jet. Every finite connected homogeneous quotient admits the resulting exact translated invariant surjection.",
      "proof_locator": "proposition-8-7",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-7",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Proposition 8.8 (ambient-root projective order).",
      "conditions": "Finite complex regular integral category O, every ambient positive-root hyperplane and the specified deformed big-projective family: nonsplit root-local two-Verma residue and exact first-order endomorphism order. Its residue projectivity is proved in Section8.15.6; Sections8.14.2–8.14.7 prove the uniform Shapovalov determinant and transverse pairing.",
      "proof_locator": "proposition-8-8",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#proposition-8-8",
      "licence_expression": "CC0-1.0",
      "status": "proved_in_named_lesson_at_stated_scope",
      "self_check": "Writing AI and bounded internal mathematical checks; no human review claimed."
    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.13 (regular integral category-O comparison and KL multiplicities).",
      "conditions": "Finite-dimensional complex semisimple Lie algebra and dominant integral lambda, including products and rank zero: actual coinvariant corner and central annihilator, full faithfulness with projective target, restriction/induction wall action, augmentation Hom and inverse projective label. The proved canonical-character theorem yields [M(w dot lambda):L(z dot lambda)]=P_(w,z)(1).",
      "proof_locator": "theorem-8-13",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-13",
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      "status": "proved_in_named_lesson_at_stated_scope",
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    },
    {
      "lesson": "RT-LIE-18",
      "name": "Theorem 8.2 (Kazhdan–Lusztig multiplicities).",
      "conditions": "Same finite complex regular integral category-O scope: general dominant-convention KL Verma multiplicities are proved in Theorem8.13; coefficient positivity is proved by the finite-Weyl Hodge theorem8.10.",
      "proof_locator": "theorem-8-2",
      "source": "src/RT-LIE-18.md",
      "source_sha256": "d30c592a2ff8551e1a8c17457dd34b3c62e06a7d7416202eaf1bd90eeddc7278",
      "owner_source_sha256": "9EE4CF2736C3B9E1D5B018EC13863127CC957648C5AEC4B33F671EB933D85897",
      "reader": "RT-LIE-18.html",
      "reader_sha256": "e74b69a565de64b147f97127095b00a222d04e473b823d6556ce96193f24a826",
      "internal_url": "https://kokunoyumeto.github.io/open-math-courses/courses/RT-LIE/RT-LIE-18.html#theorem-8-2",
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      "status": "proved_in_named_lesson_at_stated_scope",
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}