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      "proof_closure": "closed",
      "loci": [
        "tools Theorem1.1; Section1 interval-cover example",
        "Haar Theorem2.2 outer-measure construction"
      ],
      "trace": "Finite splitting, disjointization, countable union and completeness proved internally. Lebesgue interval length uses already proved compactness; Radon bridge added and checked."
    },
    {
      "id": "HAAR.convergence-Lp",
      "proof_closure": "closed",
      "loci": [
        "tools Theorems2.1–2.2,3.1–3.2",
        "Haar Propositions2.3,3.1; Theorem6.2"
      ],
      "trace": "Simple approximation and countable-additivity continuity precede monotone/Fatou/dominated convergence. Holder/Minkowski and summable subsequence completeness precede Hilbert RN. No measure/Hilbert circle found."
    },
    {
      "id": "HAAR.Radon-Nikodym",
      "proof_closure": "closed",
      "free_provenance": "tools proof closed; revised Haar13.3(5) finite-block own construction freshly checked",
      "loci": [
        "tools Theorem4.1",
        "Haar Theorem13.3(5)"
      ],
      "trace": "Finite Hilbert representation followed by sigma-finite partition gluing is proved internally. The current restricted Radon density is constructed explicitly from finite disjoint compact-block densities and nonnegative summation."
    },
    {
      "id": "HAAR.L1-dual",
      "proof_closure": "closed",
      "loci": [
        "tools Theorem4.2",
        "Haar Theorem13.3(1–4)",
        "Hilbert dependency Theorem9.1"
      ],
      "trace": "Finite bounded functionals define dominated measures and use the already proved Radon–Nikodym theorem. Sigma-finite gluing and local coset gluing are fully supplied, using the proved coset outer envelope. The commutant proof supplies its cyclic/local argument and countable-coset support."
    },
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      "proof_closure": "closed",
      "loci": [
        "tools Theorem5.1",
        "Hilbert dependency Sections1–4,8; Lemma5.0",
        "Haar tensor Theorem6.3"
      ],
      "trace": "Arbitrary orthonormal-basis coefficient sums, Parseval, finite-rank density, tensor identification and compactness are proved; product-function density supplied by topology SW and Riesz product proofs."
    },
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      "id": "HAAR.Riesz-Haar",
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      "trace": "Riesz remains a full own derivation from the internally proved measure and topology chain. Existence Theorem8.3/Lemma8.4 now uses the verified free compact-cover method, with Borel measurability and the full Radon conversion proved internally. No weighted-function template remains."
    },
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      "id": "HAAR.Weil",
      "proof_closure": "closed",
      "free_provenance": "positive-lift and invariant quotient construction explicitly adapted from verified free443P/Q/R; compact lifting now deduced without circularity; integrated revisions checked",
      "loci": [
        "quotient Lemmas1.1–1.2,2.1; Theorem3.1; Lemma4.1; Theorems5.1–5.2,6.1–6.3"
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      "trace": "LCH quotient topology, compact support of averaging, local finite orbit-shell cutoff, rho covariance, compact weighted identity, Radon representation and cocycle signs are proved. Free443P–R supplies the explicitly marked averaging and order-functional quotient methods. Compact lifting is deduced without circularity."
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        "quotient upper-half-plane example(20) and rotation calculation"
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