{
  "schema": "programme-proof-companion/v1",
  "updated_at_utc": "2026-10-05T20:50:05.281728+00:00",
  "title": "Proof companion: bounded topology and faithful states",
  "author_credit": "Original OA-MOD programme exposition; integration and checking by GPT-6 Astra (OpenAI), Ultra",
  "original_text_license": "CC0-1.0",
  "source_expression_from_human_works_reproduced": false,
  "source_sha256": "1e7c048275f6aa5cb13ea5723c8371b1607beb8e2c19a69cd192cfa7c04560a0",
  "scope": "Complete selected earlier proof passages for bounded faithful-normal topology and faithful states. Full course/general-weight/modular branches not certified.",
  "providers": [
    {
      "id": "real_Hilbert",
      "programme_source": "OA-MOD/audit/real-coercivity/real-coercive-equations.md",
      "original_source_sha256": "790f112803a3586d76218f6892f4565f48f903b99b57270f74e7c3bf69746844",
      "passages": [
        {
          "first_line": 13,
          "last_line": 45,
          "raw_excerpt_sha256": "2de22dffc64d7edc3d5d51ba27ff1f0d4b459d40454989205a045eaa223d6ff8",
          "actual_excerpt_sha256": "2de22dffc64d7edc3d5d51ba27ff1f0d4b459d40454989205a045eaa223d6ff8"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "NP1_2",
      "programme_source": "OA-MOD/audit/real-coercivity/predual-prerequisites.md",
      "original_source_sha256": "2c6775777bad72f792b47694c8aeb03726d94ce0bf8a58428ad6d421a597bc62",
      "passages": [
        {
          "first_line": 3,
          "last_line": 66,
          "raw_excerpt_sha256": "d0469ec2c7708cd259f1e7c3614bd09dbba41f8c87fd4ebbfc540f1a0a92c8f6",
          "actual_excerpt_sha256": "d0469ec2c7708cd259f1e7c3614bd09dbba41f8c87fd4ebbfc540f1a0a92c8f6"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "AB04_05",
      "programme_source": "OA-MOD/public/src/banach-second-adjoint.md",
      "original_source_sha256": "ea4c5c099f3c7b50790db1fd40d2d6bcae2a9f56d45bc40c68cb70b4e647e3ce",
      "passages": [
        {
          "first_line": 62,
          "last_line": 98,
          "raw_excerpt_sha256": "4dd484229ebc5ea287f0fc22b5746148037186256e1f35d2bf123e07b6c13569",
          "actual_excerpt_sha256": "4dd484229ebc5ea287f0fc22b5746148037186256e1f35d2bf123e07b6c13569"
        },
        {
          "first_line": 128,
          "last_line": 145,
          "raw_excerpt_sha256": "5854a3f8c9ba1525cd1257a397e8f72299f7b287fe552b7dcc86fd69e00aede5",
          "actual_excerpt_sha256": "5854a3f8c9ba1525cd1257a397e8f72299f7b287fe552b7dcc86fd69e00aede5"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "GP0",
      "programme_source": "OA-MOD/audit/real-coercivity/bounded-right-multiplier.md",
      "original_source_sha256": "eb5def25e5669834f4c1af82a6f6edfab0827451a23964ab35a55f32af31906d",
      "passages": [
        {
          "first_line": 9,
          "last_line": 53,
          "raw_excerpt_sha256": "2df908b1ec0e75a0d674083359dd4d5e9e574e8887536979fe915dc932009c78",
          "actual_excerpt_sha256": "2df908b1ec0e75a0d674083359dd4d5e9e574e8887536979fe915dc932009c78"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "BK01_03",
      "programme_source": "OA-MOD/public/src/bounded-operator-kernel.md",
      "original_source_sha256": "5f63081c17c756c935f7f4d62bdf16537b0ad75d8bbc7f9d087e5669d8ab580b",
      "passages": [
        {
          "first_line": 9,
          "last_line": 81,
          "raw_excerpt_sha256": "935c30a71a4b3338c89b55c9f3f87a80be254ab9e545d03062d723c75a86dc19",
          "actual_excerpt_sha256": "935c30a71a4b3338c89b55c9f3f87a80be254ab9e545d03062d723c75a86dc19"
        },
        {
          "first_line": 87,
          "last_line": 136,
          "raw_excerpt_sha256": "85c19ae2071f2f266c5580b55550cb71faf7ba2f61536396643148a119eb5ad6",
          "actual_excerpt_sha256": "85c19ae2071f2f266c5580b55550cb71faf7ba2f61536396643148a119eb5ad6"
        },
        {
          "first_line": 137,
          "last_line": 160,
          "owner_added_complete_BK04": true,
          "actual_excerpt_sha256": "bcf21f3ad810a21b2444200d6d507f5f4adf1f572f783e9a044a936933463cd2"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "CP01_07",
      "programme_source": "OA-MOD/public/src/concrete-preduals.md",
      "original_source_sha256": "cc473f9a648a99cc2ad88952f9a8ec8e014cdc0874e499187839447ae76240c1",
      "passages": [
        {
          "first_line": 8,
          "last_line": 247,
          "raw_excerpt_sha256": "be787cf7666fdf7c2ba8ec44fac00f8c57570609c012e286b6385b0c2f8129b7",
          "actual_excerpt_sha256": "be787cf7666fdf7c2ba8ec44fac00f8c57570609c012e286b6385b0c2f8129b7"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "HA03",
      "programme_source": "OA-MOD/public/src/hilbert-algebra-kernel.md",
      "original_source_sha256": "e1fcc4075dbc55b7a6e7e3c5dd68de003ae12d6549f2300501b754d86ba07ff1",
      "passages": [
        {
          "first_line": 102,
          "last_line": 125,
          "raw_excerpt_sha256": "faf2c18c475c47ab3e629801794cc27cd63431caca4c7fc726169d9b8b225f86",
          "actual_excerpt_sha256": "faf2c18c475c47ab3e629801794cc27cd63431caca4c7fc726169d9b8b225f86"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    },
    {
      "id": "HAP04_05",
      "programme_source": "OA-MOD/public/src/hilbert-algebra-approximation.md",
      "original_source_sha256": "0e5654cef1fecfd9e5efc4a8a29caaab61311f8385f58c1c43242aae1434b003",
      "passages": [
        {
          "first_line": 127,
          "last_line": 220,
          "raw_excerpt_sha256": "8fd5c20826d67ec8eb713c86f6d7735eecc53d5f43243da58ef60887ba52dcd6",
          "actual_excerpt_sha256": "8fd5c20826d67ec8eb713c86f6d7735eecc53d5f43243da58ef60887ba52dcd6"
        }
      ],
      "public_text_transformations": [
        "local hyperlinks resolved to supplied companion sections",
        "standalone HTML anchor markers replaced by ordinary heading anchors"
      ],
      "statements_and_proof_math_unchanged": true
    }
  ],
  "remote_route_note": "The exact needed source versions were not all present in the current private GitHub edition; these attributed passages accompany the consumer so its actual proofs are available.",
  "unformatted_companion_sha256": "7a9eacd4c9a6ccb209abd2845ab28807fe07e8ebc77a3a962d80f9fd57327101",
  "formatting": "Eighteen displays reflowed with every mathematical token, inline expression, prose, heading, link and tag preserved."
}
