{
  "kernel": "run_sabrina_projective_simplex_rank_theorem",
  "role": "post-computation assertion only",
  "schema": "ouroboros_scientific_reference_assertion_v1",
  "scientific_payload": {
    "claim_boundary": {
      "capabilities_removed": [],
      "compatibility_required": true,
      "integer_witness_matrices_claimed_as_physical_null_momenta": false,
      "non_square_distributional_jacobian_derived": false,
      "rank_deficiency_allowed_but_not_unique": true,
      "simplex_positivity_required": true,
      "source_error_claimed": false
    },
    "classification": "all_D_projective_simplex_localization_rank_theorem_exact_with_support_and_positivity_boundaries",
    "counterexamples": {
      "incompatible": {
        "augmented_rank": 2,
        "matrix_rank": 1
      },
      "rank_deficient_compatible": {
        "nullity": 1,
        "rank": 2
      },
      "unique_but_not_simplex_positive": {
        "solution": [
          "1",
          "1",
          "-1"
        ]
      }
    },
    "exact_checks": {
      "D4_n2_through_n5_have_zero_affine_moduli": true,
      "D4_n3_has_two_external_compatibility_constraints": true,
      "D4_n4_has_one_external_compatibility_constraint": true,
      "D4_n5_is_square_unique_on_support": true,
      "D4_n6_has_one_affine_modulus": true,
      "all_99_integer_witnesses_have_expected_rank": true,
      "archive_hash_matches": true,
      "finite_grid_has_99_points": true,
      "incompatible_example_fails_rank_test": true,
      "localization_statement_hash_matches": true,
      "localization_statement_unique": true,
      "member_hash_matches": true,
      "predecessor_manifest_hash_matches": true,
      "rank_deficient_example_has_one_modulus": true,
      "rank_deficient_example_is_compatible": true,
      "source_equation_blocks_match": true,
      "source_equation_markers_unique": true,
      "unique_solution_can_fail_simplex_positivity": true
    },
    "finite_exact_grid": [
      {
        "D": 2,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 2,
        "external_compatibility_codimension": 1,
        "full_rank": true,
        "n": 2,
        "rank": 2
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 3,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 1,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 4,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 2,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 5,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 3,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 6,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 4,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 7,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 5,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 8,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 6,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 9,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 7,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 10,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 8,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 11,
        "rank": 3
      },
      {
        "D": 2,
        "affine_solution_dimension_on_support": 9,
        "expected_rank": 3,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 12,
        "rank": 3
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 2,
        "external_compatibility_codimension": 2,
        "full_rank": true,
        "n": 2,
        "rank": 2
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 3,
        "external_compatibility_codimension": 1,
        "full_rank": true,
        "n": 3,
        "rank": 3
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 4,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 1,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 5,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 2,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 6,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 3,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 7,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 4,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 8,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 5,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 9,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 6,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 10,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 7,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 11,
        "rank": 4
      },
      {
        "D": 3,
        "affine_solution_dimension_on_support": 8,
        "expected_rank": 4,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 12,
        "rank": 4
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 0,
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        "external_compatibility_codimension": 3,
        "full_rank": true,
        "n": 2,
        "rank": 2
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 0,
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        "external_compatibility_codimension": 2,
        "full_rank": true,
        "n": 3,
        "rank": 3
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 0,
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        "external_compatibility_codimension": 1,
        "full_rank": true,
        "n": 4,
        "rank": 4
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 0,
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        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 5,
        "rank": 5
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 1,
        "expected_rank": 5,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 6,
        "rank": 5
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 2,
        "expected_rank": 5,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 7,
        "rank": 5
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 3,
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        "external_compatibility_codimension": 0,
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        "n": 8,
        "rank": 5
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 4,
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        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 9,
        "rank": 5
      },
      {
        "D": 4,
        "affine_solution_dimension_on_support": 5,
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        "external_compatibility_codimension": 0,
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        "n": 10,
        "rank": 5
      },
      {
        "D": 4,
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      },
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      },
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        "D": 5,
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        "n": 2,
        "rank": 2
      },
      {
        "D": 5,
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        "n": 3,
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      },
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        "rank": 6
      },
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        "n": 7,
        "rank": 6
      },
      {
        "D": 5,
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        "n": 8,
        "rank": 6
      },
      {
        "D": 5,
        "affine_solution_dimension_on_support": 3,
        "expected_rank": 6,
        "external_compatibility_codimension": 0,
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        "n": 9,
        "rank": 6
      },
      {
        "D": 5,
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        "n": 10,
        "rank": 6
      },
      {
        "D": 5,
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        "n": 11,
        "rank": 6
      },
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        "n": 12,
        "rank": 6
      },
      {
        "D": 6,
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        "n": 2,
        "rank": 2
      },
      {
        "D": 6,
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        "rank": 3
      },
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        "rank": 4
      },
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      },
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        "rank": 6
      },
      {
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        "n": 7,
        "rank": 7
      },
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      },
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        "n": 9,
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      },
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        "n": 10,
        "rank": 7
      },
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        "D": 6,
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        "n": 11,
        "rank": 7
      },
      {
        "D": 6,
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        "n": 12,
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      },
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        "D": 7,
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        "rank": 2
      },
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      },
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      },
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      },
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      },
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      },
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      },
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        "full_rank": true,
        "n": 10,
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      },
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      },
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      },
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        "D": 8,
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        "rank": 2
      },
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        "D": 8,
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        "n": 3,
        "rank": 3
      },
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        "D": 8,
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      },
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      },
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      },
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      },
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      },
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      },
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      },
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        "rank": 9
      },
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        "rank": 9
      },
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        "D": 9,
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      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 5,
        "external_compatibility_codimension": 6,
        "full_rank": true,
        "n": 5,
        "rank": 5
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 6,
        "external_compatibility_codimension": 5,
        "full_rank": true,
        "n": 6,
        "rank": 6
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 7,
        "external_compatibility_codimension": 4,
        "full_rank": true,
        "n": 7,
        "rank": 7
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 8,
        "external_compatibility_codimension": 3,
        "full_rank": true,
        "n": 8,
        "rank": 8
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 9,
        "external_compatibility_codimension": 2,
        "full_rank": true,
        "n": 9,
        "rank": 9
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 10,
        "external_compatibility_codimension": 1,
        "full_rank": true,
        "n": 10,
        "rank": 10
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 0,
        "expected_rank": 11,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 11,
        "rank": 11
      },
      {
        "D": 10,
        "affine_solution_dimension_on_support": 1,
        "expected_rank": 11,
        "external_compatibility_codimension": 0,
        "full_rank": true,
        "n": 12,
        "rank": 11
      }
    ],
    "paper": {
      "arxiv_id": "1706.03917",
      "title": "Gluon Amplitudes as 2d Conformal Correlators"
    },
    "result_version": "sabrina_projective_simplex_rank_theorem_v1",
    "source_evidence": {
      "archive_sha256": "af2756072965c969be2a74c492fe25a797fe9f85e24d61605cf088e355d06244",
      "block_hashes": {
        "delta_product_rewrite": "8a2029c1a1fe7989bbb77b99449ca37fd43965e0b9169abe60c01d969c5f6f20",
        "simplex_result": "24194f455371e8599a13a472c41489b7a6ccdbfa6911705547e0adb3ec882a9e"
      },
      "checks": {
        "archive_hash_matches": true,
        "localization_statement_hash_matches": true,
        "localization_statement_unique": true,
        "member_hash_matches": true,
        "predecessor_manifest_hash_matches": true,
        "source_equation_blocks_match": true,
        "source_equation_markers_unique": true
      },
      "localization_line_sha256": "56cabb68b24099425e4158af718ec6b892423facfa4910a1878990f84cc769e6",
      "member_sha256": "88253eb755b1b2418d0ac3480c1d4b67d9bb3985f8f6f6c3b0f79c85784f4bfe"
    },
    "status": "complete",
    "theorem": {
      "constraint_matrix_shape": "(D+1) by n",
      "external_compatibility_codimension": "D+1-r",
      "generic_n_above_D_plus_1": "n-D-1 unfixed simplex moduli",
      "generic_n_at_most_D_plus_1": "unique sigma on compatible support; D+1-n residual external constraints",
      "generic_rank": "r=min(D+1,n)",
      "on_support_affine_solution_dimension": "n-r",
      "square_distributional_jacobian": "for n=D+1 and det(M)!=0 only: delta(M sigma-b)=delta(sigma-sigma*)/abs(det(M))"
    }
  },
  "scientific_payload_sha256": "a4be65bfe6724693b679405e1985b33302d94ec8113edf75512620fcc58045f7"
}