{ "kernel": "run_sabrina_projective_simplex_positivity_theorem", "role": "post-computation assertion only", "schema": "ouroboros_scientific_reference_assertion_v1", "scientific_payload": { "claim_boundary": { "capabilities_removed": [], "invertibility_required": true, "rank_deficient_unique_localization_claimed": false, "rectangular_jacobian_claimed": false, "source_error_claimed": false, "square_n_equals_D_plus_1_only": true, "witness_matrices_claimed_as_physical_null_momenta": false }, "classification": "square_projective_simplex_signed_minor_support_and_jacobian_theorem_exact", "exact_checks": { "all_27_witnesses_classify_correctly": true, "all_coordinates_have_unit_sum": true, "all_cramer_coordinates_match_direct_solve": true, "archive_hash_matches": true, "finite_witness_grid_has_27_rows": true, "member_hash_matches": true, "orientation_reversal_preserves_absolute_jacobian": true, "orientation_reversal_preserves_all_coordinates": true, "predecessor_manifest_hash_matches": true, 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"determinant": "-5", "expected_class": "outside", "n": 7, "observed_class": "outside", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 7, "absolute_jacobian_denominator": "8", "coordinates": [ "1/8", "1/8", "1/8", "1/8", "1/8", "1/8", "1/8", "1/8" ], "cramer_matches_direct": true, "determinant": "8", "expected_class": "interior", "n": 8, "observed_class": "interior", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 7, "absolute_jacobian_denominator": "1", "coordinates": [ "0", "0", "0", "0", "0", "0", "0", "1" ], "cramer_matches_direct": true, "determinant": "1", "expected_class": "boundary", "n": 8, "observed_class": "boundary", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 7, "absolute_jacobian_denominator": "6", "coordinates": [ "1/6", "1/6", "1/6", "1/6", "1/6", 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"orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 9, "absolute_jacobian_denominator": "8", "coordinates": [ "1/8", "1/8", "1/8", "1/8", "1/8", "1/8", "1/8", "1/8", "1/8", "-1/8" ], "cramer_matches_direct": true, "determinant": "-8", "expected_class": "outside", "n": 10, "observed_class": "outside", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 10, "absolute_jacobian_denominator": "11", "coordinates": [ "1/11", "1/11", "1/11", "1/11", "1/11", "1/11", "1/11", "1/11", "1/11", "1/11", "1/11" ], "cramer_matches_direct": true, "determinant": "11", "expected_class": "interior", "n": 11, "observed_class": "interior", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 10, "absolute_jacobian_denominator": "1", "coordinates": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1" ], "cramer_matches_direct": true, "determinant": "1", "expected_class": "boundary", "n": 11, "observed_class": "boundary", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true }, { "D": 10, "absolute_jacobian_denominator": "9", "coordinates": [ "1/9", "1/9", "1/9", "1/9", "1/9", "1/9", "1/9", "1/9", "1/9", "1/9", "-1/9" ], "cramer_matches_direct": true, "determinant": "-9", "expected_class": "outside", "n": 11, "observed_class": "outside", "orientation_invariant_absolute_determinant": true, "orientation_invariant_coordinates": true, "unit_sum": true } ], "paper": { "arxiv_id": "1706.03917", "title": "Gluon Amplitudes as 2d Conformal Correlators" }, "result_version": "sabrina_projective_simplex_positivity_theorem_v1", "singular_counterexample": { "D": 4, "determinant": "0", "n": 5, "rank": 4 }, "source_evidence": { "archive_sha256": "af2756072965c969be2a74c492fe25a797fe9f85e24d61605cf088e355d06244", "block_hashes": { "delta_product_rewrite": "8a2029c1a1fe7989bbb77b99449ca37fd43965e0b9169abe60c01d969c5f6f20", "indicator": "d7e7be03a59b4e0c7702a241e5024b8fc57226ce0ac48928c307baa567ce18a0", "simplex_result": "24194f455371e8599a13a472c41489b7a6ccdbfa6911705547e0adb3ec882a9e" }, "checks": { "archive_hash_matches": true, "member_hash_matches": true, "predecessor_manifest_hash_matches": true, "source_blocks_match": true, "source_markers_unique": true }, "member_sha256": "88253eb755b1b2418d0ac3480c1d4b67d9bb3985f8f6f6c3b0f79c85784f4bfe" }, "status": "complete", "theorem": { "boundary": "all ratios are nonnegative and at least one is zero", "coordinates": "sigma_i=det(M_i)/det(M), where M_i replaces column i by b=(0,...,0,1)", "distributional_jacobian": "delta(M sigma-b)=delta(sigma-sigma*)/abs(det M)", "interior": "all signed determinant ratios are strictly positive", "orientation_invariance": "common sign reversal of det(M) and det(M_i) leaves ratios and abs(det M) unchanged", "outside": "at least one ratio is negative", "scope": "square invertible augmented matrix M with n=D+1", "unit_sum": "the final all-ones row enforces sum_i sigma_i=1" } }, "scientific_payload_sha256": "04b196f77ef38f277dfbb7ebeccf2b1964f90157335f9a1dc268c6654d548af4" }