ouroboros-pasterski-research-program / references /run_sabrina_projective_simplex_positivity_theorem.json
| { | |
| "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, | |
| "singular_counterexample_has_zero_determinant": true, | |
| "singular_counterexample_not_uniquely_localized": true, | |
| "source_blocks_match": true, | |
| "source_markers_unique": true | |
| }, | |
| "finite_exact_witnesses": [ | |
| { | |
| "D": 2, | |
| "absolute_jacobian_denominator": "3", | |
| "coordinates": [ | |
| "1/3", | |
| "1/3", | |
| "1/3" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "3", | |
| "expected_class": "interior", | |
| "n": 3, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 2, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 3, | |
| "observed_class": "boundary", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 2, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "1", | |
| "1", | |
| "-1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "-1", | |
| "expected_class": "outside", | |
| "n": 3, | |
| "observed_class": "outside", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 3, | |
| "absolute_jacobian_denominator": "4", | |
| "coordinates": [ | |
| "1/4", | |
| "1/4", | |
| "1/4", | |
| "1/4" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "4", | |
| "expected_class": "interior", | |
| "n": 4, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 3, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 4, | |
| "observed_class": "boundary", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 3, | |
| "absolute_jacobian_denominator": "2", | |
| "coordinates": [ | |
| "1/2", | |
| "1/2", | |
| "1/2", | |
| "-1/2" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "-2", | |
| "expected_class": "outside", | |
| "n": 4, | |
| "observed_class": "outside", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 4, | |
| "absolute_jacobian_denominator": "5", | |
| "coordinates": [ | |
| "1/5", | |
| "1/5", | |
| "1/5", | |
| "1/5", | |
| "1/5" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "5", | |
| "expected_class": "interior", | |
| "n": 5, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 4, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 5, | |
| "observed_class": "boundary", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 4, | |
| "absolute_jacobian_denominator": "3", | |
| "coordinates": [ | |
| "1/3", | |
| "1/3", | |
| "1/3", | |
| "1/3", | |
| "-1/3" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "-3", | |
| "expected_class": "outside", | |
| "n": 5, | |
| "observed_class": "outside", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 5, | |
| "absolute_jacobian_denominator": "6", | |
| "coordinates": [ | |
| "1/6", | |
| "1/6", | |
| "1/6", | |
| "1/6", | |
| "1/6", | |
| "1/6" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "6", | |
| "expected_class": "interior", | |
| "n": 6, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 5, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 6, | |
| "observed_class": "boundary", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 5, | |
| "absolute_jacobian_denominator": "4", | |
| "coordinates": [ | |
| "1/4", | |
| "1/4", | |
| "1/4", | |
| "1/4", | |
| "1/4", | |
| "-1/4" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "-4", | |
| "expected_class": "outside", | |
| "n": 6, | |
| "observed_class": "outside", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 6, | |
| "absolute_jacobian_denominator": "7", | |
| "coordinates": [ | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "7", | |
| "expected_class": "interior", | |
| "n": 7, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 6, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 7, | |
| "observed_class": "boundary", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 6, | |
| "absolute_jacobian_denominator": "5", | |
| "coordinates": [ | |
| "1/5", | |
| "1/5", | |
| "1/5", | |
| "1/5", | |
| "1/5", | |
| "1/5", | |
| "-1/5" | |
| ], | |
| "cramer_matches_direct": true, | |
| "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", | |
| "1/6", | |
| "1/6", | |
| "-1/6" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "-6", | |
| "expected_class": "outside", | |
| "n": 8, | |
| "observed_class": "outside", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 8, | |
| "absolute_jacobian_denominator": "9", | |
| "coordinates": [ | |
| "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": "interior", | |
| "n": 9, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 8, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 9, | |
| "observed_class": "boundary", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 8, | |
| "absolute_jacobian_denominator": "7", | |
| "coordinates": [ | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "1/7", | |
| "-1/7" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "-7", | |
| "expected_class": "outside", | |
| "n": 9, | |
| "observed_class": "outside", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 9, | |
| "absolute_jacobian_denominator": "10", | |
| "coordinates": [ | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10", | |
| "1/10" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "10", | |
| "expected_class": "interior", | |
| "n": 10, | |
| "observed_class": "interior", | |
| "orientation_invariant_absolute_determinant": true, | |
| "orientation_invariant_coordinates": true, | |
| "unit_sum": true | |
| }, | |
| { | |
| "D": 9, | |
| "absolute_jacobian_denominator": "1", | |
| "coordinates": [ | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "0", | |
| "1" | |
| ], | |
| "cramer_matches_direct": true, | |
| "determinant": "1", | |
| "expected_class": "boundary", | |
| "n": 10, | |
| "observed_class": "boundary", | |
| "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" | |
| } | |