| { |
| "status": "PASS", |
| "exact_reuse_verification": { |
| "status": "PASS", |
| "arithmetic": "fractions.Fraction exclusively", |
| "period_convention": "All exact fixture angles are measured in units of pi; T=1/2 therefore means physical period pi/2.", |
| "lemma_checked": "If each measurable A_j has measure <=2psi, then integral (sum 1_A_j -1)_+ <=2psi*(N-1)_+.", |
| "proof": "For N>=1 delete any one index j0: (sum 1_A_j -1)_+ <= sum_{j!=j0}1_A_j pointwise. Integrate and bound each remaining set's measure by 2psi. N=0 is immediate.", |
| "counts": { |
| "random_families": 600, |
| "sharp_identical_families": 24, |
| "exhaustive_pointwise_fixtures": 18434 |
| }, |
| "strict_random_cases": 549, |
| "smallest_random_slack": "0", |
| "sharpness_cases": [ |
| { |
| "psi_in_pi_units": "1/16", |
| "points": 0, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "1/16", |
| "points": 1, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "1/16", |
| "points": 2, |
| "integral_in_pi_units": "1/8" |
| }, |
| { |
| "psi_in_pi_units": "1/16", |
| "points": 3, |
| "integral_in_pi_units": "1/4" |
| }, |
| { |
| "psi_in_pi_units": "1/16", |
| "points": 8, |
| "integral_in_pi_units": "7/8" |
| }, |
| { |
| "psi_in_pi_units": "1/16", |
| "points": 20, |
| "integral_in_pi_units": "19/8" |
| }, |
| { |
| "psi_in_pi_units": "1/12", |
| "points": 0, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "1/12", |
| "points": 1, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "1/12", |
| "points": 2, |
| "integral_in_pi_units": "1/6" |
| }, |
| { |
| "psi_in_pi_units": "1/12", |
| "points": 3, |
| "integral_in_pi_units": "1/3" |
| }, |
| { |
| "psi_in_pi_units": "1/12", |
| "points": 8, |
| "integral_in_pi_units": "7/6" |
| }, |
| { |
| "psi_in_pi_units": "1/12", |
| "points": 20, |
| "integral_in_pi_units": "19/6" |
| }, |
| { |
| "psi_in_pi_units": "3/32", |
| "points": 0, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "3/32", |
| "points": 1, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "3/32", |
| "points": 2, |
| "integral_in_pi_units": "3/16" |
| }, |
| { |
| "psi_in_pi_units": "3/32", |
| "points": 3, |
| "integral_in_pi_units": "3/8" |
| }, |
| { |
| "psi_in_pi_units": "3/32", |
| "points": 8, |
| "integral_in_pi_units": "21/16" |
| }, |
| { |
| "psi_in_pi_units": "3/32", |
| "points": 20, |
| "integral_in_pi_units": "57/16" |
| }, |
| { |
| "psi_in_pi_units": "1/10", |
| "points": 0, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "1/10", |
| "points": 1, |
| "integral_in_pi_units": "0" |
| }, |
| { |
| "psi_in_pi_units": "1/10", |
| "points": 2, |
| "integral_in_pi_units": "1/5" |
| }, |
| { |
| "psi_in_pi_units": "1/10", |
| "points": 3, |
| "integral_in_pi_units": "2/5" |
| }, |
| { |
| "psi_in_pi_units": "1/10", |
| "points": 8, |
| "integral_in_pi_units": "7/5" |
| }, |
| { |
| "psi_in_pi_units": "1/10", |
| "points": 20, |
| "integral_in_pi_units": "19/5" |
| } |
| ], |
| "constant_and_half_factor_fixtures": [ |
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| "T_in_pi_units": "1/2", |
| "psi_in_pi_units": "1/16", |
| "point_membership_bound_2psi_in_pi_units": "1/8", |
| "local_redundancy_is_twice_reward": "2/7", |
| "global_redundancy_lower_bound": "8/7", |
| "disk_excess_is_half_global_redundancy": "4/7", |
| "amplification_T_over_2psi_equals_pi_over_4psi": "4" |
| }, |
| { |
| "T_in_pi_units": "1/2", |
| "psi_in_pi_units": "1/12", |
| "point_membership_bound_2psi_in_pi_units": "1/6", |
| "local_redundancy_is_twice_reward": "2/7", |
| "global_redundancy_lower_bound": "6/7", |
| "disk_excess_is_half_global_redundancy": "3/7", |
| "amplification_T_over_2psi_equals_pi_over_4psi": "3" |
| }, |
| { |
| "T_in_pi_units": "1/2", |
| "psi_in_pi_units": "3/32", |
| "point_membership_bound_2psi_in_pi_units": "3/16", |
| "local_redundancy_is_twice_reward": "2/7", |
| "global_redundancy_lower_bound": "16/21", |
| "disk_excess_is_half_global_redundancy": "8/21", |
| "amplification_T_over_2psi_equals_pi_over_4psi": "8/3" |
| }, |
| { |
| "T_in_pi_units": "1/2", |
| "psi_in_pi_units": "1/10", |
| "point_membership_bound_2psi_in_pi_units": "1/5", |
| "local_redundancy_is_twice_reward": "2/7", |
| "global_redundancy_lower_bound": "5/7", |
| "disk_excess_is_half_global_redundancy": "5/14", |
| "amplification_T_over_2psi_equals_pi_over_4psi": "5/2" |
| } |
| ], |
| "scope": "Finite reproducibility fixtures supplement the displayed elementary universal proof; they do not enumerate opaque barriers." |
| }, |
| "numerical_membership_diagnostic": { |
| "status": "DIAGNOSTIC PASS — NOT A PROOF", |
| "warning": "Floating-point midpoint samples below are sanity checks only and are not used in any rigorous certificate.", |
| "parameters_are_diagnostic_only": { |
| "a": 1.0001425248, |
| "t": 0.972532002 |
| }, |
| "radial_samples": 54, |
| "angular_midpoints_per_radius": 32768, |
| "maximum_absolute_sampling_error": 3.770831645288908e-05, |
| "sampling_error_threshold": 0.00019174760848570513, |
| "analytic_formula": "M(rho)=0 for rho<t; 2acos(t/rho) for t<=rho<=1; 2[acos(t/rho)-acos(1/rho)] for 1<rho<=a; zero outside aD.", |
| "analytic_maximum": "2acos(t), attained at rho=1. Below 1 M increases; above 1 its derivative is negative because t<1.", |
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| }, |
| "symbolic_conclusion": "Integral_alpha R_D(B intersect S_alpha) <=2psi R_D(B); each rotated family has R_D(B intersect S_alpha)>=2R; T=pi/2. Therefore delta>=W+pi/(4psi)*R with psi=acos(t).", |
| "certificate_independence": "No numerical disk-bound certificate or optimizer is imported; this file validates only the reuse lemma and its constants.", |
| "errors": [], |
| "seconds": 0.619033, |
| "script_sha256": "1ee59424aff9ada39fbd242b4df288426300ce57e2bc46fa96fd72467b1924ff" |
| } |
|
|
