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0.9995482290473038, "absolute_error": 0.0 }, { "q": 5, "u": 0.6, "budget": 2.2559265296519277, "hull_contrast": 0.999774022777356, "unrestricted_mixture_LP_contrast": 0.9997740227773559, "absolute_error": 1.1102230246251565e-16 }, { "q": 5, "u": 0.6, "budget": 2.5582017481736314, "hull_contrast": 0.9999998165074083, "unrestricted_mixture_LP_contrast": 0.9999998165074083, "absolute_error": 0.0 } ] }, "q5_example": { "q": 5, "base_retention": 0.04, "base_contrast": 0.6, "overhead_h_over_E": 0.1, "readout_E": 1.0, "ungated_expected_cost": 1.1, "gated_primitive_expected_cost": 0.14, "gate_saving_ratio": 7.857142857142857, "levels": [ { "level": 0, "m": 2, "q": 5, "initial_contrast": 0.6, "contrast": 0.6, "contrast_recursion": 0.6, "accepted_signal_correct_probability": 0.6799999999999999, "accepted_signal_error_probability": 0.32, "log_accepted_signal_error_probability": -1.1394342831883648, "contrast_rounded_to_one": false, "expected_raw_signal_calls": 1.0, "node_acceptance_probabilities": [], "scope": "independent repeated signals of ONE fixed target", "outer_retention": 0.04, "expected_logical_cost": 0.14, "uniform_prior_full_call_MAP_error": 0.7808 }, { "level": 1, "m": 2, "q": 5, "initial_contrast": 0.6, "contrast": 0.9344262295081968, "contrast_recursion": 0.9344262295081966, "accepted_signal_correct_probability": 0.9475409836065574, "accepted_signal_error_probability": 0.05245901639344263, "log_accepted_signal_error_probability": -2.9477230543676303, "contrast_rounded_to_one": false, "expected_raw_signal_calls": 4.0983606557377055, "node_acceptance_probabilities": [ 0.48799999999999993 ], "scope": "independent repeated signals of ONE fixed target", "outer_retention": 0.04, "expected_logical_cost": 0.5737704918032788, "uniform_prior_full_call_MAP_error": 0.7700983606557377 }, { "level": 2, "m": 2, "q": 5, "initial_contrast": 0.6, "contrast": 0.9990428904707782, "contrast_recursion": 0.9990428904707782, "accepted_signal_correct_probability": 0.9992343123766226, "accepted_signal_error_probability": 0.0007656876233774005, "log_accepted_signal_error_probability": -7.174736273777059, "contrast_rounded_to_one": false, "expected_raw_signal_calls": 9.12245020039481, "node_acceptance_probabilities": [ 0.48799999999999993, 0.8985219027143241 ], "scope": "independent repeated signals of ONE fixed target", "outer_retention": 0.04, "expected_logical_cost": 1.2771430280552736, "uniform_prior_full_call_MAP_error": 0.7680306275049351 }, { "level": 3, "m": 2, "q": 5, "initial_contrast": 0.6, "contrast": 0.9999998165074083, "contrast_recursion": 0.9999998165074082, "accepted_signal_correct_probability": 0.9999998532059267, "accepted_signal_error_probability": 1.4679407338109432e-07, "log_accepted_signal_error_probability": -15.73423509364436, "contrast_rounded_to_one": false, "expected_raw_signal_calls": 18.27286962981165, "node_acceptance_probabilities": [ 0.48799999999999993, 0.8985219027143241, 0.9984693576001659 ], "scope": "independent repeated signals of ONE fixed target", "outer_retention": 0.04, "expected_logical_cost": 2.5582017481736314, "uniform_prior_full_call_MAP_error": 0.768000005871763 } ], "fixed_budget_mixture": { "family": "finite-level recursive unanimous signal verifiers", "hull_levels": [ 0, 1, 2, 3 ], "left_level": 1, "right_level": 2, "right_probability": 0.7481519124997381, "budget": 1.1, "expected_logical_cost": 1.1, "unused_budget": 0.0, "contrast": 0.9827693079866993, "outer_retention": 0.04, "accepted_signal_error_probability": 0.013784553610640593, "accepted_error_relative_reduction": 0.9569232699667483, "original_uniform_prior_full_call_MAP_error": 0.7808, "uniform_prior_full_call_MAP_error": 0.7685513821444256, "old_to_new_deficiency": 0.012248617855574377, "new_to_old_deficiency": 0.0, "claims": "same outer retention and expected cost; not constant latency; not hardware evidence", "full_call_MAP_error_absolute_reduction": 0.012248617855574406, "full_call_MAP_error_relative_reduction": 0.0156872667207664 } }, "execution_status": "PASS" }, "gaussian_pure_form": { "seed": 42817, "scope": "real Gaussian coherent-quadrature model; no hardware", "energy": { "old_budget": 2.0, "new_budget": 2.0, "old_photons": 1.0, "new_photons": 1.55, "gamma": 1.55, "routing_overhead": 0.15, "fixed": 0.2, "per_detector": 0.05, "old_detectors": 16, "new_detectors": 2, "resource_units": "abstract common resource units" }, "old_binary_bayes_error": 0.15865525393145707, "channels": { "oracle": { "information_min_margin": 0.5499999999999999, "dominates": true, "retained_rank": 2, "mean_leakage_operator": 3.2001034903066126e-16, "min_principal_angle_cos_squared": 1.0000000000000002, "gamma_times_min_cos_squared": 1.5500000000000003, "reverse_kernel_available": true, "reverse_added_cov_min_eigenvalue": 0.3548387096774195, "reverse_mean_error": 1.2412670766236368e-16, "reverse_covariance_error": 1.0204274098687891e-16, "old_ols_mse": 2.0, "new_ols_mse": 1.290322580645161, "binary_bayes_error": 0.1065677204369842 }, "learned": { "information_min_margin": 0.5373823304659676, "dominates": true, "retained_rank": 2, "mean_leakage_operator": 0.09022434237743285, "min_principal_angle_cos_squared": 0.99185956804256, "gamma_times_min_cos_squared": 1.537382330465968, "reverse_kernel_available": true, "reverse_added_cov_min_eigenvalue": 0.3495437145443786, "reverse_mean_error": 1.4148301363828725e-16, "reverse_covariance_error": 8.352669508610183e-17, "old_ols_mse": 2.0, "new_ols_mse": 1.2971464295595228, "binary_bayes_error": 0.10750270861308736 }, "random": { "information_min_margin": -0.9953880532783431, "dominates": false, "retained_rank": 2, "mean_leakage_operator": 0.9985111669391368, "min_principal_angle_cos_squared": 0.002975449497843108, "gamma_times_min_cos_squared": 0.004611946721656817, "reverse_kernel_available": false, "reverse_added_cov_min_eigenvalue": -215.82817698309293, "reverse_mean_error": 4.024424951800077e-16, "reverse_covariance_error": 2.0489764566520145e-14, "old_ols_mse": 2.0, "new_ols_mse": 229.94552856530296, "binary_bayes_error": 0.3912387072868743 } }, "calibration": { "pilot_noise_std": 0.02, "alpha": 0.001, "operator_error_bound": 0.18262271502445868, "actual_operator_error": 0.09024658323436077, "h_min_singular_lower_bound": 0.8023009629081197, "pointwise_information_gain_lower_bound": 1.4696905481843314, "energy_consistency": "Pilot precision and amortized overhead are independent declared inputs; their consistency requires measured pilot energy and stable lifetime.", "scope": "Ordering on calibration event; exact reverse kernel still requires true H. This is not one robust common kernel over unknown H." }, "common_kernel": { "added_cov_min_eigenvalue": 0.3548387096774188, "covariance_error": 0.0, "true_operator_leakage": 0.09022434237743285, "actual_uniform_tv_bound": 0.07189105920717642, "calibrated_uniform_tv_bound": 0.14490593994082324, "calibration_failure_probability": 0.001, "parameter_radius": 2.0, "scope": "One fixed implementable Q-only kernel. TV bound holds for ||theta||<=B; calibrated bound holds on the declared independent calibration event." }, "likelihood_training": { "steps": 120, "batch_size": 32, "learning_rate": 0.2, "max_coupled_gradient_difference": 3.885780586188048e-16, "max_coupled_parameter_difference": 2.220446049250313e-16, "interpretation": "Learning 2 modal coefficients in a linear Gaussian model, not a neural benchmark." }, "holdout": { "basis": "oracle", "independent_replicates": 100000, "old_mse": 1.996303670221155, "new_mse": 1.2909284374715835, "empirical_ratio": 0.6466593518452889, "analytic_ratio": 0.6451612903225806 }, "approximation": { "learned_projection_leakage_operator": 0.09022434237743285, "bounded_parameter_radius": 2.0, "no_reinvestment_reconstruction_tv_upper_bound": 0.07189105920717642, "scope": "Fixed Q-fill reverse kernel at unchanged noise; a valid bound, not necessarily optimal deficiency." }, "execution_status": "PASS" }, "orbit_audit": { "status": "PASS", "seed": 8751, "scope": "Independent finite-orbit LP checks and explicit hypothesis-boundary examples; no hardware or general-RSI claim", "arbitrary_orbit_lp_cases": 96, "max_formula_lp_error": 5.551115123125783e-16, "missing_symmetry_counterexample": { "source": [ [ 1.0, 0.0 ], [ 0.5, 0.5 ] ], "target": "perfect binary experiment", "outside_hypotheses_frontier_prediction": 0.25, "actual_worst_state_deficiency_lp": 0.3333333333333334, "exact_worst_state_deficiency": "1/3", "proof": "At outcome 0 choose label 0 with probability a; at outcome 1 choose label 1. Worst-state error is max(1-a,a/2), minimized at a=2/3. Erasure cannot improve approximation to a target that always gives the correct label." }, "unknown_H_common_kernel_counterexample": { "Q": [ [ 1.0 ], [ 0.0 ] ], "sigma0": 1.0, "sigma1": 0.5, "theta": 1.0, "cases": [ { "name": "plus", "H": [ [ 1.0 ], [ 1.0 ] ], "old_information": 2.0, "new_information": 4.0, "known_H_reverse_linear": [ [ 1.0 ], [ 1.0 ] ], "known_H_added_covariance": [ [ 0.75, -0.25 ], [ -0.25, 0.75 ] ], "added_covariance_min_eigenvalue": 0.5 }, { "name": "minus", "H": [ [ 1.0 ], [ -1.0 ] ], "old_information": 2.0, "new_information": 4.0, "known_H_reverse_linear": [ [ 1.0 ], [ -1.0 ] ], "known_H_added_covariance": [ [ 0.75, 0.25 ], [ 0.25, 0.75 ] ], "added_covariance_min_eigenvalue": 0.5 } ], "common_new_law": "N(theta, 0.25)", "distinct_old_laws": "N((theta,theta),I_2) and N((theta,-theta),I_2)", "old_pair_total_variation_at_theta_1": 0.6826894921370859, "any_common_kernel_worst_error_lower_bound_at_theta_1": 0.3413447460685429, "lower_bound_as_abs_theta_goes_to_infinity": 0.5, "conclusion": "For each known H there exists a reverse kernel, but no single H-blind kernel reproduces both old laws. The triangle inequality gives the displayed lower bound." }, "principal_angle_rank_counterexample": { "H": [ [ 1.0, 0.0 ], [ 0.0, 1.0 ] ], "Q": [ [ 1.0 ], [ 0.0 ] ], "sigma0": 1.0, "sigma1": 0.1, "gamma": 100.0, "conventional_list_of_principal_angle_cosines": [ 1.0 ], "information_margin_eigenvalues": [ -1.0, 98.99999999999999 ], "extended_maximum_angle_degrees": 90.0, "dominates": false, "conclusion": "Require r>=p and full retained signal rank, or define the maximum angle using inf_{v in range(H), ||v||=1} ||Q^T v||." }, "restricted_parameter_space_counterexample": { "H": [ [ 1.0, 0.0 ], [ 0.0, 1.0 ] ], "Q": [ [ 1.0 ], [ 0.0 ] ], "sigma0": 1.0, "sigma1": 1.0, "parameter_space": [ [ -1.0, 0.0 ], [ 1.0, 0.0 ] ], "information_margin_eigenvalues": [ -1.0, 0.0 ], "nevertheless_exact_dominance": true, "reverse_kernel": "Output (T,Z) with independent Z~N(0,1).", "conclusion": "The global matrix inequality is sufficient for arbitrary parameter sets. Its necessity as stated uses theta in all R^p; a nonempty open parameter set also suffices." }, "execution_status": "PASS" }, "learning_check": { "status": "PASS", "paths": 20000, "seed": 71037, "steps": 120, "moment_checks": 12, "gamma": 1.55, "old_asymptotic_mse": 0.006944444444444445, "new_asymptotic_mse": 0.004480286738351254, "scope": "Linear Gaussian modal coefficient SGD; no neural benchmark or hardware", "execution_status": "PASS" }, "catalogue": { "status": "PASS", "seed": 864112, "theorem_scope": "Fully S_q-equivariant finite source to visible nonnegative symmetric reliability catalogue", "tv_convention": "half_l1", "formula": "(q-1)/q * max_a (E_target[(V-a)+]-E_source[(R-a)+])_+", "randomized_comparisons": 220, "visible_mixture_comparisons": 160, "arbitrary_posterior_orbit_comparisons": 60, "independent_solver": "Full unrestricted stochastic-kernel minimax TV LP", "compiler": "Explicit uniform-tie MAP collapse, scalar weak transport, label mixing", "max_formula_full_lp_error": 2.848277169675839e-13, "max_formula_kernel_tv_error": 3.625572064791527e-16, "max_kernel_stochastic_error": 2.1094237467877974e-15, "max_MAP_collapse_identity_error": 2.220446049250313e-16, "tolerance": 1e-07, "exact_rational_witness": { "q": 5, "source_values": [ "0", "1/2", "9/10" ], "source_masses": [ "1/5", "1/2", "3/10" ], "target_values": [ "1/5", "7/10", "1" ], "target_masses": [ "2/5", "2/5", "1/5" ], "exact_stoploss_gap": "3/50", "exact_deficiency": "6/125", "exact_witness_threshold": "1/2", "formula": 0.04799999999999999, "constructed_kernel_tv": 0.048000000000000084, "unrestricted_full_lp": 0.048, "coupling": [ [ 0.20000000000000004, 0.0, 0.0 ], [ 0.19999999999999998, 0.30000000000000004, 0.0 ], [ 0.0, 0.09999999999999998, 0.20000000000000007 ] ], "threshold_audit": { "gap": 0.059999999999999984, "witness_threshold": 0.5000000000000001, "thresholds": [ 0.0, 0.2, 0.5000000000000001, 0.7, 0.8999999999999999, 1.0 ], "source_stoploss": [ 0.52, 0.36000000000000004, 0.11999999999999995, 0.06, 0.0, 0.0 ], "target_stoploss": [ 0.56, 0.36, 0.17999999999999994, 0.06000000000000001, 0.020000000000000018, 0.0 ], "differences": [ 0.040000000000000036, -5.551115123125783e-17, 0.059999999999999984, 1.3877787807814457e-17, 0.020000000000000018, 0.0 ] }, "exact_fraction_check_passed": true }, "equal_mean_obstruction": { "q": 5, "source_values": [ 0.5 ], "source_masses": [ 1 ], "target_values": [ 0, 1 ], "target_masses": [ 0.5, 0.5 ], "source_mean": 0.5, "target_mean": 0.5, "stoploss_gap": 0.25, "exact_deficiency": 0.2, "interpretation": "Mean confidence alone misses coverage/purity constraints" }, "same_profile_blackwell_counterexample": { "q": 3, "A": [ [ 0.5, 0.25, 0.25 ], [ 0.25, 0.5, 0.25 ], [ 0.25, 0.25, 0.5 ] ], "B": [ [ 0.0, 0.5, 0.5 ], [ 0.5, 0.0, 0.5 ], [ 0.5, 0.5, 0.0 ] ], "confidence_A": 0.25, "confidence_B": 0.25, "deficiency_A_to_B": 0.25, "deficiency_B_to_A": 0.0, "interpretation": "Same MAP-confidence profile is not complete Blackwell order for q>=3" }, "physical_photonic_measurements": false, "world_first_priority_established": false, "execution_status": "PASS" }, "uncertainty_closure": { "seed": 1072026, "status": "PASS", "scope": "Fixed-Q, equal-Gaussian coherent-mode surrogate, unknown H, common-null bounded theta family.", "theorem": "One H-independent exact reverse kernel iff gamma U.T Q Q.T U >= I on the full mutation closure.", "novelty_status": "Present synthesis and proof; independent priority unverified.", "hardware_experiments": 0, "neural_benchmarks": 0, "counterexample": { "uncertainty_angle": 0.01, "theta": 2.0, "gamma": 1.0001000066670445, "known_H_information_margin": 0.0, "compressed_mean_difference": 0.0, "full_mean_difference": 0.03999933333666666, "exact_mutation_closure_dimension": 2, "receiver_coordinates": 1, "common_kernel_exists": false, "any_common_kernel_max_TV_error_lower_bound": 0.007978180763399623, "derivation": "Half the TV of two target laws sharing one source law." }, "exact_structured_example": { "physical_modes": 32, "pointwise_H_rank": 2, "exact_mutation_closure_dimension": 6, "bounded_nuisance_coordinates": [ -0.1, 0.1 ], "resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0099504938362078, "maximum_signal_norm": 2.0199009876724157, "photon_resource_per_exposure": 4.08, "old_worst_case_photon_resource": 4.08, "new_worst_case_photon_resource": 8.68, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 6, "old_exposure": 1.0, "new_exposure": 2.127450980392157, "gamma": 2.127450980392157, "old_budget": 11.280000000000001, "new_budget": 11.28, "receiver_coordinate_reduction_fraction": 0.8125, "minimum_exact_receiver_count": 6, "note": "The specified H(z1,z2) family has exact closure dimension six." }, "common_kernel": { "closure_dimension": 6, "receiver_coordinates": 6, "gamma": 2.127450980392157, "minimum_retained_cos_squared": 0.9999999999999992, "universal_information_margin": 1.127450980392155, "common_kernel_exists": true, "candidate_available": true, "candidate_covariance_min_eigenvalue": 0.5299539170506903, "common_mean_operator_error": 3.8453742011065983e-16, "common_covariance_identity_error": 2.681517473081736e-17 }, "independent_holdout": { "replications": 100000, "unknown_H_coefficients": [ -0.06446286983254929, 0.08115451815611968 ], "theta": [ 1.1, -0.7 ], "maximum_empirical_mean_coordinate_error": 0.004621841779348393, "maximum_empirical_covariance_entry_error": 0.010264230353982594, "interpretation": "Monte Carlo implementation check, not hardware." } }, "rotation_orbit": { "physical_modes": 32, "pointwise_H_rank": 1, "drift_parameter_interval": [ -0.2, 0.2 ], "exact_mutation_closure_dimension": 32, "exact_dimension_valid_for_every_positive_drift_radius": true, "proof": "Even and odd derivatives each form a 16-by-16 Vandermonde system in distinct j^2.", "positive_integer_vandermonde_determinant": "14355973656491146827638004051975967715503775251095756038091878465144346560214586188455148959995319764939603245797057978391666871809628140938011931649692860847934073804758712320000000000000000000000000000000000000000000000", "maximum_generator_operator_norm": 1, "seed_signal_norm": 1, "bounded_parameter_absolute_value": 2.0, "selected_derivative_order": 2, "selected_receivers": 3, "selected_rigorous_uniform_leakage": 0.0013333333333333337, "selected_rigorous_uniform_TV_upper_bound": 0.0010638457658569411, "selected_grid_leakage": 0.00022260840884392346, "selected_resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0, "maximum_signal_norm": 2.0, "photon_resource_per_exposure": 4.0, "old_worst_case_photon_resource": 4.0, "new_worst_case_photon_resource": 9.2, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 3, "old_exposure": 1.0, "new_exposure": 2.3, "gamma": 2.3, "old_budget": 11.2, "new_budget": 11.2, "receiver_coordinate_reduction_fraction": 0.90625, "minimum_exact_receiver_count": 32, "note": "The last count applies to the full 32-mode orbit class only." }, "selected_common_kernel_added_covariance_min_eigenvalue": 0.5652173913043473, "selected_common_kernel_covariance_identity_error": 0.0, "krylov_orders": [ { "derivative_order": 0, "receiver_coordinates": 1, "exact_full_class_dimension": 32, "rigorous_uniform_operator_leakage": 0.2, "rigorous_uniform_common_kernel_TV_upper_bound": 0.15851941887820598, "grid_maximum_operator_leakage": 0.12052081551261949, "grid_maximum_common_kernel_TV": 0.0959294085675193, "grid_points": 1001, "grid_status": "Sanity check only; the Taylor theorem supplies the uniform certificate.", "resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0, "maximum_signal_norm": 2.0, "photon_resource_per_exposure": 4.0, "old_worst_case_photon_resource": 4.0, "new_worst_case_photon_resource": 9.6, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 1, "old_exposure": 1.0, "new_exposure": 2.4, "gamma": 2.4, "old_budget": 11.2, "new_budget": 11.2, "receiver_coordinate_reduction_fraction": 0.96875, "minimum_exact_receiver_count": 32, "note": "The last count applies to the full 32-mode orbit class only." } }, { "derivative_order": 1, "receiver_coordinates": 2, "exact_full_class_dimension": 32, "rigorous_uniform_operator_leakage": 0.020000000000000004, "rigorous_uniform_common_kernel_TV_upper_bound": 0.015956627433803883, "grid_maximum_operator_leakage": 0.0062824669579412, "grid_maximum_common_kernel_TV": 0.005012650415097131, "grid_points": 1001, "grid_status": "Sanity check only; the Taylor theorem supplies the uniform certificate.", "resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0, "maximum_signal_norm": 2.0, "photon_resource_per_exposure": 4.0, "old_worst_case_photon_resource": 4.0, "new_worst_case_photon_resource": 9.399999999999999, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 2, "old_exposure": 1.0, "new_exposure": 2.3499999999999996, "gamma": 2.3499999999999996, "old_budget": 11.2, "new_budget": 11.2, "receiver_coordinate_reduction_fraction": 0.9375, "minimum_exact_receiver_count": 32, "note": "The last count applies to the full 32-mode orbit class only." } }, { "derivative_order": 2, "receiver_coordinates": 3, "exact_full_class_dimension": 32, "rigorous_uniform_operator_leakage": 0.0013333333333333337, "rigorous_uniform_common_kernel_TV_upper_bound": 0.0010638457658569411, "grid_maximum_operator_leakage": 0.00022260840884392346, "grid_maximum_common_kernel_TV": 0.00017761581105446567, "grid_points": 1001, "grid_status": "Sanity check only; 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the Taylor theorem supplies the uniform certificate.", "resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0, "maximum_signal_norm": 2.0, "photon_resource_per_exposure": 4.0, "old_worst_case_photon_resource": 4.0, "new_worst_case_photon_resource": 8.799999999999999, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 5, "old_exposure": 1.0, "new_exposure": 2.1999999999999997, "gamma": 2.1999999999999997, "old_budget": 11.2, "new_budget": 11.2, "receiver_coordinate_reduction_fraction": 0.84375, "minimum_exact_receiver_count": 32, "note": "The last count applies to the full 32-mode orbit class only." } }, { "derivative_order": 5, "receiver_coordinates": 6, "exact_full_class_dimension": 32, "rigorous_uniform_operator_leakage": 8.888888888888892e-08, "rigorous_uniform_common_kernel_TV_upper_bound": 7.092307208189652e-08, "grid_maximum_operator_leakage": 1.952424282607876e-09, "grid_maximum_common_kernel_TV": 1.5578092149581835e-09, "grid_points": 1001, "grid_status": "Sanity check only; the Taylor theorem supplies the uniform certificate.", "resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0, "maximum_signal_norm": 2.0, "photon_resource_per_exposure": 4.0, "old_worst_case_photon_resource": 4.0, "new_worst_case_photon_resource": 8.599999999999998, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 6, "old_exposure": 1.0, "new_exposure": 2.1499999999999995, "gamma": 2.1499999999999995, "old_budget": 11.2, "new_budget": 11.199999999999998, "receiver_coordinate_reduction_fraction": 0.8125, "minimum_exact_receiver_count": 32, "note": "The last count applies to the full 32-mode orbit class only." } }, { "derivative_order": 6, "receiver_coordinates": 7, "exact_full_class_dimension": 32, "rigorous_uniform_operator_leakage": 2.5396825396825407e-09, "rigorous_uniform_common_kernel_TV_upper_bound": 2.0263735134307126e-09, "grid_maximum_operator_leakage": 2.8825222547854315e-11, "grid_maximum_common_kernel_TV": 2.2999158133529818e-11, "grid_points": 1001, "grid_status": "Sanity check only; the Taylor theorem supplies the uniform certificate.", "resource": { "resource_units": "abstract common convertible resource units", "exposure_units": "coefficient P in variance sigma^2=1/P", "conversion_status": "declared model only; no measured physical conversion", "bounded_parameter_norm": 2.0, "transfer_operator_norm_bound": 1.0, "maximum_signal_norm": 2.0, "photon_resource_per_exposure": 4.0, "old_worst_case_photon_resource": 4.0, "new_worst_case_photon_resource": 8.399999999999999, "old_fixed_resource": 0.8, "new_fixed_resource": 1.4, "per_receiver_resource": 0.2, "old_receiver_coordinates": 32, "new_receiver_coordinates": 7, "old_exposure": 1.0, "new_exposure": 2.0999999999999996, "gamma": 2.0999999999999996, "old_budget": 11.2, "new_budget": 11.2, "receiver_coordinate_reduction_fraction": 0.78125, "minimum_exact_receiver_count": 32, "note": "The last count applies to the full 32-mode orbit class only." } } ] }, "execution_status": "PASS" }, "reference_gluing": { "status": "PASS", "seed": 731, "known_context_cases": 128, "rank_increment_identity_checks": 128, "required_reference_deletion_checks": 128, "future_interface_cases": 90, "strict_naive_future_contract_losses": 90, "max_known_context_relative_information_error": 2.0841237094270004e-15, "max_future_contract_relative_information_error": 2.908787601858244e-13, "future_interface_theorem": "For known Y=H theta+epsilon with epsilon~N(0,I), Q.T Y preserves the full joint experiment with every Z=D theta+B epsilon+eta whose B rows lie in E iff im(H)+E is contained in im(Q).", "exact_minimum_retained_modes": "dim(im(H)+E)", "exact_known_context_reference_minimum": "rank(G.T S^-1 C)=rank(J_full-J_T), conditional on keeping T fixed", "reference_counterexample": { "rho": 0.99, "current_marginal_information_before": 1.0, "current_marginal_information_after": 1.0, "future_joint_information_full": 51.25125628140696, "future_joint_information_naive": 2.0, "lost_information": 49.25125628140696, "full_unbiased_variance": 0.01951171683498385, "naive_unbiased_variance": 0.5, "conclusion": "Marginally sufficient local compression is not sufficient after correlated extension.", "unbounded_penalty_limit": "The information loss diverges as rho increases to one." }, "coordination_cost_oracle": { "local_reference_modules": 16, "full_physical_modes": 17, "fixed_semantic_modes": 1, "centralized_reference_modes": 1, "separate_local_reference_exports": 16, "coherent_to_local_reference_count_ratio": 16, "full_information": 24.999999999999996, "naive_information": 1.0, "repaired_information": 25.00000000000019, "unrestricted_global_oracle_information": 24.99999999999987, "unrestricted_global_oracle_total_modes": 1, "completed_total_modes": 2, "fixed_cost": 0.2, "per_detector_cost": 0.05, "source_cost": 1.0, "extra_mixer_cost": 0.15, "full_cost": 2.05, "repaired_cost": 1.45, "unrestricted_global_oracle_cost": 1.4, "completed_cost_reduction_fraction": 0.2926829268292683, "completed_physical_mode_reduction_fraction": 0.8823529411764706, "resource_units": "abstract common units, unchanged exposure, known covariance", "oracle_scope": "Global score oracle is stronger and uses fewer total modes; two-mode repair is minimal only while keeping the original semantic output fixed.", "hardware_scope": "Passive real-mode mixing and noiseless linear readout are model assumptions; actual insertion loss, source, detector, reference and calibration costs must be measured." }, "proof_scope": "All Gaussian means and covariances are known; linear optical mode exports; unconstrained theta; all allowed future D and positive definite independent eta covariance. No generic statistical-priority claim.", "physical_validation_percent": 0, "neural_benchmarks": 0, "execution_status": "PASS" }, "bilinear_gluing": { "experiment": "exact_prime_field_shared_bilinear_gluing", "seed": 20261017, "case_count": 91, "affine_random_cases": 60, "centered_no_compression_cases": 24, "curated_cases": 4, "direct_uniformizer_source_cases": 3, "direct_uniformizer_mixture_checks": [ "uniformized_seed_k1_L1", "uniformized_seed_k1_L2", "uniformized_seed_k2_L1" ], "seed_oracle_simulator_mixture_identity_exact_checked": true, "uniformizer_family_curve": [ { "blocks": 1, "target_dimension": 3, "uniformizer_dimension": 1, "exact_minimum_sufficient_alphabet": 8, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 3, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "3/2", "alternative_seed_oracle_expected_binary_readouts": 1.5, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.043252437039192335, "information_upper_bound_bits": 0.0874628412503394, "chi2_exact": "1/16", "chi2": 0.0625, "zero_readout_deficiency_exact": "3/32", "zero_readout_deficiency": 0.09375, "zero_readout_deficiency_upper_bound": 0.5 }, { "blocks": 2, "target_dimension": 6, "uniformizer_dimension": 2, "exact_minimum_sufficient_alphabet": 64, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 6, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "3/2", "alternative_seed_oracle_expected_binary_readouts": 1.5, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.023370230721314722, "information_upper_bound_bits": 0.04984854945056153, "chi2_exact": "9/256", "chi2": 0.03515625, "zero_readout_deficiency_exact": "69/1024", "zero_readout_deficiency": 0.0673828125, "zero_readout_deficiency_upper_bound": 0.25 }, { "blocks": 3, "target_dimension": 9, "uniformizer_dimension": 3, "exact_minimum_sufficient_alphabet": 512, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 9, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "9/8", "alternative_seed_oracle_expected_binary_readouts": 1.125, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.00985924915041796, "information_upper_bound_bits": 0.021327032624372442, "chi2_exact": "61/4096", "chi2": 0.014892578125, "zero_readout_deficiency_exact": "675/16384", "zero_readout_deficiency": 0.04119873046875, "zero_readout_deficiency_upper_bound": 0.125 }, { "blocks": 4, "target_dimension": 12, "uniformizer_dimension": 4, "exact_minimum_sufficient_alphabet": 4096, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 12, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "3/4", "alternative_seed_oracle_expected_binary_readouts": 0.75, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.00376410755660513, "information_upper_bound_bits": 0.008100301559194709, "chi2_exact": "369/65536", "chi2": 0.0056304931640625, "zero_readout_deficiency_exact": "24813/1048576", "zero_readout_deficiency": 0.02366352081298828, "zero_readout_deficiency_upper_bound": 0.0625 }, { "blocks": 5, "target_dimension": 15, "uniformizer_dimension": 5, "exact_minimum_sufficient_alphabet": 32768, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 15, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "15/32", "alternative_seed_oracle_expected_binary_readouts": 0.46875, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.001357904787553977, "information_upper_bound_bits": 0.002887792269877051, "chi2_exact": "2101/1048576", "chi2": 0.0020036697387695312, "zero_readout_deficiency_exact": "110097/8388608", "zero_readout_deficiency": 0.013124585151672363, "zero_readout_deficiency_upper_bound": 0.03125 }, { "blocks": 6, "target_dimension": 18, "uniformizer_dimension": 6, "exact_minimum_sufficient_alphabet": 262144, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 18, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "9/32", "alternative_seed_oracle_expected_binary_readouts": 0.28125, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.00047194835374961085, "information_upper_bound_bits": 0.0009910535128422298, "chi2_exact": "11529/16777216", "chi2": 0.0006871819496154785, "zero_readout_deficiency_exact": "3825333/536870912", "zero_readout_deficiency": 0.0071252379566431046, "zero_readout_deficiency_upper_bound": 0.015625 }, { "blocks": 7, "target_dimension": 21, "uniformizer_dimension": 7, "exact_minimum_sufficient_alphabet": 2097152, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 21, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "21/128", "alternative_seed_oracle_expected_binary_readouts": 0.1640625, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 0.0001597785793852491, "information_upper_bound_bits": 0.0003317862469880793, "chi2_exact": "61741/268435456", "chi2": 0.00023000314831733704, "zero_readout_deficiency_exact": "130994307/34359738368", "zero_readout_deficiency": 0.003812436101725325, "zero_readout_deficiency_upper_bound": 0.0078125 }, { "blocks": 8, "target_dimension": 24, "uniformizer_dimension": 8, "exact_minimum_sufficient_alphabet": 16777216, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 24, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "3/32", "alternative_seed_oracle_expected_binary_readouts": 0.09375, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.3067413347554113e-05, "information_upper_bound_bits": 0.0001091944378372626, "chi2_exact": "325089/4294967296", "chi2": 7.569068111479282e-05, "zero_readout_deficiency_exact": "2219440149/1099511627776", "zero_readout_deficiency": 0.0020185690564176184, "zero_readout_deficiency_upper_bound": 0.00390625 }, { "blocks": 9, "target_dimension": 27, "uniformizer_dimension": 9, "exact_minimum_sufficient_alphabet": 134217728, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 27, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "27/512", "alternative_seed_oracle_expected_binary_readouts": 0.052734375, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.7376309845441023e-05, "information_upper_bound_bits": 3.549997764882141e-05, "chi2_exact": "1690981/68719476736", "chi2": 2.4607012164779007e-05, "zero_readout_deficiency_exact": "4663345203/4398046511104", "zero_readout_deficiency": 0.0010603219386666751, "zero_readout_deficiency_upper_bound": 0.001953125 }, { "blocks": 10, "target_dimension": 30, "uniformizer_dimension": 10, "exact_minimum_sufficient_alphabet": 1073741824, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 30, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "15/512", "alternative_seed_oracle_expected_binary_readouts": 0.029296875, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.629109118957922e-06, "information_upper_bound_bits": 1.1437799468351623e-05, "chi2_exact": "8717049/1099511627776", "chi2": 7.928109880595002e-06, "zero_readout_deficiency_exact": "155803303251/281474976710656", "zero_readout_deficiency": 0.0005535245266621303, "zero_readout_deficiency_upper_bound": 0.0009765625 }, { "blocks": 11, "target_dimension": 33, "uniformizer_dimension": 11, "exact_minimum_sufficient_alphabet": 8589934592, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 33, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "33/2048", "alternative_seed_oracle_expected_binary_readouts": 0.01611328125, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.808826355650987e-06, "information_upper_bound_bits": 3.660313184683359e-06, "chi2_exact": "44633821/17592186044416", "chi2": 2.537138982461329e-06, "zero_readout_deficiency_exact": "5179548661767/18014398509481984", "zero_readout_deficiency": 0.00028752270907300703, "zero_readout_deficiency_upper_bound": 0.00048828125 }, { "blocks": 12, "target_dimension": 36, "uniformizer_dimension": 12, "exact_minimum_sufficient_alphabet": 68719476736, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 36, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "9/1024", "alternative_seed_oracle_expected_binary_readouts": 0.0087890625, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.776469617391241e-07, "information_upper_bound_bits": 1.165346681962837e-06, "chi2_exact": "227363409/281474976710656", "chi2": 8.077570932130129e-07, "zero_readout_deficiency_exact": "85743757826397/576460752303423488", "zero_readout_deficiency": 0.00014874170961992096, "zero_readout_deficiency_upper_bound": 0.000244140625 }, { "blocks": 13, "target_dimension": 39, "uniformizer_dimension": 13, "exact_minimum_sufficient_alphabet": 549755813888, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 39, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "39/8192", "alternative_seed_oracle_expected_binary_readouts": 0.0047607421875, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.835950499457662e-07, "information_upper_bound_bits": 3.6954539570420805e-07, "chi2_exact": "1153594261/4503599627370496", "chi2": 2.5614938192752845e-07, "zero_readout_deficiency_exact": "2829190226733315/36893488147419103232", "zero_readout_deficiency": 7.668535475497542e-05, "zero_readout_deficiency_upper_bound": 0.0001220703125 }, { "blocks": 14, "target_dimension": 42, "uniformizer_dimension": 14, "exact_minimum_sufficient_alphabet": 4398046511104, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 42, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "21/8192", "alternative_seed_oracle_expected_binary_readouts": 0.0025634765625, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.8138505843225554e-08, "information_upper_bound_bits": 1.1682656067768062e-07, "chi2_exact": "5835080169/72057594037927936", "chi2": 8.097800442696812e-08, "zero_readout_deficiency_exact": "46541247274219725/1180591620717411303424", "zero_readout_deficiency": 3.942196984757351e-05, "zero_readout_deficiency_upper_bound": 6.103515625e-05 }, { "blocks": 15, "target_dimension": 45, "uniformizer_dimension": 15, "exact_minimum_sufficient_alphabet": 35184372088832, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 45, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "45/32768", "alternative_seed_oracle_expected_binary_readouts": 0.001373291015625, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.8358183403894668e-08, "information_upper_bound_bits": 3.684420483445418e-08, "chi2_exact": "29443836301/1152921504606846976", "chi2": 2.5538457027081407e-08, "zero_readout_deficiency_exact": "381864694850077185/18889465931478580854784", "zero_readout_deficiency": 2.0215748620701558e-05, "zero_readout_deficiency_upper_bound": 3.0517578125e-05 }, { "blocks": 16, "target_dimension": 48, "uniformizer_dimension": 16, "exact_minimum_sufficient_alphabet": 281474976710656, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 48, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "3/4096", "alternative_seed_oracle_expected_binary_readouts": 0.000732421875, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.784058730987646e-09, "information_upper_bound_bits": 1.1597790014913883e-08, "chi2_exact": "148292923329/18446744073709551616", "chi2": 8.038975481876407e-09, "zero_readout_deficiency_exact": "12505779216837064773/1208925819614629174706176", "zero_readout_deficiency": 1.0344538113036206e-05, "zero_readout_deficiency_upper_bound": 1.52587890625e-05 }, { "blocks": 17, "target_dimension": 51, "uniformizer_dimension": 17, "exact_minimum_sufficient_alphabet": 2251799813685248, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 51, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "51/131072", "alternative_seed_oracle_expected_binary_readouts": 0.00038909912109375, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.8192136512073742e-09, "information_upper_bound_bits": 3.645303365558505e-09, "chi2_exact": "745759583941/295147905179352825856", "chi2": 2.526731753314744e-09, "zero_readout_deficiency_exact": "25549322106933038781/4835703278458516698824704", "zero_readout_deficiency": 5.283475977681871e-06, "zero_readout_deficiency_upper_bound": 7.62939453125e-06 }, { "blocks": 18, "target_dimension": 54, "uniformizer_dimension": 18, "exact_minimum_sufficient_alphabet": 18014398509481984, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 54, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "27/131072", "alternative_seed_oracle_expected_binary_readouts": 0.00020599365234375, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.714071714262822e-10, "information_upper_bound_bits": 1.1444057967061605e-09, "chi2_exact": "3745977788889/4722366482869645213696", "chi2": 7.932416517179492e-10, "zero_readout_deficiency_exact": "416891007338622809751/154742504910672534362390528", "zero_readout_deficiency": 2.6940949907672717e-06, "zero_readout_deficiency_upper_bound": 3.814697265625e-06 }, { "blocks": 19, "target_dimension": 57, "uniformizer_dimension": 19, "exact_minimum_sufficient_alphabet": 144115188075855872, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 57, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "57/524288", "alternative_seed_oracle_expected_binary_readouts": 0.0001087188720703125, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.7928546763373366e-10, "information_upper_bound_bits": 3.589389350638284e-10, "chi2_exact": "18798608421181/75557863725914323419136", "chi2": 2.4879751086363206e-10, "zero_readout_deficiency_exact": "3396263946349773448035/2475880078570760549798248448", "zero_readout_deficiency": 1.3717400837565276e-06, "zero_readout_deficiency_upper_bound": 1.9073486328125e-06 }, { "blocks": 20, "target_dimension": 60, "uniformizer_dimension": 20, "exact_minimum_sufficient_alphabet": 1152921504606846976, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 60, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "15/262144", "alternative_seed_oracle_expected_binary_readouts": 5.7220458984375e-05, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.620570673846415e-11, "information_upper_bound_bits": 1.124964480910046e-10, "chi2_exact": "94267920012849/1208925819614629174706176", "chi2": 7.797659582032825e-11, "zero_readout_deficiency_exact": "221057196795688744395387/316912650057057350374175801344", "zero_readout_deficiency": 6.975335214794655e-07, "zero_readout_deficiency_upper_bound": 9.5367431640625e-07 }, { "blocks": 21, "target_dimension": 63, "uniformizer_dimension": 21, "exact_minimum_sufficient_alphabet": 9223372036854775808, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 63, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "63/2097152", "alternative_seed_oracle_expected_binary_readouts": 3.0040740966796875e-05, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.760875460756701e-11, "information_upper_bound_bits": 3.5237147747876914e-11, "chi2_exact": "472439111692021/19342813113834066795298816", "chi2": 2.4424529612713387e-11, "zero_readout_deficiency_exact": "1796431366517053636999467/5070602400912917605986812821504", "zero_readout_deficiency": 3.5428361848951555e-07, "zero_readout_deficiency_upper_bound": 4.76837158203125e-07 }, { "blocks": 22, "target_dimension": 66, "uniformizer_dimension": 22, "exact_minimum_sufficient_alphabet": 73786976294838206464, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 66, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "33/2097152", "alternative_seed_oracle_expected_binary_readouts": 1.5735626220703125e-05, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 5.513788516663325e-12, "information_upper_bound_bits": 1.1032110600928247e-11, "chi2_exact": "2366593604971209/309485009821345068724781056", "chi2": 7.646876358688135e-12, "zero_readout_deficiency_exact": "116667385105619650393173969/649037107316853453566312041152512", "zero_readout_deficiency": 1.797545684066162e-07, "zero_readout_deficiency_upper_bound": 2.384185791015625e-07 }, { "blocks": 23, "target_dimension": 69, "uniformizer_dimension": 23, "exact_minimum_sufficient_alphabet": 590295810358705651712, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 69, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "69/8388608", "alternative_seed_oracle_expected_binary_readouts": 8.225440979003906e-06, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; this is outside the encoding lower bound.", "cheap_uniformizer_flag_contract": "If A0 is cheap observable metadata, acquire only when A0=0; this can also attain expected 3k*2^-L readouts and is excluded from the hidden-nuisance query contract.", "all_singleton_information_exactly_zero": true, "uniform_target_information_bits": 1.725806905850529e-12, "information_upper_bound_bits": 3.452660045205231e-12, "chi2_exact": "11850560210900461/4951760157141521099596496896", "chi2": 2.3932015757688426e-12, "zero_readout_deficiency_exact": "1892406978409974216532522173/20769187434139310514121985316880384", "zero_readout_deficiency": 9.111608166717847e-08, "zero_readout_deficiency_upper_bound": 1.1920928955078125e-07 }, { "blocks": 24, "target_dimension": 72, "uniformizer_dimension": 24, "exact_minimum_sufficient_alphabet": 4722366482869645213696, "minimum_expected_binary_queries_to_sufficiently_encode_Y": 72, "query_lower_bound_scope": "Prefix-free q-ary queries of one realized Y; no cheap hidden-uniformizer flag or separate seed-channel oracle.", "alternative_seed_oracle_expected_binary_readouts_exact": "9/2097152", "alternative_seed_oracle_expected_binary_readouts": 4.291534423828125e-06, "alternative_seed_oracle_contract": "If a separately available seed channel can be acquired, Bernoulli(2^-L) seed-oracle use and otherwise uniform output exactly simulates the unflagged experiment; 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no measured photonic or neural RSI result.", "execution_status": "PASS" }, "contract_audit": { "status": "PASS", "seed": 461107, "random_contracts": 60, "sufficient_joint_models": 180, "missing_mean_rejections": 60, "missing_interface_rejections": 60, "maximum_reverse_mean_operator_error": 1.9433363136495265e-15, "maximum_reverse_joint_covariance_error": 2.460481661111946e-15, "maximum_necessity_information_identity_error": 7.160936003957117e-15, "approximate_joint_KL_checks": 60, "maximum_approximate_joint_KL_identity_error": 2.1316282072803006e-14, "minimum_missing_mean_pairwise_KL_loss": 0.4999999999999991, "minimum_missing_interface_pairwise_KL_loss": 0.08571822689509778, "theorem_scope": "Fixed equal noise, fixed Q, unknown H with mean closure S, all future linear noise couplings in E; one common reverse kernel. 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