outputs/claim4_erm_query_lb.json

{
  "claim": "Corollary 3.7 (2^Omega(d) ERM queries)",
  "combined_regime": {
    "eps_prime": 0.014666666666666666,
    "note": "Corollary 3.7 as stated says rho <= 1 + O_eps(eps); the reduction gives rho <= 1 + eps/eps' * L, which is O(eps) for fixed eps' and L.",
    "rho_at_eps_1_over_12_L_1": 6.681818181818182,
    "rho_bound": "1 + (eps/eps') L"
  },
  "erm_query_drives_the_search": {
    "conclusion": "every ERM query of the constructed performative instance yields T(x) exactly, so the query counts above are simultaneously ERM query counts.",
    "max_error_recovering_T_from_ERM": 1.1102230246251565e-15
  },
  "proposition_3_2": [
    {
      "d": 2,
      "empirical_rho": 0.9651904169835266,
      "eps": 0.08333333333333333,
      "lambda": 1.0,
      "max_T_recovery_error_from_one_ERM_query": 0.0,
      "max_fixed_point_gap_identity_error": 0.0,
      "rho_bound_1_plus_lambda_L": 2.0,
      "rho_within_bound": true
    },
    {
      "d": 2,
      "empirical_rho": 0.7667395491970505,
      "eps": 0.041666666666666664,
      "lambda": 0.5,
      "max_T_recovery_error_from_one_ERM_query": 1.1102230246251565e-16,
      "max_fixed_point_gap_identity_error": 1.1102230246251565e-16,
      "rho_bound_1_plus_lambda_L": 1.5,
      "rho_within_bound": true
    },
    {
      "d": 2,
      "empirical_rho": 0.9623367010017474,
      "eps": 0.008333333333333333,
      "lambda": 0.1,
      "max_T_recovery_error_from_one_ERM_query": 6.106226635438361e-16,
      "max_fixed_point_gap_identity_error": 1.249000902703301e-16,
      "rho_bound_1_plus_lambda_L": 1.1,
      "rho_within_bound": true
    },
    {
      "d": 2,
      "empirical_rho": 0.9987650411022788,
      "eps": 0.0008333333333333333,
      "lambda": 0.01,
      "max_T_recovery_error_from_one_ERM_query": 5.551115123125783e-15,
      "max_fixed_point_gap_identity_error": 1.4051260155412137e-16,
      "rho_bound_1_plus_lambda_L": 1.01,
      "rho_within_bound": true
    },
    {
      "d": 5,
      "empirical_rho": 0.8726205536485927,
      "eps": 0.08333333333333333,
      "lambda": 1.0,
      "max_T_recovery_error_from_one_ERM_query": 0.0,
      "max_fixed_point_gap_identity_error": 0.0,
      "rho_bound_1_plus_lambda_L": 2.0,
      "rho_within_bound": true
    },
    {
      "d": 5,
      "empirical_rho": 0.8257986738803997,
      "eps": 0.041666666666666664,
      "lambda": 0.5,
      "max_T_recovery_error_from_one_ERM_query": 1.1102230246251565e-16,
      "max_fixed_point_gap_identity_error": 1.1102230246251565e-16,
      "rho_bound_1_plus_lambda_L": 1.5,
      "rho_within_bound": true
    },
    {
      "d": 5,
      "empirical_rho": 0.9606677098850223,
      "eps": 0.008333333333333333,
      "lambda": 0.1,
      "max_T_recovery_error_from_one_ERM_query": 6.106226635438361e-16,
      "max_fixed_point_gap_identity_error": 1.3877787807814457e-16,
      "rho_bound_1_plus_lambda_L": 1.1,
      "rho_within_bound": true
    },
    {
      "d": 5,
      "empirical_rho": 0.9976924785537874,
      "eps": 0.0008333333333333333,
      "lambda": 0.01,
      "max_T_recovery_error_from_one_ERM_query": 5.551115123125783e-15,
      "max_fixed_point_gap_identity_error": 1.1102230246251565e-16,
      "rho_bound_1_plus_lambda_L": 1.01,
      "rho_within_bound": true
    },
    {
      "d": 20,
      "empirical_rho": 0.8602434218272195,
      "eps": 0.08333333333333333,
      "lambda": 1.0,
      "max_T_recovery_error_from_one_ERM_query": 0.0,
      "max_fixed_point_gap_identity_error": 0.0,
      "rho_bound_1_plus_lambda_L": 2.0,
      "rho_within_bound": true
    },
    {
      "d": 20,
      "empirical_rho": 0.6776069518694584,
      "eps": 0.041666666666666664,
      "lambda": 0.5,
      "max_T_recovery_error_from_one_ERM_query": 5.551115123125783e-17,
      "max_fixed_point_gap_identity_error": 1.1102230246251565e-16,
      "rho_bound_1_plus_lambda_L": 1.5,
      "rho_within_bound": true
    },
    {
      "d": 20,
      "empirical_rho": 0.9454915757570909,
      "eps": 0.008333333333333333,
      "lambda": 0.1,
      "max_T_recovery_error_from_one_ERM_query": 3.3306690738754696e-16,
      "max_fixed_point_gap_identity_error": 8.326672684688674e-17,
      "rho_bound_1_plus_lambda_L": 1.1,
      "rho_within_bound": true
    },
    {
      "d": 20,
      "empirical_rho": 0.993858969845494,
      "eps": 0.0008333333333333333,
      "lambda": 0.01,
      "max_T_recovery_error_from_one_ERM_query": 3.941291737419306e-15,
      "max_fixed_point_gap_identity_error": 6.245004513516506e-17,
      "rho_bound_1_plus_lambda_L": 1.01,
      "rho_within_bound": true
    }
  ],
  "query_equivalence": {
    "max_recovery_error": 5.551115123125783e-15,
    "statement": "G(x) = (1-lambda)x + lambda T(x) with lambda and x known, so T(x) = (G(x) - (1-lambda)x)/lambda: the ERM oracle and the T-oracle are information-theoretically equivalent, and the Hirsch et al. bound transfers with no loss."
  },
  "seed": 20260725,
  "theorem_3_6_arithmetic": {
    "bound": "c * ((1/eps - 10) L)^(d-2)",
    "conclusion": "for any constant eps <= 1/12 the bound is at least c*2^(d-2) = 2^Omega(d); for eps > 1/11 the base drops below 1 and the bound is vacuous, so 'even when eps is a constant' means a constant below 1/11.",
    "eps_threshold_for_base_1": "[1/11]",
    "eps_threshold_for_base_2": "[1/12]",
    "rows": [
      {
        "base_at_L_1": "1",
        "base_float": 1.0,
        "base_ge_2_gives_2_to_the_d": false,
        "base_gt_1_nontrivial": false,
        "eps": "1/11"
      },
      {
        "base_at_L_1": "2",
        "base_float": 2.0,
        "base_ge_2_gives_2_to_the_d": true,
        "base_gt_1_nontrivial": true,
        "eps": "1/12"
      },
      {
        "base_at_L_1": "10",
        "base_float": 10.0,
        "base_ge_2_gives_2_to_the_d": true,
        "base_gt_1_nontrivial": true,
        "eps": "1/20"
      },
      {
        "base_at_L_1": "90",
        "base_float": 90.0,
        "base_ge_2_gives_2_to_the_d": true,
        "base_gt_1_nontrivial": true,
        "eps": "1/100"
      },
      {
        "base_at_L_1": "0",
        "base_float": 0.0,
        "base_ge_2_gives_2_to_the_d": false,
        "base_gt_1_nontrivial": false,
        "eps": "1/10"
      }
    ]
  },
  "toy_hidden_path": {
    "R2": 0.9975602300281715,
    "blocker": "the true Hirsch-Papadimitriou-Vavasis instance is a continuous Brouwer map on [0,1]^d whose construction we do not implement; this is a discrete hidden-path family with the same information-hiding mechanism, run at m=3 and d<=9.",
    "implied_base": 1.737024441318484,
    "label": "toy",
    "log_queries_slope_per_dimension": 0.5521735581550697,
    "measurements": [
      {
        "cells": 9,
        "d": 2,
        "grid_side": 3,
        "mean_queries": 4.2,
        "median_queries": 4.0,
        "path_length": 3,
        "trials": 40
      },
      {
        "cells": 27,
        "d": 3,
        "grid_side": 3,
        "mean_queries": 6.925,
        "median_queries": 6.0,
        "path_length": 5,
        "trials": 40
      },
      {
        "cells": 81,
        "d": 4,
        "grid_side": 3,
        "mean_queries": 12.875,
        "median_queries": 10.0,
        "path_length": 9,
        "trials": 40
      },
      {
        "cells": 243,
        "d": 5,
        "grid_side": 3,
        "mean_queries": 21.25,
        "median_queries": 19.0,
        "path_length": 16,
        "trials": 40
      },
      {
        "cells": 729,
        "d": 6,
        "grid_side": 3,
        "mean_queries": 33.075,
        "median_queries": 32.5,
        "path_length": 27,
        "trials": 40
      },
      {
        "cells": 2187,
        "d": 7,
        "grid_side": 3,
        "mean_queries": 58.725,
        "median_queries": 58.0,
        "path_length": 47,
        "trials": 40
      },
      {
        "cells": 6561,
        "d": 8,
        "grid_side": 3,
        "mean_queries": 122.2,
        "median_queries": 92.5,
        "path_length": 81,
        "trials": 40
      },
      {
        "cells": 19683,
        "d": 9,
        "grid_side": 3,
        "mean_queries": 199.05,
        "median_queries": 186.5,
        "path_length": 140,
        "trials": 40
      }
    ],
    "predicted_base_sqrt_m": 1.7320508075688772,
    "predicted_slope_half_log_m": 0.5493061443340549
  },
  "verdict": {
    "empirical_family": "toy",
    "exponential_arithmetic_verified": true,
    "reduction_and_query_equivalence_verified": true
  }
}