{ "version": "20261029-binomial-log-v1", "generated_display": "29/10/2026", "inputs": { "path": "data/features-2026.npz", "sha256": "5350c7e67dfec5f9358a4e6ab353a81a8031891d36d001212f7b4cecf69a4c7c", "post_hash_unchanged": true }, "observed": { "sections": 497897, "states": 27, "all_paired": true, "first_turn": { "lula": 53722151, "flavio": 55960603, "p_lula": 0.4897957886797773, "universe": "Brazil domestic; exterior excluded" }, "exclusive_bands": { "definition": [ "[70%,80%)", "[80%,90%)", "[90%,100%)", "100%" ], "lula": [ 51578, 29070, 6779, 106 ], "flavio": [ 37745, 6850, 370, 4 ], "ratios_lula_flavio": [ 1.3664856272353954, 4.2437956204379566, 18.32162162162162, 26.5 ] }, "flips": { "f_to_l": 31341, "l_to_f": 555 }, "growth": { "sections": 380771, "lula_net": 8886745, "flavio_net": 3546231, "turnout_delta": 3582746, "valid2": 97390626, "lula2": 50072520, "lula1": 41185775, "others1": 7417084, "blank_reduction": 712032, "null_reduction": 721114, "retention_incompatible": 34618 } }, "model": { "id": "homogeneous_first_turn_finalists", "formula": "independent X_s ~ Bin(valid2_s, sum(L1)/sum(L1+F1))", "assumptions": [ "identical national probability at every section", "independent ballots and sections", "second-turn valid totals held fixed" ], "interpretation": "conditional hypothetical model; not probability of fraud or of a legitimate election", "geographic_limitation": "ignores real territorial preferences; ordinary geographic concentration can reject this null", "joint_event": "four exclusive Lula counts>=observed AND F_to_L>=31341 AND L_to_F<=555 AND sum_growth X>=50072520", "growth_constant": 41185775, "growth_scale": 100, "constraints": 7, "conceptual_factors": 6 }, "bounds": [ { "method": "finite_support_multivariate_chernoff_upper_bound", "indices": [ 0 ], "features": [ "lula_70_80" ], "targets": [ 51578.0 ], "log_bound_natural": -792173.511539377, "theta": [ 18.64844098218529 ], "expected_null": [ 7.593065558724618 ], "tilted_means": [ 51578.0000000011 ], "log_mgf": 169675.7774397757, "KKT": { "projected_gradient_inf": 1.0986695997416973e-09, "relative_projected_gradient_inf": 2.130112838306443e-14, "relative_tolerance": 1e-08 }, "iterations": 6, "evaluations": 13, "status": "converged", "groups": 2005, "support_size": 430801, "independent_units": 497897, "null_log_mgf_numerical": 1.4206635867708428e-11, "numerics": "float64; numerical bound, not certified interval arithmetic", "id": "band70_80", "observed": 51578, "scale": { "negative_log10": 344036.5847714734, "probability_type": "chernoff_upper_bound", "public_exponent": 344035, "mega_sena_equivalent": 44682.63040615063, "heads_equivalent": 1142859.5321245736, "analogy_relation": "at_least_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "one decimal order slack" } }, { "method": "finite_support_multivariate_chernoff_upper_bound", "indices": [ 1 ], "features": [ "lula_80_90" ], "targets": [ 29070.0 ], "log_bound_natural": -834198.6261362422, "theta": [ 35.697821618234315 ], "expected_null": [ 0.6677536243128859 ], "tilted_means": [ 29070.00000507039 ], "log_mgf": 203537.04830582935, "KKT": { "projected_gradient_inf": 5.07038930663839e-06, "relative_projected_gradient_inf": 1.744199967883863e-10, "relative_tolerance": 1e-08 }, "iterations": 7, "evaluations": 15, "status": "converged", "groups": 2005, "support_size": 430801, "independent_units": 497897, "null_log_mgf_numerical": 1.4206635867708428e-11, "numerics": "float64; numerical bound, not certified interval arithmetic", "id": "band80_90", "observed": 29070, "scale": { "negative_log10": 362287.86014224374, "probability_type": "chernoff_upper_bound", "public_exponent": 362286, "mega_sena_equivalent": 47053.03657861173, "heads_equivalent": 1203488.0417843629, "analogy_relation": "at_least_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "one decimal order slack" } }, { "method": "finite_support_multivariate_chernoff_upper_bound", "indices": [ 2 ], "features": [ "lula_90_100" ], "targets": [ 6779.0 ], "log_bound_natural": -227102.81921797944, "theta": [ 44.716753596167464 ], "expected_null": [ 0.05742202368887793 ], "tilted_means": [ 6779.000000000103 ], "log_mgf": 76032.0534104398, "KKT": { "projected_gradient_inf": 1.0277290130034089e-10, "relative_projected_gradient_inf": 1.516048108870643e-14, "relative_tolerance": 1e-08 }, "iterations": 7, "evaluations": 15, "status": "converged", "groups": 2005, "support_size": 430801, "independent_units": 497897, "null_log_mgf_numerical": 1.4206635867708428e-11, "numerics": "float64; numerical bound, not certified interval arithmetic", "id": "band90_100", "observed": 6779, "scale": { "negative_log10": 98629.50121104023, "probability_type": "chernoff_upper_bound", "public_exponent": 98628, "mega_sena_equivalent": 12809.622485205937, "heads_equivalent": 327635.12414255075, "analogy_relation": "at_least_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "one decimal order slack" } }, { "method": "finite_support_multivariate_chernoff_upper_bound", "indices": [ 3 ], "features": [ "lula_100" ], "targets": [ 106.0 ], "log_bound_natural": -1364.221019080059, "theta": [ 17.332140021745342 ], "expected_null": [ 0.006336788690288073 ], "tilted_means": [ 106.0000000000005 ], "log_mgf": 472.98582322494735, "KKT": { "projected_gradient_inf": 4.973799150320701e-13, "relative_projected_gradient_inf": 4.692263349359152e-15, "relative_tolerance": 1e-08 }, "iterations": 6, "evaluations": 13, "status": "converged", "groups": 2005, "support_size": 430801, "independent_units": 497897, "null_log_mgf_numerical": 1.4206635867708428e-11, "numerics": "float64; numerical bound, not certified interval arithmetic", "id": "unanimous", "observed": 106, "scale": { "negative_log10": 592.4736606829005, "probability_type": "chernoff_upper_bound", "public_exponent": 591, "mega_sena_equivalent": 76.75798848964502, "heads_equivalent": 1963.259504078431, "analogy_relation": "at_least_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "one decimal order slack" } }, { "method": "finite_support_multivariate_chernoff_upper_bound", "indices": [ 4, 5 ], "features": [ "flavio_to_lula", "negative_lula_to_flavio" ], "targets": [ 31341.0, -555.0 ], "log_bound_natural": -202080.26008867373, "theta": [ 0.0, 6.4285235993018395 ], "expected_null": [ 100635.60504694306, -133105.52636579386 ], "tilted_means": [ 100635.60504694306, -555.0000000001329 ], "log_mgf": -205648.09068628625, "KKT": { "projected_gradient_inf": 1.3289991329656914e-10, "relative_projected_gradient_inf": 2.394593032370615e-13, "relative_tolerance": 1e-08 }, "iterations": 10, "evaluations": 21, "status": "converged", "groups": 2005, "support_size": 430801, "independent_units": 497897, "null_log_mgf_numerical": 1.4206635867708428e-11, "numerics": "float64; numerical bound, not certified interval arithmetic", "id": "flips_both_directions", "scale": { "negative_log10": 87762.34185808492, "probability_type": "chernoff_upper_bound", "public_exponent": 87761, "mega_sena_equivalent": 11398.236595329503, "heads_equivalent": 291535.7315354098, "analogy_relation": "at_least_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "one decimal order slack" } }, { "method": "finite_support_multivariate_chernoff_upper_bound", "indices": [ 0, 1, 2, 3, 4, 5, 6 ], "features": [ "lula_70_80", "lula_80_90", "lula_90_100", "lula_100", "flavio_to_lula", "negative_lula_to_flavio", "growth_lula_div_100" ], "targets": [ 51578.0, 29070.0, 6779.0, 106.0, 31341.0, -555.0, 500725.2 ], "log_bound_natural": -2312334.693980583, "theta": [ 20.159718181790144, 39.48584925729304, 59.19514954981529, 64.73489390886765, 0.0, 6.194121043643355, 0.0 ], "expected_null": [ 7.593065558724616, 0.667753624312885, 0.05742202368887793, 0.006336788690288073, 100635.60504694303, -133105.52636579375, 477015.1847168718 ], "tilted_means": [ 51577.99999771783, 29069.999999917745, 6778.999999999999, 106.00000000000011, 123658.59848772461, -555.0000413867256, 510459.36201712594 ], "log_mgf": 280024.96868261416, "KKT": { "projected_gradient_inf": 4.138672557019163e-05, "relative_projected_gradient_inf": 7.457067670304798e-08, "relative_tolerance": 1e-08 }, "iterations": 58, "evaluations": 589, "status": "line_search_stalled", "groups": 2005, "support_size": 430801, "independent_units": 497897, "null_log_mgf_numerical": 1.4206635867708428e-11, "numerics": "float64; numerical bound, not certified interval arithmetic", "id": "joint_six_factors", "scale": { "negative_log10": 1004234.1979092115, "probability_type": "chernoff_upper_bound", "public_exponent": 1004233, "mega_sena_equivalent": 130427.92733489285, "heads_equivalent": 3335989.8165130205, "analogy_relation": "at_least_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "one decimal order slack" } } ], "tails": [ { "log_probability": -115469.6370553916, "terms": 308, "omitted_relative_bound": 9.340418018605319e-14, "rounding": "double precision lgamma, not interval arithmetic", "id": "full_second_turn_growth_subset", "n": 97390626, "k": 50072520, "p": 0.4897957886797773, "explanation": "Main full-vote model. Gain threshold equals L2 >= L1+8886745; no retention assumption.", "scale": { "negative_log10": 50147.82620052782, "probability_type": "analytic_binomial_tail", "public_exponent": 50147.82620052782, "mega_sena_equivalent": 6513.107049544557, "heads_equivalent": 166587.47275306194, "analogy_relation": "same_logarithmic_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "approximate exponent; numerical rounding remains" } }, { "log_probability": -1296763.7703292575, "terms": 32, "omitted_relative_bound": 4.627646492975412e-14, "rounding": "double precision lgamma, not interval arithmetic", "id": "grok_style_net_gains", "n": 12432976, "k": 8886745, "p": 0.4897957886797773, "explanation": "Counterfactual: treats net gains as independent new choices; not identified ballots. Retention contradicted in 34618 sections.", "scale": { "negative_log10": 563177.3497860522, "probability_type": "analytic_binomial_tail", "public_exponent": 563177.3497860522, "mega_sena_equivalent": 73144.43406515496, "heads_equivalent": 1870834.660658494, "analogy_relation": "same_logarithmic_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "approximate exponent; numerical rounding remains" } }, { "log_probability": -1076.9014442730156, "terms": 2, "omitted_relative_bound": 0.0, "rounding": "double precision lgamma, not interval arithmetic", "id": "silva_jardim_net_gains", "n": 985, "k": 984, "p": 0.33253624271409354, "explanation": "Counterfactual 985 independent net-gain choices, actual finalist ratio1t; equivalent to P(Flavio<=1).", "scale": { "negative_log10": 467.6923548014129, "probability_type": "analytic_binomial_tail", "public_exponent": 467.6923548014129, "mega_sena_equivalent": 60.743019266568226, "heads_equivalent": 1553.6403731788419, "analogy_relation": "same_logarithmic_scale_under_null", "mega_sena_denominator": 50063860, "public_rounding": "approximate exponent; numerical rounding remains" } } ], "notes": [ "This is analytical binomial/MGF evaluation, not additional Monte Carlo draws.", "Exclusive section categories remain dependent; joint MGF retains dependence.", "Chernoff upper bounds are not exact joint probabilities. Display adds one decimal order slack, not interval arithmetic certification.", "First-turn finalist proportion is an assumption for second-turn choices, not an identified migration probability.", "Neither these conditional probabilities nor the analogies measure fraud probability." ] }