{
  "plans": [
    {
      "status": "complete",
      "n": 32,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 793,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          4118,
          4119
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "8/25",
        "time_factor_if_at_most_2_to_60": 1.2483305489016119,
        "space_vs_quarter_exponent_log2_factor": "8/5",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        384,
        408
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 33,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 249,
          "per_call_error_bound": "23250276051918256523088869753771339503260149185929629295596596957269153206899718307694197164670614352103013171/784706782373648623826199635272746945272676651642585750769726974941918703843510959491279949161485148051119497030990643",
          "conditional_all_call_error_bound": "255753036571100821753977567291484734535861641045225922251562566529960685275896901384636168811376757873133144881/261568927457882874608733211757582315090892217214195250256575658313972901281170319830426649720495049350373165676996881",
          "maximum_base_oracle_calls": 8217
        },
        "low_space_one_sided_OR": {
          "repetitions": 16,
          "conditional_all_call_error_bound": "11/14348907",
          "maximum_base_oracle_calls": 528
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 64,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 842,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          4386,
          4387
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "16/25",
        "time_factor_if_at_most_2_to_60": 1.5583291593209998,
        "space_vs_quarter_exponent_log2_factor": "16/5",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        512,
        540
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 65,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 261,
          "per_call_error_bound": "17884477477710750259813284894211970484787200011336991590978213573351499193262983528204923556005736520397597217166345/1251076071394302594884458081106951728027969657276762241924443419870324653667897923469015922386986485696304989864940094919689",
          "conditional_all_call_error_bound": "1162491036051198766887863518123778081511168000736904453413583882267847447562093929333320031140372873825843819115812425/1251076071394302594884458081106951728027969657276762241924443419870324653667897923469015922386986485696304989864940094919689",
          "maximum_base_oracle_calls": 16965
        },
        "low_space_one_sided_OR": {
          "repetitions": 17,
          "conditional_all_call_error_bound": "65/129140163",
          "maximum_base_oracle_calls": 1105
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 100,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 885,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          4623,
          4624
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "1",
        "time_factor_if_at_most_2_to_60": 2.0,
        "space_vs_quarter_exponent_log2_factor": "5",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        656,
        684
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 101,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 269,
          "per_call_error_bound": "8024436522478053132794521189676793846024156051935826413174009134924703044297918148084107975380131744771941569864240545/912034456046446591670769941126967809732389880154759674362919253085466672523897586208912607420113148072606337611541329196453281",
          "conditional_all_call_error_bound": "810468088770283366412246640157356178448439761245518467730574922627395007474089732956494905513393306221966098556288295045/912034456046446591670769941126967809732389880154759674362919253085466672523897586208912607420113148072606337611541329196453281",
          "maximum_base_oracle_calls": 27169
        },
        "low_space_one_sided_OR": {
          "repetitions": 17,
          "conditional_all_call_error_bound": "101/129140163",
          "maximum_base_oracle_calls": 1717
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 200,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 966,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          5095,
          5096
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "2",
        "time_factor_if_at_most_2_to_60": 4.0,
        "space_vs_quarter_exponent_log2_factor": "10",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        1056,
        1088
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 201,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 279,
          "per_call_error_bound": "6978219222085346956560174257054094828787973706555532396444517042413014371631758224153733530411182860573959058949364271476403/1454077510067338869372316944847370699315973030897976908309312512336980481738317971337352174999857054574561953999845406588476984323763",
          "conditional_all_call_error_bound": "467540687879718246089531675222624353528794238339220670561782641841671962899327801018300146537549251658455256949607406188919001/484692503355779623124105648282456899771991010299325636103104170778993493912772657112450724999952351524853984666615135529492328107921",
          "maximum_base_oracle_calls": 56079
        },
        "low_space_one_sided_OR": {
          "repetitions": 18,
          "conditional_all_call_error_bound": "67/129140163",
          "maximum_base_oracle_calls": 3618
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 599999,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 2303,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          311901,
          311902
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "599999/100",
        "time_factor_if_at_most_2_to_60": null,
        "space_vs_quarter_exponent_log2_factor": "599999/20",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        2400252,
        2400332
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 600000,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 413,
          "per_call_error_bound": "688942456096409680065411246748286143813638984200102370034691290827786805224776748657964634481570616869014037256014508594058387009867464475847098194955486422354664293665477568697755361/462883740318876369999478085809708197976508657409497578720198095475968543303935631001316777651702647211654642268789160137758926372344240982080285899106205821360447432009191414855620381954951374561",
          "conditional_all_call_error_bound": "137788491219281936013082249349657228762727796840020474006938258165557361044955349731592926896314123373802807451202901718811677401973492895169419638991097284470932858733095513739551072200000/154294580106292123333159361936569399325502885803165859573399365158656181101311877000438925883900882403884880756263053379252975457448080327360095299702068607120149144003063804951873460651650458187",
          "maximum_base_oracle_calls": 247800000
        },
        "low_space_one_sided_OR": {
          "repetitions": 25,
          "conditional_all_call_error_bound": "200000/282429536481",
          "maximum_base_oracle_calls": 15000000
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 600000,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": true,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "unknown",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 2303,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          311901,
          311902
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "6000",
        "time_factor_if_at_most_2_to_60": null,
        "space_vs_quarter_exponent_log2_factor": "30000",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        2400256,
        2400336
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 600001,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 413,
          "per_call_error_bound": "688942456096409680065411246748286143813638984200102370034691290827786805224776748657964634481570616869014037256014508594058387009867464475847098194955486422354664293665477568697755361/462883740318876369999478085809708197976508657409497578720198095475968543303935631001316777651702647211654642268789160137758926372344240982080285899106205821360447432009191414855620381954951374561",
          "conditional_all_call_error_bound": "413366162600301904448926813460218434574327204159045622123184809187962910921671273971527438653576851692025291367645961170943626264307488552972734764071486808899220930863580206696221914355361/462883740318876369999478085809708197976508657409497578720198095475968543303935631001316777651702647211654642268789160137758926372344240982080285899106205821360447432009191414855620381954951374561",
          "maximum_base_oracle_calls": 247800413
        },
        "low_space_one_sided_OR": {
          "repetitions": 25,
          "conditional_all_call_error_bound": "600001/847288609443",
          "maximum_base_oracle_calls": 15000025
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 35999999999,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": true,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "unknown",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": false,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "known_guard_failed",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 4208,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          18000021742,
          18000021743
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "35999999999/100",
        "time_factor_if_at_most_2_to_60": null,
        "space_vs_quarter_exponent_log2_factor": "35999999999/20",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        144000000252,
        144000000396
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 36000000000,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 595,
          "per_call_error_bound": "709858201223764251586849278882151652810779971026475903688609210034897833574806683653311517626910814488789132373418832745083287588652882607043276336905824495773027864920350684494853967667822153316393579616925395096904569252413444325350893714080458196134806254909627977/25705455471670699433133086310504970403715321030168424633699464997547202967294600382340356864840920242534609860043703580766898611376020646207324507842758072816714600293371782640955963124826622986068174849236342614311579299762272094845817375769121376914784382735886412671731445651447369",
          "conditional_all_call_error_bound": "2839432804895057006347397115528606611243119884105903614754436840139591334299226734613246070507643257955156529493675330980333150354611530428173105347623297983092111459681402737979415870671288613265574318467701580387618277009653777301403574856321832784539225019638511908000000000/2856161719074522159237009590056107822635035670018713848188829444171911440810511153593372984982324471392734428893744842307433179041780071800813834204750896979634955588152420293439551458314069220674241649915149179367953255529141343871757486196569041879420486970654045852414605072383041",
          "maximum_base_oracle_calls": 21420000000000
        },
        "low_space_one_sided_OR": {
          "repetitions": 35,
          "conditional_all_call_error_bound": "4000000000/5559060566555523",
          "maximum_base_oracle_calls": 1260000000000
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 36000000000,
      "maximum_input_bit_length": 64,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": true,
        "minimum_n_for_this_known_guard": 600000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "unknown",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": true,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "unknown",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 4208,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          18000021742,
          18000021743
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "360000000",
        "time_factor_if_at_most_2_to_60": null,
        "space_vs_quarter_exponent_log2_factor": "1800000000",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        144000000256,
        144000000400
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 36000000001,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 595,
          "per_call_error_bound": "709858201223764251586849278882151652810779971026475903688609210034897833574806683653311517626910814488789132373418832745083287588652882607043276336905824495773027864920350684494853967667822153316393579616925395096904569252413444325350893714080458196134806254909627977/25705455471670699433133086310504970403715321030168424633699464997547202967294600382340356864840920242534609860043703580766898611376020646207324507842758072816714600293371782640955963124826622986068174849236342614311579299762272094845817375769121376914784382735886412671731445651447369",
          "conditional_all_call_error_bound": "25554895244765371258350338291344308780070230609763912503816407464944931218727938445094021318222100839223319579931867111196417185936587061442210830735652958184734827632905652506735093520536451487057991019525707803105489888183788564965045618032247388774933483372881413426909627977/25705455471670699433133086310504970403715321030168424633699464997547202967294600382340356864840920242534609860043703580766898611376020646207324507842758072816714600293371782640955963124826622986068174849236342614311579299762272094845817375769121376914784382735886412671731445651447369",
          "maximum_base_oracle_calls": 21420000000595
        },
        "low_space_one_sided_OR": {
          "repetitions": 35,
          "conditional_all_call_error_bound": "36000000001/50031545098999707",
          "maximum_base_oracle_calls": 1260000000035
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    },
    {
      "status": "complete",
      "n": 1000000000000,
      "maximum_input_bit_length": 4096,
      "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "fast_decision": {
        "known_guard": "b^100000 <= 2^n",
        "known_guard_passes": true,
        "minimum_n_for_this_known_guard": 1200000,
        "fixed_source_cutoff_n_star": "effective but no evaluated numerical value recorded",
        "main_branch_eligibility": "unknown",
        "fallback": "exact meet-in-the-middle",
        "error_model": "two_sided_at_most_one_third_per_fixed_input",
        "main_time_formula": "K0*2^(0.489995*n)*(n+b+2)^K; K0,K unselected here",
        "ordinary_time_claim": "O_c(2^(0.49*n)) for every fixed polynomial-bit class, eventually"
      },
      "low_space_decision": {
        "known_guard": "n+b+2 <= 2^floor(n/1000000000)",
        "known_guard_passes": true,
        "minimum_n_for_this_known_guard": 36000000000,
        "fixed_source_cutoff_n_zero": "no evaluated numerical value recorded",
        "main_branch_eligibility": "unknown",
        "fallback": "exact indexed subset enumeration with O(n) writable words",
        "error_model": "no_false_YES; each_YES_found_with_probability_at_least_two_thirds",
        "rounds_formula": "(n+b+2)^120",
        "rounds_floor_log2": 4783,
        "total_trial_cap_formula": "2*(n+b+2)^620*2^ceil(n/2)",
        "total_trial_cap_log2_interval": [
          500000024716,
          500000024717
        ],
        "space_formula": "A_c*(n+b+2)^15*2^(0.199*n) writable words",
        "ordinary_space_claim": "O_c(2^(n/5)); polynomial factor remains at exponent 0.199"
      },
      "ideal_exponent_only_comparison": {
        "time_vs_half_exponent_log2_factor": "10000000000",
        "time_factor_if_at_most_2_to_60": null,
        "space_vs_quarter_exponent_log2_factor": "50000000000",
        "ignored_factors": "All constants, polynomial overhead, thresholds, arithmetic and actual implementation; neither comparison is a measured speedup."
      },
      "word_width_enclosure_bits": [
        4000000016384,
        4000000016544
      ],
      "conditional_witness_plan": {
        "maximum_adaptive_decision_questions": 1000000000001,
        "error_budget": "1/1000000",
        "fast_two_sided_majority": {
          "status": "complete",
          "repetitions": 651,
          "per_call_error_bound": "39427201715520624899868116110309183751163039935157671296041563453941879522085076974413297372349257819495572810745179859318010215255376866449321781999358362821445486694057627615452753621925130366584963041495861663326514960384513779351230153346098324426698926407830707054053502727268983396944379/40358667830967222524422656041993338729632946355906157332566651747227817102094523924973377132281452181359887366411681049282574198438282321958668433640196569701709615030534041108081870611585956649224590904724957046932925439860985375673614273855486128268241780068361316594507421380121649747437639522939549819914747",
          "conditional_all_call_error_bound": "39427201715560052101583636735209051867273349118908834335976721125237921085539018853935382449323671116867922068564675432128755395114694876664577158865807684603444845056879073102146811249540583120206888171862446626368010822047840294311614667125449554580045024732257405980461333434323036899671647983396944379/40358667830967222524422656041993338729632946355906157332566651747227817102094523924973377132281452181359887366411681049282574198438282321958668433640196569701709615030534041108081870611585956649224590904724957046932925439860985375673614273855486128268241780068361316594507421380121649747437639522939549819914747",
          "maximum_base_oracle_calls": 651000000000651
        },
        "low_space_one_sided_OR": {
          "repetitions": 38,
          "conditional_all_call_error_bound": "1000000000001/1350851717672992089",
          "maximum_base_oracle_calls": 38000000000038
        },
        "assumptions": "Fresh independent runs conditional on each adaptive history; correct source/input/machine correspondence and proved base error <=1/3 are prerequisites, not checked by this planner.",
        "final_witness_check": "Check distinct original item IDs and exact integer sum. Invalid recovered output is unknown, never an exact negative certificate.",
        "negative_answer": "Amplified randomized NO remains probabilistic unless separately established by a complete exact method."
      },
      "source_algorithms_implemented": false,
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Exact bounded guard and error-budget algebra for supplied n,b. Displayed caps are permitted upper schedules, not runtime lower bounds. Unknown source cutoffs prevent an implementation-ready main-branch badge. Separate faster-time and lower-space results cannot be combined into one backend guarantee."
    }
  ],
  "source_program_execution": "not_run",
  "interpretation": "Exact supplied-parameter guard/error algebra; all main cutoffs and actual source implementations remain unvalidated."
}