{
"certificate_version": 1,
"self_test": {
"passed": true,
"checks": [
"literal/full-state = count Bellman; floor bound, K=1",
"literal/full-state = count Bellman; floor bound, K=1",
"literal/full-state = count Bellman; floor bound, K=2",
"literal/full-state = count Bellman; floor bound, K=2",
"literal/full-state = count Bellman; floor bound, K=3",
"literal/full-state = count Bellman; floor bound, K=3",
"n=1 randomized mixture has certified TV <= 1e-20",
"rational logarithm intervals and reciprocal identity"
]
},
"witnesses": [
{
"id": "n8_B1_50_single_d1_4",
"n": 8,
"tau": "1/4",
"weights": [
"7477325/8388608",
"911283/8388608"
],
"probabilities": [
"1/4",
"1/2"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "31892171151/549755813888",
"float": 0.05801152137973986
},
"kl_lower": {
"rational": "27222543162586797511423518666869436249/1361129467683753853853498429727072845824",
"float": 0.019999966064147917
},
"kl_upper": {
"rational": "1742242762405555040731105194679648179555/87112285931760246646623899502532662132736",
"float": 0.019999966064147917
},
"kl_enclosure_width": {
"rational": "4259619/87112285931760246646623899502532662132736",
"float": 4.88980280386258e-35
},
"kl_budget": {
"rational": "1/50",
"float": 0.02
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "29161056/124121549",
"float": 0.23493951078551234
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 36
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.06857508646216329,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "107211178603/274877906944",
"float": 0.3900319956410385
},
"signed_test_gap_lower": {
"rational": "18849776235/274877906944",
"float": 0.06857508646135102
},
"tv_lower_certified": {
"rational": "18849776235/274877906944",
"float": 0.06857508646135102
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 175,
"bellman_action_count": 201,
"execution_gap_lower_certified": {
"rational": "5807381319/549755813888",
"float": 0.01056356508161116
},
"tv_upper_certified": {
"rational": "20725548151603712731371/302231454903657293676544",
"float": 0.06857508646216331
},
"policy_tv_lower_certified": {
"rational": "82902192606414850925481/1208925819614629174706176",
"float": 0.06857508646216331
},
"terminal_subprobability": [
"90104193325519206878433/1208925819614629174706176",
"138194192431059592902769/604462909807314587353088",
"370913491419736807202449/1208925819614629174706176",
"142218937090295279373491/604462909807314587353088",
"34081778054354936462929/302231454903657293676544",
"20908699677586509473317/604462909807314587353088",
"4008499438379181885411/604462909807314587353088",
"439135534019177312623/604462909807314587353088",
"42094309273932878353/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "3/1208925819614629174706176",
"float": 2.481541837659083e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 10,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "245502028011/302231454903657293676544",
"float": 8.122980716525982e-13
},
"elapsed_seconds": 0.0011809607967734337
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.06259081240102633,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "211132467749/549755813888",
"float": 0.3840477215799183
},
"signed_test_gap_lower": {
"rational": "34409663013/549755813888",
"float": 0.06259081240023079
},
"tv_lower_certified": {
"rational": "34409663013/549755813888",
"float": 0.06259081240023079
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 340,
"bellman_action_count": 3195,
"execution_gap_lower_certified": {
"rational": "1258745931/274877906944",
"float": 0.004579291020490928
},
"tv_upper_certified": {
"rational": "75667649182256060543797/1208925819614629174706176",
"float": 0.06259081240102617
},
"policy_tv_lower_certified": {
"rational": "75667649182256060543783/1208925819614629174706176",
"float": 0.06259081240102617
},
"terminal_subprobability": [
"99065223477741488470205/1208925819614629174706176",
"142694174226927215420171/604462909807314587353088",
"45023380137492258869695/151115727451828646838272",
"267362971323026055587043/1208925819614629174706176",
"133664857094335964781663/1208925819614629174706176",
"48215900570884532850883/1208925819614629174706176",
"3164729411939274658227/302231454903657293676544",
"275451989188239513583/151115727451828646838272",
"89472016792808238447/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "7/604462909807314587353088",
"float": 1.1580528575742387e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 31,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "961559245621/1208925819614629174706176",
"float": 7.95383165798806e-13
},
"elapsed_seconds": 0.005016230046749115
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.05936763744329711,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "209360508577/549755813888",
"float": 0.3808245466225344
},
"signed_test_gap_lower": {
"rational": "32637703841/549755813888",
"float": 0.0593676374428469
},
"tv_lower_certified": {
"rational": "32637703841/549755813888",
"float": 0.0593676374428469
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 505,
"bellman_action_count": 6189,
"execution_gap_lower_certified": {
"rational": "372766345/274877906944",
"float": 0.001356116063107038
},
"tv_upper_certified": {
"rational": "71771069754721867984653/1208925819614629174706176",
"float": 0.059367637443296914
},
"policy_tv_lower_certified": {
"rational": "17942767438680466996157/302231454903657293676544",
"float": 0.059367637443296914
},
"terminal_subprobability": [
"101193215450337461998277/1208925819614629174706176",
"287757255883880304459605/1208925819614629174706176",
"359586721124850416369369/1208925819614629174706176",
"260967860786898737370765/1208925819614629174706176",
"131243295608074040234171/1208925819614629174706176",
"25192135308212033912905/604462909807314587353088",
"14832482662924525289179/1208925819614629174706176",
"2748606232536243811169/1208925819614629174706176",
"106055624351688673903/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "25/1208925819614629174706176",
"float": 2.0679515313825692e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 52,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "544033009421/1208925819614629174706176",
"float": 4.5001355798192986e-13
},
"elapsed_seconds": 0.007540580816566944
}
],
"transition_kernel_count": 486,
"elapsed_seconds": 0.015316233970224857
},
{
"id": "n8_B1_20_single_d1_4",
"n": 8,
"tau": "1/4",
"weights": [
"3376889/4194304",
"817415/4194304"
],
"probabilities": [
"1/4",
"1/2"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "28607072755/274877906944",
"float": 0.10407192441562074
},
"kl_lower": {
"rational": "4355608445233033786297949466539709005519/87112285931760246646623899502532662132736",
"float": 0.049999932829739045
},
"kl_upper": {
"rational": "4355608445233033786297949466539713352799/87112285931760246646623899502532662132736",
"float": 0.049999932829739045
},
"kl_enclosure_width": {
"rational": "271705/5444517870735015415413993718908291383296",
"float": 4.9904326967214e-35
},
"kl_budget": {
"rational": "1/20",
"float": 0.05
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "26157280/108006009",
"float": 0.24218356221272838
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 36
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.12351995927084782,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "61157155119/137438953472",
"float": 0.44497686844988493
},
"signed_test_gap_lower": {
"rational": "16976453935/137438953472",
"float": 0.12351995927019743
},
"tv_lower_certified": {
"rational": "16976453935/137438953472",
"float": 0.12351995927019743
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 175,
"bellman_action_count": 201,
"execution_gap_lower_certified": {
"rational": "5345835115/274877906944",
"float": 0.01944803485457669
},
"tv_upper_certified": {
"rational": "37331617000068875630031/302231454903657293676544",
"float": 0.12351995927084798
},
"policy_tv_lower_certified": {
"rational": "149326468000275502520119/1208925819614629174706176",
"float": 0.12351995927084798
},
"terminal_subprobability": [
"70716817080678089219941/1208925819614629174706176",
"240984509728919339405567/1208925819614629174706176",
"11227514606372409989571/37778931862957161709568",
"19130224001768604665157/75557863725914323419136",
"1273258302219401430777/9444732965739290427392",
"6942292719279191426119/151115727451828646838272",
"2957190871305580204127/302231454903657293676544",
"719809780671094283567/604462909807314587353088",
"76653887935527839829/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "5/1208925819614629174706176",
"float": 4.1359030627651384e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 10,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "196614632911/302231454903657293676544",
"float": 6.505432499528386e-13
},
"elapsed_seconds": 0.0011199777945876122
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.11316914770479522,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "238938201639/549755813888",
"float": 0.4346260568836442
},
"signed_test_gap_lower": {
"rational": "62215396903/549755813888",
"float": 0.1131691477039567
},
"tv_lower_certified": {
"rational": "62215396903/549755813888",
"float": 0.1131691477039567
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 340,
"bellman_action_count": 3195,
"execution_gap_lower_certified": {
"rational": "5001251393/549755813888",
"float": 0.009097223288335954
},
"tv_upper_certified": {
"rational": "34203276161027158325599/302231454903657293676544",
"float": 0.11316914770479526
},
"policy_tv_lower_certified": {
"rational": "136813104644108633302383/1208925819614629174706176",
"float": 0.11316914770479526
},
"terminal_subprobability": [
"83859004576445417825693/1208925819614629174706176",
"127985286541664821668993/604462909807314587353088",
"343665579909906356345829/1208925819614629174706176",
"139469416985366287246779/604462909807314587353088",
"157404591656010677748733/1208925819614629174706176",
"65522896034426297983407/1208925819614629174706176",
"4890532173105763960627/302231454903657293676544",
"1850474604914739782087/604462909807314587353088",
"301262481525671564275/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "13/1208925819614629174706176",
"float": 1.075334796318936e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 31,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "253438736735/302231454903657293676544",
"float": 8.385584379884913e-13
},
"elapsed_seconds": 0.00508461706340313
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.10733143138973233,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "235728883155/549755813888",
"float": 0.42878834056864434
},
"signed_test_gap_lower": {
"rational": "59006078419/549755813888",
"float": 0.10733143138895684
},
"tv_lower_certified": {
"rational": "59006078419/549755813888",
"float": 0.10733143138895684
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 505,
"bellman_action_count": 6189,
"execution_gap_lower_certified": {
"rational": "1791932909/549755813888",
"float": 0.0032595069733361015
},
"tv_upper_certified": {
"rational": "129755738663243386767121/1208925819614629174706176",
"float": 0.10733143138973224
},
"policy_tv_lower_certified": {
"rational": "129755738663243386767101/1208925819614629174706176",
"float": 0.10733143138973224
},
"terminal_subprobability": [
"88387080350582410564087/1208925819614629174706176",
"16370875935215478358579/75557863725914323419136",
"42528928529564574967929/151115727451828646838272",
"265741419059171645194729/1208925819614629174706176",
"153293259504747566813563/1208925819614629174706176",
"35043274889330590117655/604462909807314587353088",
"23873840448810755952159/1208925819614629174706176",
"4982194588075893216151/1208925819614629174706176",
"396032684615469249461/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "5/302231454903657293676544",
"float": 1.6543612251060553e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 50,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "937397634833/1208925819614629174706176",
"float": 7.753971497869203e-13
},
"elapsed_seconds": 0.007409163750708103
}
],
"transition_kernel_count": 486,
"elapsed_seconds": 0.015387684106826782
},
{
"id": "n8_B1_10_single_d1_4",
"n": 8,
"tau": "1/4",
"weights": [
"11681887/16777216",
"5095329/16777216"
],
"probabilities": [
"1/4",
"1/2"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "178321229013/1099511627776",
"float": 0.16218221300096047
},
"kl_lower": {
"rational": "2177804514094715606365965147713960173491/21778071482940061661655974875633165533184",
"float": 0.09999987904350059
},
"kl_upper": {
"rational": "8711218056378862425463860590855845053641/87112285931760246646623899502532662132736",
"float": 0.09999987904350059
},
"kl_enclosure_width": {
"rational": "4359677/87112285931760246646623899502532662132736",
"float": 5.004663754794783e-35
},
"kl_budget": {
"rational": "1/10",
"float": 0.1
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "163050528/663883999",
"float": 0.2456009306529468
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 36
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.1920175515263912,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "35285696257/68719476736",
"float": 0.5134744607057655
},
"signed_test_gap_lower": {
"rational": "13195345665/68719476736",
"float": 0.19201755152607802
},
"tv_lower_certified": {
"rational": "13195345665/68719476736",
"float": 0.19201755152607802
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 175,
"bellman_action_count": 201,
"execution_gap_lower_certified": {
"rational": "32804301627/1099511627776",
"float": 0.02983533852511755
},
"tv_upper_certified": {
"rational": "232134975859436881173085/1208925819614629174706176",
"float": 0.1920175515263913
},
"policy_tv_lower_certified": {
"rational": "29016871982429610146635/151115727451828646838272",
"float": 0.1920175515263913
},
"terminal_subprobability": [
"51529819230208105163327/1208925819614629174706176",
"49831123082963144704573/302231454903657293676544",
"21082435924518280353575/75557863725914323419136",
"326199152300534579115465/1208925819614629174706176",
"98576843167423946809187/604462909807314587353088",
"76261782065421293387255/1208925819614629174706176",
"18436947312248374846447/1208925819614629174706176",
"1273511981599758465903/604462909807314587353088",
"153941284024347168005/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "5/1208925819614629174706176",
"float": 4.1359030627651384e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 10,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "378718116445/1208925819614629174706176",
"float": 3.1326828354590396e-13
},
"elapsed_seconds": 0.0011385204270482063
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.17691617291856207,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "547966998737/1099511627776",
"float": 0.49837308209771436
},
"signed_test_gap_lower": {
"rational": "194521389265/1099511627776",
"float": 0.17691617291802686
},
"tv_lower_certified": {
"rational": "194521389265/1099511627776",
"float": 0.17691617291802686
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 340,
"bellman_action_count": 3195,
"execution_gap_lower_certified": {
"rational": "4050040063/274877906944",
"float": 0.014733959917066386
},
"tv_upper_certified": {
"rational": "213878529348656087054107/1208925819614629174706176",
"float": 0.17691617291856204
},
"policy_tv_lower_certified": {
"rational": "213878529348656087054093/1208925819614629174706176",
"float": 0.17691617291856204
},
"terminal_subprobability": [
"17736301643095396407103/302231454903657293676544",
"54970838006880813291811/302231454903657293676544",
"315601174265229124962141/1208925819614629174706176",
"287421588478439043524295/1208925819614629174706176",
"187108409945965359397587/1208925819614629174706176",
"22458607804671067465759/302231454903657293676544",
"481443221671493114985/18889465931478580854784",
"6661126504388599190749/1208925819614629174706176",
"329082207521189806829/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "7/604462909807314587353088",
"float": 1.1580528575742387e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 31,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "647004829467/1208925819614629174706176",
"float": 5.351898511632803e-13
},
"elapsed_seconds": 0.00531412661075592
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.1680395090603633,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "67275875451/137438953472",
"float": 0.48949641823855927
},
"signed_test_gap_lower": {
"rational": "23095174267/137438953472",
"float": 0.16803950905887177
},
"tv_lower_certified": {
"rational": "23095174267/137438953472",
"float": 0.16803950905887177
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 505,
"bellman_action_count": 6189,
"execution_gap_lower_certified": {
"rational": "6440165123/1099511627776",
"float": 0.005857296057911299
},
"tv_upper_certified": {
"rational": "203147301218439449942273/1208925819614629174706176",
"float": 0.16803950906036316
},
"policy_tv_lower_certified": {
"rational": "50786825304609862485563/302231454903657293676544",
"float": 0.16803950906036316
},
"terminal_subprobability": [
"18084556722364139041999/302231454903657293676544",
"228332817508176621229871/1208925819614629174706176",
"79122479149429355867941/302231454903657293676544",
"68247621024298060686547/302231454903657293676544",
"179877868852668188884329/1208925819614629174706176",
"23570145559431663149765/302231454903657293676544",
"2232830719162450749379/75557863725914323419136",
"2041464055955342879557/302231454903657293676544",
"724775701270906098655/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "21/1208925819614629174706176",
"float": 1.737079286361358e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 50,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1802988820737/1208925819614629174706176",
"float": 1.4913973971634927e-12
},
"elapsed_seconds": 0.007418840192258358
}
],
"transition_kernel_count": 486,
"elapsed_seconds": 0.01542369369417429
},
{
"id": "n12_B1_50_relaxed_d1_64",
"n": 12,
"tau": "1/4",
"weights": [
"3540019/4194304",
"654285/8388608",
"654285/8388608"
],
"probabilities": [
"1/4",
"1/2",
"17/64"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "1054240316350457439026956575/19807040628566084398385987584",
"float": 0.05322553409770929
},
"kl_lower": {
"rational": "223007156188572961189015159161941986733899/11150372599265311570767859136324180752990208",
"float": 0.019999973472031484
},
"kl_upper": {
"rational": "55751789047143240297253789790485617442125/2787593149816327892691964784081045188247552",
"float": 0.019999973472031484
},
"kl_enclosure_width": {
"rational": "483034601/11150372599265311570767859136324180752990208",
"float": 4.332004125421125e-35
},
"kl_budget": {
"rational": "1/50",
"float": 0.02
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "377192046908521076503485/1518085017019264923397952",
"float": 0.2484656937390315
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 78
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.06478036477409799,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "14293711201/34359738368",
"float": 0.4160017473914195
},
"signed_test_gap_lower": {
"rational": "2225836385/34359738368",
"float": 0.06478036477346905
},
"tv_lower_certified": {
"rational": "2225836385/34359738368",
"float": 0.06478036477346905
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 469,
"bellman_action_count": 533,
"execution_gap_lower_certified": {
"rational": "228867000650975120627054305/19807040628566084398385987584",
"float": 0.011554830675759752
},
"tv_upper_certified": {
"rational": "78314655579461102282461/1208925819614629174706176",
"float": 0.06478036477409801
},
"policy_tv_lower_certified": {
"rational": "39157327789730551141227/604462909807314587353088",
"float": 0.06478036477409801
},
"terminal_subprobability": [
"27361897209574175304855/1208925819614629174706176",
"15235577256785501843869/151115727451828646838272",
"124423680042617094543395/604462909807314587353088",
"153958345419243342813103/604462909807314587353088",
"16073750167950169217125/75557863725914323419136",
"76374544154443329146807/604462909807314587353088",
"33076295895997030973195/604462909807314587353088",
"5262129487502142445295/302231454903657293676544",
"2441703869191595781185/604462909807314587353088",
"805678763776035422371/1208925819614629174706176",
"89723172012839060185/1208925819614629174706176",
"3027843202749188919/604462909807314587353088",
"93664190274009709/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "7/1208925819614629174706176",
"float": 5.790264287871194e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 14,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "760384530141/1208925819614629174706176",
"float": 6.289753414178785e-13
},
"elapsed_seconds": 0.0038053346797823906
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.05596104930915014,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "447701818529/1099511627776",
"float": 0.4071824319262305
},
"signed_test_gap_lower": {
"rational": "61529824417/1099511627776",
"float": 0.05596104930828005
},
"tv_lower_certified": {
"rational": "61529824417/1099511627776",
"float": 0.05596104930828005
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 924,
"bellman_action_count": 19084,
"execution_gap_lower_certified": {
"rational": "54182460915835546117846753/19807040628566084398385987584",
"float": 0.0027355152105707596
},
"tv_upper_certified": {
"rational": "33826378701279539295007/604462909807314587353088",
"float": 0.0559610493091502
},
"policy_tv_lower_certified": {
"rational": "67652757402559078589987/1208925819614629174706176",
"float": 0.0559610493091502
},
"terminal_subprobability": [
"30976426967657425229333/1208925819614629174706176",
"131438098710979415964049/1208925819614629174706176",
"254757410142247872623237/1208925819614629174706176",
"299500528543596374644631/1208925819614629174706176",
"241473084786766276328353/1208925819614629174706176",
"72225057137627128836449/604462909807314587353088",
"34086103359548140912421/604462909807314587353088",
"13283938308091264015145/604462909807314587353088",
"4358486939881467428301/604462909807314587353088",
"73131168297938341779/37778931862957161709568",
"234927156336890310445/604462909807314587353088",
"59826650849127294151/1208925819614629174706176",
"3220624028872679945/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "27/1208925819614629174706176",
"float": 2.2333876538931747e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 57,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "525969191711/604462909807314587353088",
"float": 8.701430363670517e-13
},
"elapsed_seconds": 0.033102791756391525
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.052261417262724964,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "443634030075/1099511627776",
"float": 0.4034827998793844
},
"signed_test_gap_lower": {
"rational": "57462035963/1099511627776",
"float": 0.05226141726143396
},
"tv_lower_certified": {
"rational": "57462035963/1099511627776",
"float": 0.05226141726143396
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 1379,
"bellman_action_count": 37635,
"execution_gap_lower_certified": {
"rational": "-19096301346790077918365983/19807040628566084398385987584",
"float": -0.0009641168362753311
},
"tv_upper_certified": {
"rational": "3948761043660120275943/75557863725914323419136",
"float": 0.05226141726272498
},
"policy_tv_lower_certified": {
"rational": "31590088349280962207523/604462909807314587353088",
"float": 0.05226141726272498
},
"terminal_subprobability": [
"15796596108272157578419/604462909807314587353088",
"66261761509085447015901/604462909807314587353088",
"256901027331181445254397/1208925819614629174706176",
"300127302502581588193139/1208925819614629174706176",
"236268167053041715375567/1208925819614629174706176",
"140348853442163471528731/1208925819614629174706176",
"2162767250541302668511/37778931862957161709568",
"29035227768897835918945/1208925819614629174706176",
"9816900744477631958545/1208925819614629174706176",
"2530889882818153675975/1208925819614629174706176",
"497092802633756823921/1208925819614629174706176",
"70184560163015009657/1208925819614629174706176",
"4906274633666386265/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "21/604462909807314587353088",
"float": 3.474158572722716e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 86,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "97546419175/75557863725914323419136",
"float": 1.2910161082484945e-12
},
"elapsed_seconds": 0.05083287041634321
}
],
"transition_kernel_count": 1807,
"elapsed_seconds": 0.09192118886858225
},
{
"id": "n12_B1_20_relaxed_d1_64",
"n": 12,
"tau": "1/4",
"weights": [
"5958593/8388608",
"2430015/16777216",
"2430015/16777216"
],
"probabilities": [
"1/4",
"1/2",
"17/64"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "3915449356681502454889061925/39614081257132168796771975168",
"float": 0.09883983756348155
},
"kl_lower": {
"rational": "1115036137322709812999505992481644419256043/22300745198530623141535718272648361505980416",
"float": 0.04999994966070365
},
"kl_upper": {
"rational": "1115036137322709812999505992481645387611879/22300745198530623141535718272648361505980416",
"float": 0.04999994966070365
},
"kl_enclosure_width": {
"rational": "242088959/5575186299632655785383929568162090376495104",
"float": 4.342257746901678e-35
},
"kl_budget": {
"rational": "1/20",
"float": 0.05
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "1400891556230709619996815/5621980656856424271193024",
"float": 0.24918114126241575
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 78
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.12048457682771604,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "518646187301/1099511627776",
"float": 0.4717059594449893
},
"signed_test_gap_lower": {
"rational": "132474193189/1099511627776",
"float": 0.12048457682703884
},
"tv_lower_certified": {
"rational": "132474193189/1099511627776",
"float": 0.12048457682703884
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 469,
"bellman_action_count": 533,
"execution_gap_lower_certified": {
"rational": "857436459975997538572952027/39614081257132168796771975168",
"float": 0.02164473926355729
},
"tv_upper_certified": {
"rational": "145656915792368471653929/1208925819614629174706176",
"float": 0.12048457682771613
},
"policy_tv_lower_certified": {
"rational": "72828457896184235826961/604462909807314587353088",
"float": 0.12048457682771613
},
"terminal_subprobability": [
"636128749759260943269/37778931862957161709568",
"12379423310394147602103/151115727451828646838272",
"110417331470187985974153/604462909807314587353088",
"298442136558846192447597/1208925819614629174706176",
"136121527955177267533741/604462909807314587353088",
"44150025102426413903261/302231454903657293676544",
"83531809288260451134451/1208925819614629174706176",
"14514071408132919980943/604462909807314587353088",
"1838880170379586768429/302231454903657293676544",
"1325394168026847000983/1208925819614629174706176",
"161205725505070413129/1208925819614629174706176",
"5941579736787681467/604462909807314587353088",
"401480847157681209/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "7/1208925819614629174706176",
"float": 5.790264287871194e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 14,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "818789236265/1208925819614629174706176",
"float": 6.772865820055084e-13
},
"elapsed_seconds": 0.0037703383713960648
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.1044457093193377,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "501011265979/1099511627776",
"float": 0.45566709193644783
},
"signed_test_gap_lower": {
"rational": "114839271867/1099511627776",
"float": 0.10444570931849739
},
"tv_lower_certified": {
"rational": "114839271867/1099511627776",
"float": 0.10444570931849739
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 924,
"bellman_action_count": 19084,
"execution_gap_lower_certified": {
"rational": "222071459220259820940026331/39614081257132168796771975168",
"float": 0.005605871755015845
},
"tv_upper_certified": {
"rational": "126267114744111919307737/1208925819614629174706176",
"float": 0.10444570931933793
},
"policy_tv_lower_certified": {
"rational": "63133557372055959653855/604462909807314587353088",
"float": 0.10444570931933793
},
"terminal_subprobability": [
"12823201763648209037511/604462909807314587353088",
"114273038897886335756205/1208925819614629174706176",
"14489747457944860714013/75557863725914323419136",
"71575676317656930622023/302231454903657293676544",
"123244163945184960451303/604462909807314587353088",
"161404752042121977670335/1208925819614629174706176",
"85660053902527655796595/1208925819614629174706176",
"9526513653570033397073/302231454903657293676544",
"7052913402898027849501/604462909807314587353088",
"1032193888400344252733/302231454903657293676544",
"430519227246056544081/604462909807314587353088",
"26827250142366372615/302231454903657293676544",
"2787163971113858119/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "27/1208925819614629174706176",
"float": 2.2333876538931747e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 57,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1016146729945/1208925819614629174706176",
"float": 8.405368745196611e-13
},
"elapsed_seconds": 0.032609621062874794
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.09770444760959102,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "493599170343/1099511627776",
"float": 0.4489258302264716
},
"signed_test_gap_lower": {
"rational": "107427176231/1099511627776",
"float": 0.09770444760852115
},
"tv_lower_certified": {
"rational": "107427176231/1099511627776",
"float": 0.09770444760852115
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 1379,
"bellman_action_count": 37635,
"execution_gap_lower_certified": {
"rational": "-44977429934332815514017317/39614081257132168796771975168",
"float": -0.0011353899549603973
},
"tv_upper_certified": {
"rational": "118117429406419732791777/1208925819614629174706176",
"float": 0.09770444760959128
},
"policy_tv_lower_certified": {
"rational": "118117429406419732791735/1208925819614629174706176",
"float": 0.09770444760959128
},
"terminal_subprobability": [
"13661542896682786790495/604462909807314587353088",
"117707315994824900390073/1208925819614629174706176",
"117934487963494330217575/604462909807314587353088",
"142654207322720649926637/604462909807314587353088",
"3711274543293762725633/18889465931478580854784",
"153577367828812458162577/1208925819614629174706176",
"85266740306583372770991/1208925819614629174706176",
"5256111777440328217879/151115727451828646838272",
"17516815061711563467191/1208925819614629174706176",
"2744131320115405980441/604462909807314587353088",
"35750759574931626107/37778931862957161709568",
"143881498623755386239/1208925819614629174706176",
"10470621325526479799/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "21/604462909807314587353088",
"float": 3.474158572722716e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 88,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1293706199521/1208925819614629174706176",
"float": 1.0701286865834302e-12
},
"elapsed_seconds": 0.0509194228798151
}
],
"transition_kernel_count": 1807,
"elapsed_seconds": 0.09152637887746096
},
{
"id": "n12_B1_10_relaxed_d1_64",
"n": 12,
"tau": "1/4",
"weights": [
"4503979/8388608",
"3884629/16777216",
"3884629/16777216"
],
"probabilities": [
"1/4",
"1/2",
"17/64"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "6259248654430655037040200055/39614081257132168796771975168",
"float": 0.15800564990520213
},
"kl_lower": {
"rational": "2230072137627751909849626532269652900817165/22300745198530623141535718272648361505980416",
"float": 0.0999998931773226
},
"kl_upper": {
"rational": "2230072137627751909849626532269653911902373/22300745198530623141535718272648361505980416",
"float": 0.0999998931773226
},
"kl_enclosure_width": {
"rational": "126385651/2787593149816327892691964784081045188247552",
"float": 4.5338628776702024e-35
},
"kl_budget": {
"rational": "1/10",
"float": 0.1
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "2239469289361977304839109/8976007232657848933347648",
"float": 0.2494950406472497
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 78
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.1913868702876772,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "149151020849/274877906944",
"float": 0.5426082529047562
},
"signed_test_gap_lower": {
"rational": "52608022321/274877906944",
"float": 0.1913868702868058
},
"tv_lower_certified": {
"rational": "52608022321/274877906944",
"float": 0.1913868702868058
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 469,
"bellman_action_count": 533,
"execution_gap_lower_certified": {
"rational": "1322366376659083718314718857/39614081257132168796771975168",
"float": 0.033381220381603655
},
"tv_upper_certified": {
"rational": "231372529026008951383045/1208925819614629174706176",
"float": 0.19138687028767726
},
"policy_tv_lower_certified": {
"rational": "231372529026008951383037/1208925819614629174706176",
"float": 0.19138687028767726
},
"terminal_subprobability": [
"6857910607124186213577/604462909807314587353088",
"9308375944758854177293/151115727451828646838272",
"5790785176297640833007/37778931862957161709568",
"279464738327187503166497/1208925819614629174706176",
"4445190761055799204669/18889465931478580854784",
"102972487636059834431991/604462909807314587353088",
"108707330402973430809099/1208925819614629174706176",
"42157309057735160878133/1208925819614629174706176",
"11921028867534313435667/1208925819614629174706176",
"1198566311112615066343/604462909807314587353088",
"325367490412287674403/1208925819614629174706176",
"6691327976452297203/302231454903657293676544",
"1009141120908956351/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "1/151115727451828646838272",
"float": 6.617444900424222e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 14,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1053545030661/1208925819614629174706176",
"float": 8.714720238143643e-13
},
"elapsed_seconds": 0.004949675872921944
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.16710497662826604,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "569905858973/1099511627776",
"float": 0.5183263592452931
},
"signed_test_gap_lower": {
"rational": "183733864861/1099511627776",
"float": 0.1671049766273427
},
"tv_lower_certified": {
"rational": "183733864861/1099511627776",
"float": 0.1671049766273427
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 924,
"bellman_action_count": 19084,
"execution_gap_lower_certified": {
"rational": "360461468156070313700128393/39614081257132168796771975168",
"float": 0.009099326722140562
},
"tv_upper_certified": {
"rational": "202017520832010176771623/1208925819614629174706176",
"float": 0.1671049766282662
},
"policy_tv_lower_certified": {
"rational": "202017520832010176771593/1208925819614629174706176",
"float": 0.1671049766282662
},
"terminal_subprobability": [
"20997090218079294819309/1208925819614629174706176",
"96144843881236589604119/1208925819614629174706176",
"50369858049901311098731/302231454903657293676544",
"263686334636108861461289/1208925819614629174706176",
"247681633127759682707505/1208925819614629174706176",
"11363539515530937647729/75557863725914323419136",
"6852742613879949157835/75557863725914323419136",
"27627712428109924333247/604462909807314587353088",
"5721247328299344788517/302231454903657293676544",
"7384343918913699547857/1208925819614629174706176",
"1697018520814277992831/1208925819614629174706176",
"239556262350995165555/1208925819614629174706176",
"14638609769112303171/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "15/604462909807314587353088",
"float": 2.481541837659083e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 57,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1116458792487/1208925819614629174706176",
"float": 9.235130678596103e-13
},
"elapsed_seconds": 0.03230403736233711
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.1568670253958722,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "5892517/16777216",
"float": 0.35122138261795044
},
"minimum_policy_test_mean_lower": {
"rational": "139662278137/274877906944",
"float": 0.5080884080125543
},
"signed_test_gap_lower": {
"rational": "43119279609/274877906944",
"float": 0.15686702539460384
},
"tv_lower_certified": {
"rational": "43119279609/274877906944",
"float": 0.15686702539460384
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 1379,
"bellman_action_count": 37635,
"execution_gap_lower_certified": {
"rational": "-45105563884202990287686007/39614081257132168796771975168",
"float": -0.0011386245105982895
},
"tv_upper_certified": {
"rational": "189640597247213938061277/1208925819614629174706176",
"float": 0.15686702539587244
},
"policy_tv_lower_certified": {
"rational": "189640597247213938061229/1208925819614629174706176",
"float": 0.15686702539587244
},
"terminal_subprobability": [
"22256860361478985640701/1208925819614629174706176",
"6300999142737188477227/75557863725914323419136",
"104297906465878292901075/604462909807314587353088",
"16438497808924727619469/75557863725914323419136",
"59022740077903418288901/302231454903657293676544",
"170681177577759986847579/1208925819614629174706176",
"107332315335501144887269/1208925819614629174706176",
"929220789784114481555/18889465931478580854784",
"6948741973784572421487/302231454903657293676544",
"2504919928746344477989/302231454903657293676544",
"2477492127327407420737/1208925819614629174706176",
"2761722240627098749/9444732965739290427392",
"2621392436683793457/151115727451828646838272"
],
"unrepresented_mass_upper_exact": {
"rational": "3/75557863725914323419136",
"float": 3.970466940254533e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 96,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1533638782941/1208925819614629174706176",
"float": 1.2685962679082162e-12
},
"elapsed_seconds": 0.051309531554579735
}
],
"transition_kernel_count": 1807,
"elapsed_seconds": 0.09289911482483149
},
{
"id": "n16_B1_50_single_d1_8",
"n": 16,
"tau": "1/4",
"weights": [
"13991273/16777216",
"2785943/16777216"
],
"probabilities": [
"1/4",
"3/8"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "319614993586267905949/4722366482869645213696",
"float": 0.0676811075010102
},
"kl_lower": {
"rational": "114179692462173356561010258998347799209312859/5708990770823839524233143877797980545530986496",
"float": 0.019999978463040427
},
"kl_upper": {
"rational": "14272461557771669570126282374793511177283619/713623846352979940529142984724747568191373312",
"float": 0.019999978463040427
},
"kl_enclosure_width": {
"rational": "290208956093/5708990770823839524233143877797980545530986496",
"float": 5.083367056330365e-35
},
"kl_budget": {
"rational": "1/50",
"float": 0.02
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "39975237014301/323469624383720",
"float": 0.1235826612482162
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 136
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.07539204222422739,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"0"
],
"teacher_test_mean_exact": {
"rational": "794169141/2147483648",
"float": 0.3698138245381415
},
"minimum_policy_test_mean_lower": {
"rational": "30594314175/68719476736",
"float": 0.44520586634462234
},
"signed_test_gap_lower": {
"rational": "5180901663/68719476736",
"float": 0.07539204180648085
},
"tv_lower_certified": {
"rational": "5180901663/68719476736",
"float": 0.07539204180648085
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 987,
"bellman_action_count": 1105,
"execution_gap_lower_certified": {
"rational": "36413857715764306019/4722366482869645213696",
"float": 0.007710934305470646
},
"tv_upper_certified": {
"rational": "91143386661115643479881/1208925819614629174706176",
"float": 0.07539204240849909
},
"policy_tv_lower_certified": {
"rational": "45571693330557821739937/604462909807314587353088",
"float": 0.07539204240849909
},
"terminal_subprobability": [
"3866579844331243631959/604462909807314587353088",
"45939276354705348127427/1208925819614629174706176",
"127924168266888973192011/1208925819614629174706176",
"221649405964032814169917/1208925819614629174706176",
"133729470731577784814687/604462909807314587353088",
"238328265290315860920433/1208925819614629174706176",
"40556846975301317829391/302231454903657293676544",
"21511630667297113998727/302231454903657293676544",
"35941238254572006213651/1208925819614629174706176",
"2965430229436657369683/302231454903657293676544",
"3082853807942473364295/1208925819614629174706176",
"2438814223429471571/4722366482869645213696",
"96586100290647931401/1208925819614629174706176",
"11034139223196243977/1208925819614629174706176",
"877887503849172643/1208925819614629174706176",
"43459506712368729/1208925819614629174706176",
"1008490934595013/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "7/1208925819614629174706176",
"float": 5.790264287871194e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 18,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "727795397216073/1208925819614629174706176",
"float": 6.020182424824654e-10
},
"elapsed_seconds": 0.009230627678334713
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.07160591128077236,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "242673066259/549755813888",
"float": 0.4414197360511025
},
"signed_test_gap_lower": {
"rational": "39365766035/549755813888",
"float": 0.07160591128013039
},
"tv_lower_certified": {
"rational": "39365766035/549755813888",
"float": 0.07160591128013039
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 1956,
"bellman_action_count": 75701,
"execution_gap_lower_certified": {
"rational": "18534361818357276771/4722366482869645213696",
"float": 0.003924803779120184
},
"tv_upper_certified": {
"rational": "86566234984360008304919/1208925819614629174706176",
"float": 0.07160591128077225
},
"policy_tv_lower_certified": {
"rational": "10820779373045001038109/151115727451828646838272",
"float": 0.07160591128077225
},
"terminal_subprobability": [
"2217501860714381273753/302231454903657293676544",
"50271062076533741164525/1208925819614629174706176",
"133715499929728635027193/1208925819614629174706176",
"222342028067656347053865/1208925819614629174706176",
"130041752948712289608507/604462909807314587353088",
"228558121375429656418919/1208925819614629174706176",
"157357054736452493232829/1208925819614629174706176",
"1363555113686342899605/18889465931478580854784",
"39647897911016411165797/1208925819614629174706176",
"3713189117618000537379/302231454903657293676544",
"4567974005199236016191/1208925819614629174706176",
"567545942989895618431/604462909807314587353088",
"221384635225279908797/1208925819614629174706176",
"32384668659121965757/1208925819614629174706176",
"1654364544421640639/604462909807314587353088",
"104227826270314857/604462909807314587353088",
"1512831756392035/302231454903657293676544"
],
"unrepresented_mass_upper_exact": {
"rational": "47/1208925819614629174706176",
"float": 3.88774887899923e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 89,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "775961528599/1208925819614629174706176",
"float": 6.418603325441045e-13
},
"elapsed_seconds": 0.14215601421892643
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.06921703005006644,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "482719529827/1099511627776",
"float": 0.4390308548199755
},
"signed_test_gap_lower": {
"rational": "76104929379/1099511627776",
"float": 0.06921703004900337
},
"tv_lower_certified": {
"rational": "76104929379/1099511627776",
"float": 0.06921703004900337
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 2925,
"bellman_action_count": 150297,
"execution_gap_lower_certified": {
"rational": "7253189160926683235/4722366482869645213696",
"float": 0.0015359225479931685
},
"tv_upper_certified": {
"rational": "41839127392283458657583/604462909807314587353088",
"float": 0.06921703005006638
},
"policy_tv_lower_certified": {
"rational": "10459781848070864664387/151115727451828646838272",
"float": 0.06921703005006638
},
"terminal_subprobability": [
"4598124473553128182279/604462909807314587353088",
"25558469207121420654361/604462909807314587353088",
"135470885111723889549523/1208925819614629174706176",
"223794921099875408951979/1208925819614629174706176",
"64647772510261380593145/302231454903657293676544",
"224655490141228396149751/1208925819614629174706176",
"154560304266718287483133/1208925819614629174706176",
"87286452027764522450495/1208925819614629174706176",
"20601644111248257448293/604462909807314587353088",
"16217680827276911061991/1208925819614629174706176",
"162831523058634144875/37778931862957161709568",
"661690444616874741189/604462909807314587353088",
"257096768543931547227/1208925819614629174706176",
"18614322360985052513/604462909807314587353088",
"3919863301752718577/1208925819614629174706176",
"276301091366857555/1208925819614629174706176",
"9310381560770025/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "35/604462909807314587353088",
"float": 5.790264287871194e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 146,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "642551242031/604462909807314587353088",
"float": 1.0630118599598226e-12
},
"elapsed_seconds": 0.23544943053275347
}
],
"transition_kernel_count": 4828,
"elapsed_seconds": 0.3911705482751131
},
{
"id": "n16_B1_20_single_d1_8",
"n": 16,
"tau": "1/4",
"weights": [
"11998679/16777216",
"4778537/16777216"
],
"probabilities": [
"1/4",
"3/8"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "548213682981577110691/4722366482869645213696",
"float": 0.11608876290525498
},
"kl_lower": {
"rational": "285449202594226692071090448272190082923635385/5708990770823839524233143877797980545530986496",
"float": 0.04999994115475418
},
"kl_upper": {
"rational": "285449202594226692071090448272190377330131545/5708990770823839524233143877797980545530986496",
"float": 0.04999994115475418
},
"kl_enclosure_width": {
"rational": "9200203005/178405961588244985132285746181186892047843328",
"float": 5.156892136953227e-35
},
"kl_budget": {
"rational": "1/20",
"float": 0.05
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "68566783009059/551679645780248",
"float": 0.12428731698463166
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 136
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.12916895370035303,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "548637366989/1099511627776",
"float": 0.49898277847114514
},
"signed_test_gap_lower": {
"rational": "142022766541/1099511627776",
"float": 0.129168953700173
},
"tv_lower_certified": {
"rational": "142022766541/1099511627776",
"float": 0.129168953700173
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 987,
"bellman_action_count": 1105,
"execution_gap_lower_certified": {
"rational": "61769454599460932445/4722366482869645213696",
"float": 0.013080190794918022
},
"tv_upper_certified": {
"rational": "39038920805240788650299/302231454903657293676544",
"float": 0.12916895370035283
},
"policy_tv_lower_certified": {
"rational": "156155683220963154601187/1208925819614629174706176",
"float": 0.12916895370035283
},
"terminal_subprobability": [
"5564611161667608982285/1208925819614629174706176",
"35594072614982899365199/1208925819614629174706176",
"6670245826036442302787/75557863725914323419136",
"99554635041444023424893/604462909807314587353088",
"129350384050738024609735/604462909807314587353088",
"248217179722839670458303/1208925819614629174706176",
"181926577110156193638191/1208925819614629174706176",
"103901300945167893171349/1208925819614629174706176",
"46730064959694381589093/1208925819614629174706176",
"2075758152169998824583/151115727451828646838272",
"4647160329540246818241/1208925819614629174706176",
"506686230926333517251/604462909807314587353088",
"5275100706770579323/37778931862957161709568",
"20764441260686761999/1208925819614629174706176",
"1778837782792898369/1208925819614629174706176",
"47409782531291135/604462909807314587353088",
"1184597624678759/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "9/1208925819614629174706176",
"float": 7.444625512977249e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 18,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "54353889595/302231454903657293676544",
"float": 1.7984193475932695e-13
},
"elapsed_seconds": 0.00987230334430933
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.12284333701315131,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "1057973199/2147483648",
"float": 0.492657161783427
},
"signed_test_gap_lower": {
"rational": "527608115/4294967296",
"float": 0.12284333701245487
},
"tv_lower_certified": {
"rational": "527608115/4294967296",
"float": 0.12284333701245487
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 1956,
"bellman_action_count": 75701,
"execution_gap_lower_certified": {
"rational": "31897574369899891549/4722366482869645213696",
"float": 0.0067545741071998846
},
"tv_upper_certified": {
"rational": "148508481882820082244351/1208925819614629174706176",
"float": 0.12284333701315132
},
"policy_tv_lower_certified": {
"rational": "148508481882820082244305/1208925819614629174706176",
"float": 0.12284333701315132
},
"terminal_subprobability": [
"3595272817530085422021/604462909807314587353088",
"20981005801161223006777/604462909807314587353088",
"57761815944327434701695/604462909807314587353088",
"200629422964728230943171/1208925819614629174706176",
"62008561106243759103505/302231454903657293676544",
"58325823049348911422315/302231454903657293676544",
"173876437567443416483859/1208925819614629174706176",
"52602835641437523254927/604462909807314587353088",
"26150290283045437598655/604462909807314587353088",
"21389954057607932223331/1208925819614629174706176",
"7136461771683908373957/1208925819614629174706176",
"238465391350682993973/151115727451828646838272",
"6204320917725877569/18889465931478580854784",
"61637331801998505877/1208925819614629174706176",
"1667262610095229687/302231454903657293676544",
"445809436835570769/1208925819614629174706176",
"3448643115374697/302231454903657293676544"
],
"unrepresented_mass_upper_exact": {
"rational": "23/604462909807314587353088",
"float": 3.8050308177439273e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 91,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "841969670911/1208925819614629174706176",
"float": 6.964609881352321e-13
},
"elapsed_seconds": 0.14361178502440453
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.11887869238694684,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "537323105021/1099511627776",
"float": 0.4886925171567782
},
"signed_test_gap_lower": {
"rational": "130708504573/1099511627776",
"float": 0.11887869238580606
},
"tv_lower_certified": {
"rational": "130708504573/1099511627776",
"float": 0.11887869238580606
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 2925,
"bellman_action_count": 150297,
"execution_gap_lower_certified": {
"rational": "13175069468524333917/4722366482869645213696",
"float": 0.0027899294805510786
},
"tv_upper_certified": {
"rational": "71857760314302674213855/604462909807314587353088",
"float": 0.11887869238694705
},
"policy_tv_lower_certified": {
"rational": "143715520628605348427651/1208925819614629174706176",
"float": 0.11887869238694705
},
"terminal_subprobability": [
"3711888156594546669297/604462909807314587353088",
"21182122239952072269741/604462909807314587353088",
"29292862918075943703163/302231454903657293676544",
"50818366299949668499277/302231454903657293676544",
"123949940052379900373491/604462909807314587353088",
"57284662500927821848447/302231454903657293676544",
"42391791513567434298207/302231454903657293676544",
"104076318143768939064157/1208925819614629174706176",
"53730464978959842199635/1208925819614629174706176",
"23196925080461756779399/1208925819614629174706176",
"8202243492038196391315/1208925819614629174706176",
"575577145524180731559/302231454903657293676544",
"493223692150634538771/1208925819614629174706176",
"76896954677845843225/1208925819614629174706176",
"2057225222523883399/302231454903657293676544",
"556570394643569689/1208925819614629174706176",
"4847813496459665/302231454903657293676544"
],
"unrepresented_mass_upper_exact": {
"rational": "59/1208925819614629174706176",
"float": 4.880365614062863e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 122,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "689689304031/604462909807314587353088",
"float": 1.1409952419592678e-12
},
"elapsed_seconds": 0.23320357128977776
}
],
"transition_kernel_count": 4828,
"elapsed_seconds": 0.3911970295011997
},
{
"id": "n16_B1_10_single_d1_8",
"n": 16,
"tau": "1/4",
"weights": [
"150175/262144",
"111969/262144"
],
"probabilities": [
"1/4",
"3/8"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "12845550399581337867/73786976294838206464",
"float": 0.17408967062497657
},
"kl_lower": {
"rational": "142724606159260562266146605506359524186786733/1427247692705959881058285969449495136382746624",
"float": 0.09999988571616808
},
"kl_upper": {
"rational": "570898424637042249064586422025438392665696901/5708990770823839524233143877797980545530986496",
"float": 0.09999988571616808
},
"kl_enclosure_width": {
"rational": "295918549969/5708990770823839524233143877797980545530986496",
"float": 5.183377620459829e-35
},
"kl_budget": {
"rational": "1/10",
"float": 0.1
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "1606632767883/12892429618264",
"float": 0.12461830822073841
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 136
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.19209433932893072,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "308912280085/549755813888",
"float": 0.5619081640998047
},
"signed_test_gap_lower": {
"rational": "105604979861/549755813888",
"float": 0.19209433932883258
},
"tv_lower_certified": {
"rational": "105604979861/549755813888",
"float": 0.19209433932883258
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 987,
"bellman_action_count": 1105,
"execution_gap_lower_certified": {
"rational": "1328510062847837941/73786976294838206464",
"float": 0.018004668703856
},
"tv_upper_certified": {
"rational": "116113903308278996523691/604462909807314587353088",
"float": 0.1920943393289305
},
"policy_tv_lower_certified": {
"rational": "232227806616557993047375/1208925819614629174706176",
"float": 0.1920943393289305
},
"terminal_subprobability": [
"1858639477362147827731/604462909807314587353088",
"25903515375394028245509/1208925819614629174706176",
"84612307774696735029227/1208925819614629174706176",
"171970475521118466773877/1208925819614629174706176",
"243416954156069317111071/1208925819614629174706176",
"254434287475244743252001/1208925819614629174706176",
"101578231204587184607163/604462909807314587353088",
"126399737639376699736609/1208925819614629174706176",
"61931595087339619513965/1208925819614629174706176",
"23975811415748074855541/1208925819614629174706176",
"228420430856699489105/37778931862957161709568",
"868214560579566293053/604462909807314587353088",
"315107971295377782637/1208925819614629174706176",
"10556734714651117813/302231454903657293676544",
"1970454420848601987/604462909807314587353088",
"114424415028309719/604462909807314587353088",
"3114671786501907/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "7/1208925819614629174706176",
"float": 5.790264287871194e-24
},
"flow_grid_bits": 80,
"flow_rounded_edges": 18,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "59188304555/604462909807314587353088",
"float": 9.791883603556012e-14
},
"elapsed_seconds": 0.010766145773231983
},
{
"K": 3,
"lp_optimum_uncertified_input": 0.18319063160286606,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "304017414997/549755813888",
"float": 0.5530044563729462
},
"signed_test_gap_lower": {
"rational": "100710114773/549755813888",
"float": 0.18319063160197402
},
"tv_lower_certified": {
"rational": "100710114773/549755813888",
"float": 0.18319063160197402
},
"bellman_rounding_error_at_most": {
"rational": "3/1099511627776",
"float": 2.7284841053187847e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 1956,
"bellman_action_count": 75701,
"execution_gap_lower_certified": {
"rational": "671532391869957877/73786976294838206464",
"float": 0.009100960976997443
},
"tv_upper_certified": {
"rational": "221463884456216645583917/1208925819614629174706176",
"float": 0.18319063160286622
},
"policy_tv_lower_certified": {
"rational": "55365971114054161395969/302231454903657293676544",
"float": 0.18319063160286622
},
"terminal_subprobability": [
"1374358506940578314001/302231454903657293676544",
"33058868081696511594969/1208925819614629174706176",
"95034252465849900986747/1208925819614629174706176",
"21854535899077882869657/151115727451828646838272",
"231957612174412401483635/1208925819614629174706176",
"14778307859141929155051/75557863725914323419136",
"192031798538071298015583/1208925819614629174706176",
"126704347628024701634805/1208925819614629174706176",
"68445955941448707378559/1208925819614629174706176",
"15112100113530432479287/604462909807314587353088",
"337685143100787281675/37778931862957161709568",
"3074048133414772611161/1208925819614629174706176",
"338728421366944043483/604462909807314587353088",
"111076152816416105393/1208925819614629174706176",
"6350911492993788977/604462909807314587353088",
"900371244416027871/1208925819614629174706176",
"14844493936268121/604462909807314587353088"
],
"unrepresented_mass_upper_exact": {
"rational": "41/1208925819614629174706176",
"float": 3.3914405114674135e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 91,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "1078616114221/1208925819614629174706176",
"float": 8.922103380708932e-13
},
"elapsed_seconds": 0.1420437004417181
},
{
"K": 4,
"lp_optimum_uncertified_input": 0.17812359402230588,
"terminal_test_dyadic": [
"0",
"0",
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "1588338283/4294967296",
"float": 0.36981382477097213
},
"minimum_policy_test_mean_lower": {
"rational": "75307945407/137438953472",
"float": 0.5479374187925714
},
"signed_test_gap_lower": {
"rational": "24481120351/137438953472",
"float": 0.17812359402159927
},
"tv_lower_certified": {
"rational": "24481120351/137438953472",
"float": 0.17812359402159927
},
"bellman_rounding_error_at_most": {
"rational": "1/274877906944",
"float": 3.637978807091713e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 2925,
"bellman_action_count": 150297,
"execution_gap_lower_certified": {
"rational": "297651010041792245/73786976294838206464",
"float": 0.004033923396622698
},
"tv_upper_certified": {
"rational": "53834552974029919110315/302231454903657293676544",
"float": 0.17812359402230596
},
"policy_tv_lower_certified": {
"rational": "215338211896119676441185/1208925819614629174706176",
"float": 0.17812359402230596
},
"terminal_subprobability": [
"346529101447255965903/75557863725914323419136",
"32986126992406592745767/1208925819614629174706176",
"48316467449471654205205/604462909807314587353088",
"44613264421158493737673/302231454903657293676544",
"232893541303301187859951/1208925819614629174706176",
"232089306565746457919515/1208925819614629174706176",
"46554139289344938078275/302231454903657293676544",
"123822380296639068050067/1208925819614629174706176",
"68948423511535962271661/1208925819614629174706176",
"32325341788790646248715/1208925819614629174706176",
"6455215887208226393975/604462909807314587353088",
"2223135548866647493005/604462909807314587353088",
"40640932897296896331/37778931862957161709568",
"302069735392337103231/1208925819614629174706176",
"1545764105362473943/37778931862957161709568",
"4754154295520296189/1208925819614629174706176",
"182726173423459627/1208925819614629174706176"
],
"unrepresented_mass_upper_exact": {
"rational": "75/1208925819614629174706176",
"float": 6.203854594147708e-23
},
"flow_grid_bits": 80,
"flow_rounded_edges": 152,
"visited_fallback_states": 0,
"fallback_rule": "empty action until the last round, then complete every remaining coordinate",
"policy_interpretation": "exact normalized decimal weights; uniformly lifted fair-pair actions",
"optimality_gap_upper_certified": {
"rational": "213578171563/302231454903657293676544",
"float": 7.066708911257519e-13
},
"elapsed_seconds": 0.23372914921492338
}
],
"transition_kernel_count": 4828,
"elapsed_seconds": 0.3908644877374172
}
]
}
{
"certificate_version": 1,
"self_test": {
"passed": true,
"checks": [
"literal/full-state = count Bellman; floor bound, K=1",
"literal/full-state = count Bellman; floor bound, K=1",
"literal/full-state = count Bellman; floor bound, K=2",
"literal/full-state = count Bellman; floor bound, K=2",
"literal/full-state = count Bellman; floor bound, K=3",
"literal/full-state = count Bellman; floor bound, K=3",
"n=1 randomized mixture has certified TV <= 1e-20",
"rational logarithm intervals and reciprocal identity"
]
},
"witnesses": [
{
"id": "n8_B1_50_single_d1_4",
"n": 8,
"tau": "1/4",
"weights": [
"7477325/8388608",
"911283/8388608"
],
"probabilities": [
"1/4",
"1/2"
],
"arithmetic": "integer/Fraction; floats are display-only",
"model": {
"model_tv_exact": {
"rational": "31892171151/549755813888",
"float": 0.05801152137973986
},
"kl_lower": {
"rational": "27222543162586797511423518666869436249/1361129467683753853853498429727072845824",
"float": 0.019999966064147917
},
"kl_upper": {
"rational": "1742242762405555040731105194679648179555/87112285931760246646623899502532662132736",
"float": 0.019999966064147917
},
"kl_enclosure_width": {
"rational": "4259619/87112285931760246646623899502532662132736",
"float": 4.88980280386258e-35
},
"kl_budget": {
"rational": "1/50",
"float": 0.02
},
"kl_budget_certified": true,
"all_context_head_error_exact": {
"rational": "29161056/124121549",
"float": 0.23493951078551234
},
"head_bound": {
"rational": "1/4",
"float": 0.25
},
"head_bound_certified": true,
"head_contexts_checked": 36
},
"cases": [
{
"K": 2,
"lp_optimum_uncertified_input": 0.06857508646216329,
"terminal_test_dyadic": [
"0",
"0",
"0",
"1",
"1",
"1",
"1",
"1",
"1"
],
"teacher_test_mean_exact": {
"rational": "21067/65536",
"float": 0.3214569091796875
},
"minimum_policy_test_mean_lower": {
"rational": "107211178603/274877906944",
"float": 0.3900319956410385
},
"signed_test_gap_lower": {
"rational": "18849776235/274877906944",
"float": 0.06857508646135102
},
"tv_lower_certified": {
"rational": "18849776235/274877906944",
"float": 0.06857508646135102
},
"bellman_rounding_error_at_most": {
"rational": "1/549755813888",
"float": 1.8189894035458565e-12
},
"bellman_grid_bits": 40,
"bellman_state_count": 175,
"bellman_action_count": 201,