files / problems / hilbert16-limit-cycles / compute / q2 / p-harnack-recurrence / certs / table_replay.json
{
"papers": {
"harnack": "arXiv:2510.11705v2 Corollary 2",
"increment": "arXiv:2407.13465v2",
"chebyshev_table": "arXiv:2604.12883v1 Table 1 / Appendix A"
},
"seed_citations": {
"small_n H(1)=0, H(2)>=4, H(3)>=13": {
"1": 0,
"2": 4,
"3": 13
},
"Prohens\u2013Torregrosa 2019 Thm 1": {
"4": 28,
"5": 37,
"6": 53,
"7": 74,
"8": 96,
"9": 120,
"10": 142
},
"Han\u2013Li 2012 Thm 1.2(i) as quoted in 2604.12883v1 App A": {
"11": 153,
"12": 157,
"14": 194,
"15": 345,
"16": 351,
"18": 372,
"19": 503,
"20": 509
},
"Prohens\u2013Torregrosa 2019 Cor 2(a)": {
"13": 212,
"17": 384,
"21": 568,
"31": 1184,
"35": 1536,
"39": 1920,
"43": 2272
},
"Han\u2013Li Table 1 rows not in App A seeds": {
"23": 833,
"24": 843,
"25": 870,
"26": 880,
"27": 1023,
"29": 1060
},
"Chebyshev L_Ch already on 2604.12883v1": {
"11": 148,
"13": 212,
"14": 252,
"15": 296,
"17": 384,
"19": 480,
"20": 477,
"21": 568,
"23": 666,
"24": 700,
"25": 628,
"26": 864,
"27": 848,
"29": 1080,
"31": 1380,
"35": 1536,
"39": 2012,
"43": 2272
}
},
"PAPER_L_CH_included_as_published": true,
"N_max": 50,
"n_pairs": 1225,
"Har_1_to_6": [
0,
1,
1,
4,
6,
11
],
"Har": {
"1": 0,
"2": 1,
"3": 1,
"4": 4,
"5": 6,
"6": 11,
"7": 15,
"8": 22,
"9": 28,
"10": 37,
"11": 45,
"12": 56,
"13": 66,
"14": 79,
"15": 91,
"16": 106,
"17": 120,
"18": 137,
"19": 153,
"20": 172,
"21": 190,
"22": 211,
"23": 231,
"24": 254,
"25": 276,
"26": 301,
"27": 325,
"28": 352,
"29": 378,
"30": 407,
"31": 435,
"32": 466,
"33": 496,
"34": 529,
"35": 561,
"36": 596,
"37": 630,
"38": 667,
"39": 703,
"40": 742,
"41": 780,
"42": 821,
"43": 861,
"44": 904,
"45": 946,
"46": 991,
"47": 1035,
"48": 1082,
"49": 1128,
"50": 1177
},
"L_pub": {
"1": 0,
"2": 4,
"3": 13,
"4": 28,
"5": 37,
"6": 53,
"7": 74,
"8": 96,
"9": 120,
"10": 142,
"11": 153,
"12": 157,
"13": 212,
"14": 252,
"15": 345,
"16": 351,
"17": 384,
"18": 372,
"19": 503,
"20": 509,
"21": 568,
"23": 833,
"24": 843,
"25": 870,
"26": 880,
"27": 1023,
"29": 1080,
"31": 1380,
"35": 1536,
"39": 2012,
"43": 2272
},
"recorded_N": [
{
"N": 2,
"L_pub": 4,
"L_closed": 4,
"L_Ch": null,
"harnack_best": 0,
"argmax": {
"n": 1,
"m": 1,
"L_pub_n": 0,
"Har_m": 0
},
"slack_quoted": 4,
"slack_closed": 4
},
{
"N": 3,
"L_pub": 13,
"L_closed": 13,
"L_Ch": null,
"harnack_best": 4,
"argmax": {
"n": 2,
"m": 1,
"L_pub_n": 4,
"Har_m": 0
},
"slack_quoted": 9,
"slack_closed": 9
},
{
"N": 4,
"L_pub": 28,
"L_closed": 28,
"L_Ch": null,
"harnack_best": 13,
"argmax": {
"n": 3,
"m": 1,
"L_pub_n": 13,
"Har_m": 0
},
"slack_quoted": 15,
"slack_closed": 15
},
{
"N": 5,
"L_pub": 37,
"L_closed": 37,
"L_Ch": null,
"harnack_best": 28,
"argmax": {
"n": 4,
"m": 1,
"L_pub_n": 28,
"Har_m": 0
},
"slack_quoted": 9,
"slack_closed": 9
},
{
"N": 6,
"L_pub": 53,
"L_closed": 53,
"L_Ch": null,
"harnack_best": 37,
"argmax": {
"n": 5,
"m": 1,
"L_pub_n": 37,
"Har_m": 0
},
"slack_quoted": 16,
"slack_closed": 16
},
{
"N": 7,
"L_pub": 74,
"L_closed": 74,
"L_Ch": null,
"harnack_best": 53,
"argmax": {
"n": 6,
"m": 1,
"L_pub_n": 53,
"Har_m": 0
},
"slack_quoted": 21,
"slack_closed": 21
},
{
"N": 8,
"L_pub": 96,
"L_closed": 96,
"L_Ch": null,
"harnack_best": 74,
"argmax": {
"n": 7,
"m": 1,
"L_pub_n": 74,
"Har_m": 0
},
"slack_quoted": 22,
"slack_closed": 22
},
{
"N": 9,
"L_pub": 120,
"L_closed": 120,
"L_Ch": 112,
"harnack_best": 96,
"argmax": {
"n": 8,
"m": 1,
"L_pub_n": 96,
"Har_m": 0
},
"slack_quoted": 24,
"slack_closed": 24
},
{
"N": 10,
"L_pub": 142,
"L_closed": 142,
"L_Ch": null,
"harnack_best": 120,
"argmax": {
"n": 9,
"m": 1,
"L_pub_n": 120,
"Har_m": 0
},
"slack_quoted": 22,
"slack_closed": 22
},
{
"N": 11,
"L_pub": 153,
"L_closed": 153,
"L_Ch": 148,
"harnack_best": 142,
"argmax": {
"n": 10,
"m": 1,
"L_pub_n": 142,
"Har_m": 0
},
"slack_quoted": 11,
"slack_closed": 11
},
{
"N": 12,
"L_pub": 157,
"L_closed": 157,
"L_Ch": null,
"harnack_best": 153,
"argmax": {
"n": 11,
"m": 1,
"L_pub_n": 153,
"Har_m": 0
},
"slack_quoted": 4,
"slack_closed": 4
},
{
"N": 13,
"L_pub": 212,
"L_closed": 212,
"L_Ch": 212,
"harnack_best": 157,
"argmax": {
"n": 12,
"m": 1,
"L_pub_n": 157,
"Har_m": 0
},
"slack_quoted": 55,
"slack_closed": 55
},
{
"N": 14,
"L_pub": 252,
"L_closed": 252,
"L_Ch": 252,
"harnack_best": 212,
"argmax": {
"n": 13,
"m": 1,
"L_pub_n": 212,
"Har_m": 0
},
"slack_quoted": 40,
"slack_closed": 40
},
{
"N": 15,
"L_pub": 345,
"L_closed": 345,
"L_Ch": 296,
"harnack_best": 252,
"argmax": {
"n": 14,
"m": 1,
"L_pub_n": 252,
"Har_m": 0
},
"slack_quoted": 93,
"slack_closed": 93
},
{
"N": 16,
"L_pub": 351,
"L_closed": 351,
"L_Ch": null,
"harnack_best": 345,
"argmax": {
"n": 15,
"m": 1,
"L_pub_n": 345,
"Har_m": 0
},
"slack_quoted": 6,
"slack_closed": 6
},
{
"N": 17,
"L_pub": 384,
"L_closed": 384,
"L_Ch": 384,
"harnack_best": 351,
"argmax": {
"n": 16,
"m": 1,
"L_pub_n": 351,
"Har_m": 0
},
"slack_quoted": 33,
"slack_closed": 33
},
{
"N": 18,
"L_pub": 372,
"L_closed": 385,
"L_Ch": null,
"harnack_best": 384,
"argmax": {
"n": 17,
"m": 1,
"L_pub_n": 384,
"Har_m": 0
},
"slack_quoted": -12,
"slack_closed": 1
},
{
"N": 19,
"L_pub": 503,
"L_closed": 503,
"L_Ch": 480,
"harnack_best": 385,
"argmax": {
"n": 17,
"m": 2,
"L_pub_n": 384,
"Har_m": 1
},
"slack_quoted": 118,
"slack_closed": 118
},
{
"N": 20,
"L_pub": 509,
"L_closed": 509,
"L_Ch": 477,
"harnack_best": 503,
"argmax": {
"n": 19,
"m": 1,
"L_pub_n": 503,
"Har_m": 0
},
"slack_quoted": 6,
"slack_closed": 6
},
{
"N": 21,
"L_pub": 568,
"L_closed": 568,
"L_Ch": 568,
"harnack_best": 509,
"argmax": {
"n": 20,
"m": 1,
"L_pub_n": 509,
"Har_m": 0
},
"slack_quoted": 59,
"slack_closed": 59
},
{
"N": 23,
"L_pub": 833,
"L_closed": 833,
"L_Ch": 666,
"harnack_best": 569,
"argmax": {
"n": 21,
"m": 2,
"L_pub_n": 568,
"Har_m": 1
},
"slack_quoted": 264,
"slack_closed": 264
},
{
"N": 24,
"L_pub": 843,
"L_closed": 843,
"L_Ch": 700,
"harnack_best": 833,
"argmax": {
"n": 23,
"m": 1,
"L_pub_n": 833,
"Har_m": 0
},
"slack_quoted": 10,
"slack_closed": 10
},
{
"N": 25,
"L_pub": 870,
"L_closed": 870,
"L_Ch": 628,
"harnack_best": 843,
"argmax": {
"n": 24,
"m": 1,
"L_pub_n": 843,
"Har_m": 0
},
"slack_quoted": 27,
"slack_closed": 27
},
{
"N": 26,
"L_pub": 880,
"L_closed": 880,
"L_Ch": 864,
"harnack_best": 870,
"argmax": {
"n": 25,
"m": 1,
"L_pub_n": 870,
"Har_m": 0
},
"slack_quoted": 10,
"slack_closed": 10
},
{
"N": 27,
"L_pub": 1023,
"L_closed": 1023,
"L_Ch": 848,
"harnack_best": 880,
"argmax": {
"n": 26,
"m": 1,
"L_pub_n": 880,
"Har_m": 0
},
"slack_quoted": 143,
"slack_closed": 143
},
{
"N": 29,
"L_pub": 1080,
"L_closed": 1080,
"L_Ch": 1080,
"harnack_best": 1024,
"argmax": {
"n": 27,
"m": 2,
"L_pub_n": 1023,
"Har_m": 1
},
"slack_quoted": 56,
"slack_closed": 56
},
{
"N": 31,
"L_pub": 1380,
"L_closed": 1380,
"L_Ch": 1380,
"harnack_best": 1081,
"argmax": {
"n": 29,
"m": 2,
"L_pub_n": 1080,
"Har_m": 1
},
"slack_quoted": 299,
"slack_closed": 299
},
{
"N": 35,
"L_pub": 1536,
"L_closed": 1536,
"L_Ch": 1536,
"harnack_best": 1384,
"argmax": {
"n": 31,
"m": 4,
"L_pub_n": 1380,
"Har_m": 4
},
"slack_quoted": 152,
"slack_closed": 152
},
{
"N": 39,
"L_pub": 2012,
"L_closed": 2012,
"L_Ch": 2012,
"harnack_best": 1540,
"argmax": {
"n": 35,
"m": 4,
"L_pub_n": 1536,
"Har_m": 4
},
"slack_quoted": 472,
"slack_closed": 472
},
{
"N": 43,
"L_pub": 2272,
"L_closed": 2272,
"L_Ch": 2272,
"harnack_best": 2016,
"argmax": {
"n": 39,
"m": 4,
"L_pub_n": 2012,
"Har_m": 4
},
"slack_quoted": 256,
"slack_closed": 256
}
],
"unrecorded_N_le_50": [
{
"N": 22,
"harnack_best": 568,
"argmax": {
"n": 21,
"m": 1,
"L_pub_n": 568,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 28,
"harnack_best": 1023,
"argmax": {
"n": 27,
"m": 1,
"L_pub_n": 1023,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 30,
"harnack_best": 1080,
"argmax": {
"n": 29,
"m": 1,
"L_pub_n": 1080,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 32,
"harnack_best": 1380,
"argmax": {
"n": 31,
"m": 1,
"L_pub_n": 1380,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 33,
"harnack_best": 1381,
"argmax": {
"n": 31,
"m": 2,
"L_pub_n": 1380,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 34,
"harnack_best": 1381,
"argmax": {
"n": 31,
"m": 3,
"L_pub_n": 1380,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 36,
"harnack_best": 1536,
"argmax": {
"n": 35,
"m": 1,
"L_pub_n": 1536,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 37,
"harnack_best": 1537,
"argmax": {
"n": 35,
"m": 2,
"L_pub_n": 1536,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 38,
"harnack_best": 1537,
"argmax": {
"n": 35,
"m": 3,
"L_pub_n": 1536,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 40,
"harnack_best": 2012,
"argmax": {
"n": 39,
"m": 1,
"L_pub_n": 2012,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 41,
"harnack_best": 2013,
"argmax": {
"n": 39,
"m": 2,
"L_pub_n": 2012,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 42,
"harnack_best": 2013,
"argmax": {
"n": 39,
"m": 3,
"L_pub_n": 2012,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 44,
"harnack_best": 2272,
"argmax": {
"n": 43,
"m": 1,
"L_pub_n": 2272,
"Har_m": 0
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 45,
"harnack_best": 2273,
"argmax": {
"n": 43,
"m": 2,
"L_pub_n": 2272,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 46,
"harnack_best": 2273,
"argmax": {
"n": 43,
"m": 3,
"L_pub_n": 2272,
"Har_m": 1
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 47,
"harnack_best": 2276,
"argmax": {
"n": 43,
"m": 4,
"L_pub_n": 2272,
"Har_m": 4
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 48,
"harnack_best": 2278,
"argmax": {
"n": 43,
"m": 5,
"L_pub_n": 2272,
"Har_m": 6
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 49,
"harnack_best": 2283,
"argmax": {
"n": 43,
"m": 6,
"L_pub_n": 2272,
"Har_m": 11
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
},
{
"N": 50,
"harnack_best": 2287,
"argmax": {
"n": 43,
"m": 7,
"L_pub_n": 2272,
"Har_m": 15
},
"note": "no recorded L_pub at this N in the six seed tables; lift is Cor. 2 arithmetic, not a new H(N)"
}
],
"m1_quoted_exceedances": [
{
"n": 17,
"m": 1,
"N": 18,
"lift": 384,
"L_pub_N": 372,
"L_pub_n": 384,
"Har_m": 0,
"increment_would_give": 385,
"is_dent": false
}
],
"n_beats_quoted_m_ge_2": 0,
"n_beats_closed_recorded": 0,
"n_beats_claimed": 0,
"four_chebyshev": {
"14": 252,
"29": 1080,
"31": 1380,
"39": 2012
},
"harnack_vs_four": {
"14": {
"chebyshev_already_on_2604_12883": 252,
"harnack_best": 212,
"argmax": {
"n": 13,
"m": 1
},
"beats_chebyshev": false
},
"29": {
"chebyshev_already_on_2604_12883": 1080,
"harnack_best": 1024,
"argmax": {
"n": 27,
"m": 2
},
"beats_chebyshev": false
},
"31": {
"chebyshev_already_on_2604_12883": 1380,
"harnack_best": 1081,
"argmax": {
"n": 29,
"m": 2
},
"beats_chebyshev": false
},
"39": {
"chebyshev_already_on_2604_12883": 2012,
"harnack_best": 1540,
"argmax": {
"n": 35,
"m": 4
},
"beats_chebyshev": false
}
},
"H_K_5": 28,
"L_pub_4": 28,
"L_pub_5": 37,
"H_K_5_equals_L_pub_4": true,
"H_K_5_equals_9": false,
"H_K_5_beats_H_5": false,
"Har_1": 0,
"weaker_than_increment": true,
"do_not_claim_252_1080_1380_2012_as_ours": true,
"do_not_claim_HK5_28_as_ours": true,
"example_H6_from_H2": {
"n": 2,
"m": 4,
"lift": 8,
"L_pub_6": 53
}
}