files / problems / hilbert16-limit-cycles / compute / q2 / i-lienard / certs / arithmetic.json
{
"beats_B_n": false,
"claim": "Replay B(n) against Han-Romanovski, the Xiong-Han +2 bound, and Xiong H(N,3). This is their Li\u00e9nard H(2n+1,5), not planar H(n).",
"delta1_negative_n": [
2,
3
],
"delta1_positive_iff_n_ge": 7,
"delta1_zero_n": [
4,
5,
6
],
"delta2_positive_iff_n_ge": 13,
"delta2_zero_n": [
10,
11,
12
],
"delta3_positive_n": [
21,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40
],
"formulas": {
"B": "2n + n//3 + (n+1)//3 - 2",
"HR": "2*((N-1)//3) + (N-1)//2",
"HR2": "2*((N-1)//3) + (N//2) + 2",
"N": "2n+1",
"Xiong": "N + N//4",
"delta1": "B(n) - (2*((2n)//3) + n)",
"delta2": "B(n) - (2*((2n)//3) + n + 2)",
"delta3": "B(n) - (N + N//4)"
},
"hn_moved": false,
"n_max": 40,
"n_min": 2,
"paper": "arXiv:2608.17773v1 Theorem 3",
"paper_delta3": "positive at n=21 and for all n>=23",
"rows": [
{
"B": 3,
"HR": 4,
"HR2": 6,
"N": 5,
"Xiong": 6,
"delta1": -1,
"delta2": -3,
"delta3": -3,
"n": 2
},
{
"B": 6,
"HR": 7,
"HR2": 9,
"N": 7,
"Xiong": 8,
"delta1": -1,
"delta2": -3,
"delta3": -2,
"n": 3
},
{
"B": 8,
"HR": 8,
"HR2": 10,
"N": 9,
"Xiong": 11,
"delta1": 0,
"delta2": -2,
"delta3": -3,
"n": 4
},
{
"B": 11,
"HR": 11,
"HR2": 13,
"N": 11,
"Xiong": 13,
"delta1": 0,
"delta2": -2,
"delta3": -2,
"n": 5
},
{
"B": 14,
"HR": 14,
"HR2": 16,
"N": 13,
"Xiong": 16,
"delta1": 0,
"delta2": -2,
"delta3": -2,
"n": 6
},
{
"B": 16,
"HR": 15,
"HR2": 17,
"N": 15,
"Xiong": 18,
"delta1": 1,
"delta2": -1,
"delta3": -2,
"n": 7
},
{
"B": 19,
"HR": 18,
"HR2": 20,
"N": 17,
"Xiong": 21,
"delta1": 1,
"delta2": -1,
"delta3": -2,
"n": 8
},
{
"B": 22,
"HR": 21,
"HR2": 23,
"N": 19,
"Xiong": 23,
"delta1": 1,
"delta2": -1,
"delta3": -1,
"n": 9
},
{
"B": 24,
"HR": 22,
"HR2": 24,
"N": 21,
"Xiong": 26,
"delta1": 2,
"delta2": 0,
"delta3": -2,
"n": 10
},
{
"B": 27,
"HR": 25,
"HR2": 27,
"N": 23,
"Xiong": 28,
"delta1": 2,
"delta2": 0,
"delta3": -1,
"n": 11
},
{
"B": 30,
"HR": 28,
"HR2": 30,
"N": 25,
"Xiong": 31,
"delta1": 2,
"delta2": 0,
"delta3": -1,
"n": 12
},
{
"B": 32,
"HR": 29,
"HR2": 31,
"N": 27,
"Xiong": 33,
"delta1": 3,
"delta2": 1,
"delta3": -1,
"n": 13
},
{
"B": 35,
"HR": 32,
"HR2": 34,
"N": 29,
"Xiong": 36,
"delta1": 3,
"delta2": 1,
"delta3": -1,
"n": 14
},
{
"B": 38,
"HR": 35,
"HR2": 37,
"N": 31,
"Xiong": 38,
"delta1": 3,
"delta2": 1,
"delta3": 0,
"n": 15
},
{
"B": 40,
"HR": 36,
"HR2": 38,
"N": 33,
"Xiong": 41,
"delta1": 4,
"delta2": 2,
"delta3": -1,
"n": 16
},
{
"B": 43,
"HR": 39,
"HR2": 41,
"N": 35,
"Xiong": 43,
"delta1": 4,
"delta2": 2,
"delta3": 0,
"n": 17
},
{
"B": 46,
"HR": 42,
"HR2": 44,
"N": 37,
"Xiong": 46,
"delta1": 4,
"delta2": 2,
"delta3": 0,
"n": 18
},
{
"B": 48,
"HR": 43,
"HR2": 45,
"N": 39,
"Xiong": 48,
"delta1": 5,
"delta2": 3,
"delta3": 0,
"n": 19
},
{
"B": 51,
"HR": 46,
"HR2": 48,
"N": 41,
"Xiong": 51,
"delta1": 5,
"delta2": 3,
"delta3": 0,
"n": 20
},
{
"B": 54,
"HR": 49,
"HR2": 51,
"N": 43,
"Xiong": 53,
"delta1": 5,
"delta2": 3,
"delta3": 1,
"n": 21
},
{
"B": 56,
"HR": 50,
"HR2": 52,
"N": 45,
"Xiong": 56,
"delta1": 6,
"delta2": 4,
"delta3": 0,
"n": 22
},
{
"B": 59,
"HR": 53,
"HR2": 55,
"N": 47,
"Xiong": 58,
"delta1": 6,
"delta2": 4,
"delta3": 1,
"n": 23
},
{
"B": 62,
"HR": 56,
"HR2": 58,
"N": 49,
"Xiong": 61,
"delta1": 6,
"delta2": 4,
"delta3": 1,
"n": 24
},
{
"B": 64,
"HR": 57,
"HR2": 59,
"N": 51,
"Xiong": 63,
"delta1": 7,
"delta2": 5,
"delta3": 1,
"n": 25
},
{
"B": 67,
"HR": 60,
"HR2": 62,
"N": 53,
"Xiong": 66,
"delta1": 7,
"delta2": 5,
"delta3": 1,
"n": 26
},
{
"B": 70,
"HR": 63,
"HR2": 65,
"N": 55,
"Xiong": 68,
"delta1": 7,
"delta2": 5,
"delta3": 2,
"n": 27
},
{
"B": 72,
"HR": 64,
"HR2": 66,
"N": 57,
"Xiong": 71,
"delta1": 8,
"delta2": 6,
"delta3": 1,
"n": 28
},
{
"B": 75,
"HR": 67,
"HR2": 69,
"N": 59,
"Xiong": 73,
"delta1": 8,
"delta2": 6,
"delta3": 2,
"n": 29
},
{
"B": 78,
"HR": 70,
"HR2": 72,
"N": 61,
"Xiong": 76,
"delta1": 8,
"delta2": 6,
"delta3": 2,
"n": 30
},
{
"B": 80,
"HR": 71,
"HR2": 73,
"N": 63,
"Xiong": 78,
"delta1": 9,
"delta2": 7,
"delta3": 2,
"n": 31
},
{
"B": 83,
"HR": 74,
"HR2": 76,
"N": 65,
"Xiong": 81,
"delta1": 9,
"delta2": 7,
"delta3": 2,
"n": 32
},
{
"B": 86,
"HR": 77,
"HR2": 79,
"N": 67,
"Xiong": 83,
"delta1": 9,
"delta2": 7,
"delta3": 3,
"n": 33
},
{
"B": 88,
"HR": 78,
"HR2": 80,
"N": 69,
"Xiong": 86,
"delta1": 10,
"delta2": 8,
"delta3": 2,
"n": 34
},
{
"B": 91,
"HR": 81,
"HR2": 83,
"N": 71,
"Xiong": 88,
"delta1": 10,
"delta2": 8,
"delta3": 3,
"n": 35
},
{
"B": 94,
"HR": 84,
"HR2": 86,
"N": 73,
"Xiong": 91,
"delta1": 10,
"delta2": 8,
"delta3": 3,
"n": 36
},
{
"B": 96,
"HR": 85,
"HR2": 87,
"N": 75,
"Xiong": 93,
"delta1": 11,
"delta2": 9,
"delta3": 3,
"n": 37
},
{
"B": 99,
"HR": 88,
"HR2": 90,
"N": 77,
"Xiong": 96,
"delta1": 11,
"delta2": 9,
"delta3": 3,
"n": 38
},
{
"B": 102,
"HR": 91,
"HR2": 93,
"N": 79,
"Xiong": 98,
"delta1": 11,
"delta2": 9,
"delta3": 4,
"n": 39
},
{
"B": 104,
"HR": 92,
"HR2": 94,
"N": 81,
"Xiong": 101,
"delta1": 12,
"delta2": 10,
"delta3": 3,
"n": 40
}
],
"schema": "hilbert16-i-lienard-arithmetic/v1"
}