files / problems / kissing-5d / compute / q5 / n1_struct.json
{
"n": 1480,
"n_d5": 40,
"n_extras": 1440,
"n_groups": 240,
"n_extra_edges": 884720,
"n_intra_group_edges": 0,
"groups_edgeless": true,
"type_split": {
"n_four_seeds": 160,
"n_six_seeds": 80,
"four_seed_group_size": 5,
"six_seed_group_size": 8,
"type_A": {
"abs": [
4,
2,
2,
2,
2
],
"n": 160
},
"type_B": {
"abs": [
5,
2,
1,
1,
1
],
"n": 640
},
"type_C": {
"abs": [
3,
3,
3,
2,
1
],
"n": 640
},
"four_seed_type_split_unique": [
"{\"(1, 1, 1, 2, 5)\": 4, \"(2, 2, 2, 2, 4)\": 1}"
],
"six_seed_type_split_unique": [
"{\"(1, 2, 3, 3, 3)\": 8}"
],
"four_seeds_inside_a_six_seed": 0,
"comment": "Each four-seed is 1 type-A (4,2,2,2,2) plus 4 type-B (5,2,1,1,1). Each six-seed is 8 type-C (3,3,3,2,1)."
},
"proved": false,
"argument": "A missed-root colouring of extras does not prove omega(pool(U)) <= |U|. Two extras that miss the same D5 root can still kiss: 236960 kissing pairs share at least one root (287360 root-edge incidences). An explicit pair is recorded under coloring.example_pair (inner product 14 <= 16). Same-missed extras \u2014 those with the same full seed \u2014 are edgeless, but sharing a single root is weaker and is not a stable set. A principal-root colouring (the unique missed root of largest inner product, when it exists) does not cover the graph: only 640 extras have a unique principal root (the 640 type-B points); the 800 type-A and type-C extras have a tie, and 0 kissing pairs among the type-B points share a principal root. A tight star 8-clique on e0- has missed-set rank 5 over Q and 4 over GF(2), so the characteristic vectors are dependent and do not force |E| <= |U|. Hall matching of a clique into U is equivalent to the inequality itself, not a separate reason. The tests below found no U with omega > |U|, but that is a finite sample (plus the ten stars and every single seed), not a proof for every U. Leftover |U| >= 19 of the 1480-graph remains open. This does not claim tau_5 = 40.",
"counterexample": null,
"n_leads": 0,
"leads": [],
"coloring": {
"same_missed_edgeless": true,
"shared_root_kissing_pairs": 236960,
"shared_root_incidences": 287360,
"shared_root_is_stable": false,
"example_pair": {
"extra_i": [
-5,
-2,
-1,
-1,
1
],
"extra_j": [
-4,
2,
-2,
2,
-2
],
"inner": 14,
"thresh": 16,
"missed_i": 4,
"missed_j": 4,
"n_shared_roots": 1,
"shared_root_indices": [
4
]
},
"per_root_edge_min": 7184,
"per_root_edge_max": 7184,
"n_unique_principal": 640,
"principal_conflicts": 0,
"principal_example": null
},
"linear_algebra": {
"checked": true,
"star": "e0-",
"clique_size": 8,
"rank_Q": 5,
"rank_GF2": 4,
"nodes": 8,
"complete": true,
"best": 8
},
"tests": {
"node_limit": 200000,
"rng_seed": 20260827,
"stars": {
"n": 10,
"omega_eq_U": 10,
"omega_lt_U": 0,
"omega_gt_U": 0,
"incomplete": 0,
"all_complete": true,
"rows": [
{
"name": "e0-",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 10,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e0+",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e1-",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e1+",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e2-",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e2+",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e3-",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e3+",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e4-",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
},
{
"name": "e4+",
"n_pool": 80,
"k": 8,
"best": 8,
"nodes": 9,
"complete": true,
"n_seeds_in_U": 16,
"cmp": "eq"
}
]
},
"single_seeds": {
"n": 240,
"omega_eq_U": 0,
"omega_lt_U": 240,
"omega_gt_U": 0,
"incomplete": 0,
"all_complete": true,
"four_seed_best_unique": [
1
],
"six_seed_best_unique": [
1
],
"four_seed_n_pool_unique": [
5
],
"six_seed_n_pool_unique": [
8
]
},
"random_unions": {
"n": 200,
"omega_eq_U": 0,
"omega_lt_U": 200,
"omega_gt_U": 0,
"incomplete": 0,
"all_complete": true,
"n_chosen_hist": {
"3": 47,
"4": 47,
"5": 55,
"6": 51
},
"k_hist": {
"9": 5,
"10": 4,
"11": 6,
"12": 18,
"13": 6,
"14": 20,
"15": 14,
"16": 15,
"17": 15,
"18": 25,
"19": 14,
"20": 11,
"21": 15,
"22": 10,
"23": 11,
"24": 6,
"25": 4,
"26": 1
},
"best_minus_k_hist": {
"-14": 6,
"-13": 19,
"-12": 27,
"-11": 36,
"-10": 33,
"-9": 36,
"-8": 22,
"-7": 13,
"-6": 7,
"-5": 1
},
"sample": [
{
"n_pool": 30,
"k": 16,
"best": 4,
"complete": true,
"n_seeds_chosen": 4,
"n_seeds_in_U": 6,
"cmp": "lt"
},
{
"n_pool": 20,
"k": 11,
"best": 3,
"complete": true,
"n_seeds_chosen": 3,
"n_seeds_in_U": 4,
"cmp": "lt"
},
{
"n_pool": 20,
"k": 12,
"best": 3,
"complete": true,
"n_seeds_chosen": 3,
"n_seeds_in_U": 4,
"cmp": "lt"
},
{
"n_pool": 79,
"k": 17,
"best": 6,
"complete": true,
"n_seeds_chosen": 5,
"n_seeds_in_U": 14,
"cmp": "lt"
},
{
"n_pool": 66,
"k": 14,
"best": 5,
"complete": true,
"n_seeds_chosen": 4,
"n_seeds_in_U": 12,
"cmp": "lt"
},
{
"n_pool": 46,
"k": 14,
"best": 5,
"complete": true,
"n_seeds_chosen": 4,
"n_seeds_in_U": 8,
"cmp": "lt"
},
{
"n_pool": 109,
"k": 18,
"best": 7,
"complete": true,
"n_seeds_chosen": 6,
"n_seeds_in_U": 20,
"cmp": "lt"
},
{
"n_pool": 15,
"k": 11,
"best": 3,
"complete": true,
"n_seeds_chosen": 3,
"n_seeds_in_U": 3,
"cmp": "lt"
},
{
"n_pool": 204,
"k": 25,
"best": 14,
"complete": true,
"n_seeds_chosen": 6,
"n_seeds_in_U": 36,
"cmp": "lt"
},
{
"n_pool": 102,
"k": 18,
"best": 8,
"complete": true,
"n_seeds_chosen": 4,
"n_seeds_in_U": 18,
"cmp": "lt"
},
{
"n_pool": 81,
"k": 15,
"best": 6,
"complete": true,
"n_seeds_chosen": 4,
"n_seeds_in_U": 15,
"cmp": "lt"
},
{
"n_pool": 28,
"k": 10,
"best": 3,
"complete": true,
"n_seeds_chosen": 3,
"n_seeds_in_U": 5,
"cmp": "lt"
}
]
}
},
"stats": {
"n": 450,
"omega_eq_U": 10,
"omega_lt_U": 440,
"omega_gt_U": 0,
"incomplete": 0,
"all_complete": true
},
"comment": "q4 emptied every seed-union with |U| <= 18; leftover is |U| >= 19. A 41-set needs |E| >= |U| + 1. This file asks whether omega(pool(U)) <= |U| always. A yes with a proof would empty the 1480-graph. A single U with omega > |U| is a 41-lead. Tests used cliqueutil.clique_search with node_limit=200000. No claim that tau_5 = 40."
}