data/global_integer_problem.json

{
  "name": "Phi(n,m,N)",
  "formal_definition": "SAELYRA manuscript, Definition 8.1 and Theorem 8.3",
  "dimension_of_original_target": 5,
  "maximum_directions_of_original_target": 32,
  "level": null,
  "universal_level_verified": false,
  "body_inputs": "Every subset W of {0,...,N}^n containing N*e_i for every coordinate i",
  "body": "P_W = (1/N) conv {sigma*w : w in W, sigma in {+1,-1}^n}",
  "body_test_points": "Y_W = {b in Z^n : b/N in P_W}",
  "normal_test_points": "Z_N = {z in Z^n : sum_i abs(z_i) = N}",
  "support_integer": "s_W(z) = max_{w in W} sum_i abs(z_i)*w_i",
  "candidate_directions": "At most m integer vectors a_r in [-N,N]^n",
  "test": "For every b in Y_W, some r satisfies z dot a_r <= -4*(n-1)*N for every z in Z_N with s_W(z) - z dot b <= (4*n-3)*N",
  "five_dimensional_near_threshold": "17*N",
  "five_dimensional_inward_threshold": "-16*N",
  "scope_warning": "No value of N is certified universally. The two-dimensional cube sanity check is one template only. The definition is finite but no practical complexity bound is asserted."
}