{
  "residual_sum_counts": {
    "-5": 0,
    "-4": 1,
    "-3": 2,
    "-2": 3,
    "-1": 4,
    "0": 5,
    "1": 4,
    "2": 3,
    "3": 2,
    "4": 1,
    "5": 0
  },
  "proof": "For fixed z, u lies between max(-2,z-2) and min(2,z+2), and v=z-u. Integer count is max(0,5-abs(z)).",
  "scope": "B,R>=5; apply nonnegative intersection separately when either rounded component is zero"
}