{
  "protocol": "Each geometry is applied to each modality independently, so no transform crosses the gauge or uses the pairing. The bound is the anchor upper bound; the last column re-computes it with a rank comparison operator instead of a linear one.",
  "label": "natural-best",
  "rows": [
    {
      "geometry": "euclidean (current)",
      "correlation": 0.7157944505313568,
      "anchor_bound": 0.3359375,
      "anchor_bound_rank_operator": 0.1453125
    },
    {
      "geometry": "global rank (copula)",
      "correlation": 0.6453667782045476,
      "anchor_bound": 0.171875,
      "anchor_bound_rank_operator": 0.1453125
    },
    {
      "geometry": "row rank",
      "correlation": 0.6374571651517471,
      "anchor_bound": 0.165625,
      "anchor_bound_rank_operator": 0.1484375
    },
    {
      "geometry": "local scaling",
      "correlation": 0.7162509779848556,
      "anchor_bound": 0.3625,
      "anchor_bound_rank_operator": 0.1703125
    },
    {
      "geometry": "diffusion t=3",
      "correlation": 0.5659469014331384,
      "anchor_bound": 0.0546875,
      "anchor_bound_rank_operator": 0.0453125
    },
    {
      "geometry": "heat kernel",
      "correlation": 0.7381033036202107,
      "anchor_bound": 0.31875,
      "anchor_bound_rank_operator": 0.1546875
    },
    {
      "geometry": "hyperbolic d=16",
      "correlation": 0.7135662664726405,
      "anchor_bound": 0.246875,
      "anchor_bound_rank_operator": 0.1703125
    }
  ],
  "best": {
    "geometry": "local scaling",
    "correlation": 0.7162509779848556,
    "anchor_bound": 0.3625,
    "anchor_bound_rank_operator": 0.1703125
  }
}