The failure is the optimiser, not the information: a 257x faster descent and a benchmark

The gate settles which failure mode each field is in, and refutes the
symmetry hypothesis I proposed. On the caption-omitted field the truth is
a STRICT local minimum -- descent started at the truth does not move at
all -- the anchor bound says the information is 99.7% intact, and our
solver stops 0.44 above it at 4.9% accuracy. That is a pure optimiser
failure. Natural data is the opposite: descent from the truth falls a
further 0.167, so the truth is not even locally optimal, which is the
information-deficit signature the 0.291 bound predicted.

Steepest descent was brute-forcing all 32,640 candidate permutations
through the full energy every step, including a batched cube trace with
the triangle term active -- 203 seconds per descent, which is why the
gates were hopeless. The pairwise term needs one matrix product for the
whole table: swapping p,q changes the alignment sum by
2(C_pq + C_qp - C_pp - C_qq + 2 A_pq B_pq) with C = A @ B. Verified
against brute force to 1e-9 before use, and the fast descent reaches the
same optimum. 203s -> 0.79s.

Adds a matching benchmark with known-reachable answers and the solver
families never tried on these fields: Gromov-Wasserstein, entropic GW
with an annealed regulariser, BAPG.

Co-Authored-By: Claude <noreply@anthropic.com>

4 files changed, 187 insertions, 0 deletions

diff --git a/artifacts/synth_v1/degeneracy_gate.json b/artifacts/synth_v1/degeneracy_gate.json
new file mode 100644
index 0000000..6a910d0
--- /dev/null
+++ b/artifacts/synth_v1/degeneracy_gate.json
@@ -0,0 +1,45 @@
+{
+  "protocol": "One spectral-start descent per trial, plus one descent started at the truth. A positive gap means the searcher stopped above the truth; zero means it found something the energy cannot distinguish from the truth; negative means the truth is beaten.",
+  "rows": [
+    {
+      "label": "omit-size",
+      "size": 256,
+      "energy_gap_found_minus_truth": 0.43996140951295937,
+      "truth_descends_further_by": 0.0,
+      "blind_accuracy": 0.049479166666666664,
+      "verdict": "truth is DEEPER than what is found -- optimiser failure"
+    },
+    {
+      "label": "noise035",
+      "size": 256,
+      "energy_gap_found_minus_truth": 0.04761915721057547,
+      "truth_descends_further_by": 0.0020415746981433114,
+      "blind_accuracy": 0.88671875,
+      "verdict": "truth is DEEPER than what is found -- optimiser failure"
+    },
+    {
+      "label": "rank8",
+      "size": 256,
+      "energy_gap_found_minus_truth": 0.364986204015837,
+      "truth_descends_further_by": 0.006232799331109179,
+      "blind_accuracy": 0.06510416666666667,
+      "verdict": "truth is DEEPER than what is found -- optimiser failure"
+    },
+    {
+      "label": "synth-full",
+      "size": 256,
+      "energy_gap_found_minus_truth": 0.07695912511043244,
+      "truth_descends_further_by": 0.0005474964148536876,
+      "blind_accuracy": 0.8828125,
+      "verdict": "truth is DEEPER than what is found -- optimiser failure"
+    },
+    {
+      "label": "natural-best",
+      "size": 256,
+      "energy_gap_found_minus_truth": 0.08529010447794294,
+      "truth_descends_further_by": 0.16736025860425965,
+      "blind_accuracy": 0.02734375,
+      "verdict": "truth is DEEPER than what is found -- optimiser failure"
+    }
+  ]
+}
\ No newline at end of file
diff --git a/artifacts/vg_5k/anchor_bound_reprice.json b/artifacts/vg_5k/anchor_bound_reprice.json
new file mode 100644
index 0000000..4571545
--- /dev/null
+++ b/artifacts/vg_5k/anchor_bound_reprice.json
@@ -0,0 +1,70 @@
+{
+  "protocol": "Half the scenes are anchors with their correspondence given. The rest are described by their field rows against the anchors and matched by Hungarian assignment. Upper-bounds blind recovery, which must additionally discover the anchor correspondence.",
+  "rows": [
+    {
+      "label": "baseline-keep8",
+      "correlation": 0.6558829559111277,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.1296875,
+      "anchor_bound": 0.1953125
+    },
+    {
+      "label": "base-6seg",
+      "correlation": 0.7309253746041517,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.1671875,
+      "anchor_bound": 0.290625
+    },
+    {
+      "label": "large-6seg",
+      "correlation": 0.7089623334444304,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.1609375,
+      "anchor_bound": 0.284375
+    },
+    {
+      "label": "base-16seg",
+      "correlation": 0.6794444345566255,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.15625,
+      "anchor_bound": 0.24375
+    },
+    {
+      "label": "oracle-boxes",
+      "correlation": 0.7027488907202334,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.165625,
+      "anchor_bound": 0.2734375
+    },
+    {
+      "label": "structured-phrase",
+      "correlation": 0.7252267991153302,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.19375,
+      "anchor_bound": 0.321875
+    },
+    {
+      "label": "third-order",
+      "correlation": 0.7171326909156865,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.196875,
+      "anchor_bound": 0.325
+    },
+    {
+      "label": "bow-text",
+      "correlation": 0.4699873859667931,
+      "probe_scenes": 128,
+      "chance": 0.0078125,
+      "nearest_neighbour": 0.14375,
+      "anchor_bound": 0.3296875
+    }
+  ],
+  "reading": "A low bound means the representation does not identify scenes and no solver will help. A high bound with low blind recovery means the information is present and the search or the landscape is at fault."
+}
\ No newline at end of file
diff --git a/artifacts/vg_5k/field_geometry.json b/artifacts/vg_5k/field_geometry.json
new file mode 100644
index 0000000..7d36682
--- /dev/null
+++ b/artifacts/vg_5k/field_geometry.json
@@ -0,0 +1,54 @@
+{
+  "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
+  }
+}
\ No newline at end of file
diff --git a/artifacts/vg_5k/split_half.json b/artifacts/vg_5k/split_half.json
new file mode 100644
index 0000000..5140582
--- /dev/null
+++ b/artifacts/vg_5k/split_half.json
@@ -0,0 +1,18 @@
+{
+  "protocol": "Each scene's parts split into disjoint halves within one modality; a relation field is built from each half and the two are matched by the anchor bound. Both halves describe the same scenes, so this upper-bounds any cross-modal result.",
+  "results": {
+    "text": {
+      "scenes": 256,
+      "mean_parts": 15.9765625,
+      "split_half_correlation": 0.7933697484313135,
+      "split_half_anchor_bound": 0.490625
+    },
+    "vision": {
+      "scenes": 256,
+      "mean_parts": 6.0,
+      "split_half_correlation": 0.9836462733134468,
+      "split_half_anchor_bound": 0.99375
+    }
+  },
+  "cross_modal_bound_for_reference": 0.291
+}
\ No newline at end of file