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+# The search was the narrow part
+
+*2026-08-01. What the project had tried for matching was one family, and the
+cost of that was measured rather than guessed.*
+
+**Result in one line.** On a field whose information is 99.7% intact, fifteen
+solvers spanning spectral, Birkhoff, Gromov-Wasserstein and convex-concave
+relaxations all return under 5%; the instance yields not to a sixteenth solver
+but to amplification — a diverse pool of cheap descents, voted, rounded, and
+fed back — because exact descent on that field multiplies a partial answer by
+about four.
+
+## A benchmark with known-reachable answers
+
+Natural data is a bad place to test solvers: when everything fails there, no
+one can tell a weak method from an impossible instance. The anchor bound fixes
+that by pricing each field's information content separately, so a benchmark can
+be built where the answer is known to be reachable.
+
+| Field | what it is | anchor bound | previous best blind |
+|---|---|---|---|
+| synth-full | the world that recovers | 0.99 | 0.958 |
+| rank8 | same, truncated to rank 8 | 0.93 | 0.076 |
+| omit-size | same, captions never state size | 0.997 | 0.044 |
+| natural | Visual Genome | 0.29 | 0.000 |
+
+Natural data is kept in as a negative control: a method that scores there is
+reporting a bug, not a result. None did.
+
+## Two solvers beat the pipeline we spent months on
+
+| Method | synth-full | rank8 | omit-size |
+|---|---|---|---|
+| Umeyama | 0.540 | 0.016 | 0.014 |
+| GRAMPA + exact descent *(the pipeline)* | 0.884 | 0.076 | 0.044 |
+| Gromov-Wasserstein | 0.397 | **0.881** | 0.022 |
+| GW, 12 restarts | 0.397 | 0.881 | 0.022 |
+| GW, KL loss | 0.089 | 0.251 | 0.000 |
+| entropic GW, annealed | **0.961** | **0.887** | 0.034 |
+| entropic GW, deep anneal | — | 0.013 | 0.034 |
+| BAPG GW | 0.418 | 0.233 | 0.012 |
+| semirelaxed GW | — | 0.044 | 0.013 |
+| FAQ (Frank-Wolfe) | — | — | 0.021 |
+| FAQ, 30 restarts | — | — | 0.021 |
+| PATH convex-concave | **0.961** | 0.866 | 0.004 |
+| moment ladder tr(M^k) | 0.112 | 0.214 | 0.012 |
+| consensus vote, 65 descents | 0.932 | — | 0.127 |
+
+Two things fall out immediately. **Entropic GW with an annealed regulariser and
+PATH both reach 0.961 on the reference field, above the 0.958 that the
+project's own composed pipeline reached after months of tuning** — and neither
+had ever been run. And on the rank-8 field, GW reaches 0.881 where GRAMPA
+reaches 0.076.
+
+That second number retires a conclusion drawn earlier the same day. A
+rank-truncation ladder had appeared to show that the width of the shared
+spectrum governs recovery; the ladder was run entirely with GRAMPA, and GRAMPA
+degrades on fields with clustered eigenvalues, which is exactly what truncation
+produces. **The rank ladder was measuring a spectral solver's sensitivity, not
+an information limit** — the fourth time in this project that a solver
+limitation has been read as an intrinsic one, and the reason the standing rule
+is now to check the anchor bound before concluding anything is impossible.
+
+More annealing is not better. Extending the entropic schedule from five steps
+ending at 2e-3 to eleven ending at 1e-3 takes rank8 from 0.887 to 0.013 and
+costs 29× the time: once the coupling sharpens into a near-deterministic
+transport plan it locks, and stopping while it is still soft is what works.
+
+## The hard instance, and why it is hard
+
+Deleting one word class from the synthetic captions produces a field that
+defeats everything. Its diagnosis is unusual and each part was measured.
+
+**The information is intact.** Anchor bound 0.997 — give the solver half the
+correspondence and Hungarian assignment recovers essentially all of the rest.
+Five percent of the correspondence, twelve pairs, is worth 62%.
+
+**The truth is a strict local minimum.** Descent started exactly at the truth
+does not move at all, and every solver lands at an energy 0.44 above it against
+E(truth) = 0.34.
+
+**Its basin is wide, not a needle.** Perturb the truth by *k* random
+transpositions and descend: at k=16 it returns to the exact truth in 100% of
+trials, at k=32 in 67%, and at k=128 it still reaches 0.90 accuracy.
+
+Those three look contradictory next to fifteen failures until the basin is
+measured in the right coordinate. 128 random transpositions leave about 94 of
+256 scenes in place, so "k=128 works" means "a start that is already 37%
+correct works". Re-running the probe in accuracy rather than distance shows
+what descent actually does:
+
+| start accuracy | 0.05 | 0.10 | 0.15 | 0.20 | 0.30 | 0.50 |
+|---|---|---|---|---|---|---|
+| after descent | 0.152 | 0.420 | 0.614 | 0.783 | 0.806 | 0.875 |
+
+**Descent is an amplifier with a gain near four, and no sharp threshold.** The
+failure was never the local search and never the information. It was that
+every initialiser — spectral, GW, Birkhoff, path-following — lands in the same
+systematically wrong region and cannot clear the entry price.
+
+## Amplification
+
+If the field supplies an amplifier, the job is to feed it, not to replace it.
+Run a diverse pool of cheap descents from random and spectral starts, let them
+vote on pairings, round the vote matrix to a permutation by Hungarian
+assignment, descend from that, and rebuild the pool by perturbing the result.
+Each round hands the amplifier a better start than the last.
+
+Two details earn their place. **Rounding the vote matrix beats taking its
+argmax** — 0.207 against 0.127 on the first round — because the argmax is not a
+permutation and discards the assignment constraint. And ladders vary widely, so
+several are run and the best is chosen **by energy, never by accuracy**, which
+keeps the selection blind; on the first trial of the tuned run this picked
+0.188 out of {0.094, 0.043, 0.188}, the best of the three.
+
+On the instance that defeated fifteen solvers, the ladder climbs 0.13 → 0.68 →
+0.72 across rounds, against 0.044 for the best solver and 0.127 for voting
+alone.
+
+One negative result on the design. Injecting 25% fresh random starts into each
+rebuilt pool, intended to rescue a plateaued ladder, instead dilutes the vote
+with descents that agree on nothing and drops the first trial to 0.188. The
+same shape as the annealing result: the more conservative option was worse.
+
+## What this changes
+
+The register's search rows were closed on the strength of tempering
+equilibrating and spectral methods stalling, both of which are now known to be
+statements about two particular algorithms. They are reopened. More usefully,
+the diagnosis generalises past this instance: **wherever the anchor bound is
+high and blind recovery is low, the gap is amplification, and the question is
+how to buy an initial 10–20% rather than how to build a better optimiser.**
+That is the same statement the anchor curve makes from the other side, where
+twelve correct pairs are worth the whole problem.
+
+Engineering note, because it gated everything above. `steepest_descent` was
+forming all 32,640 candidate permutations each step and scoring them through
+the full energy, including a batched cube trace when the third-order term was
+active — 203 seconds per descent, which is why three earlier gate runs produced
+no output in 50 to 89 minutes. The pairwise term needs one matrix product for
+the whole table: with C = A @ B, swapping p and q changes the alignment sum by
+2(C_pq + C_qp − C_pp − C_qq + 2 A_pq B_pq). Checked against brute force to 1e-9
+before use. **203 s → 0.79 s**, and every experiment in this document is
+downstream of that.