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The caption-omitted field returned under 5% from every solver tried:
Umeyama, GRAMPA, five Gromov-Wasserstein variants, FAQ with and without
restarts, PATH convex-concave, a moment ladder, semirelaxed GW. The
measurements said the answer was not a sixteenth solver.
Descent on that field amplifies: a start 10% correct comes out 42%, one
20% correct comes out 78%. What no initialiser could do was clear the
entry price, since all of them land in the same wrong region. So run a
diverse pool of cheap descents, let them vote, round the vote matrix to a
permutation by Hungarian assignment rather than argmax, descend from
that, and rebuild the pool around the result. Each round feeds the
amplifier a better start.
0.044 -> 0.72 on the best run, 0.54 mean over two. Competitive elsewhere:
0.977 on the easy field against 0.961 for the best GW variant.
Also adds the diagnostics that led here: the basin-width probe (k=16
transpositions still returns to the exact truth 100% of the time) and the
capture-threshold curve that measures amplification directly.
Co-Authored-By: Claude <noreply@anthropic.com>
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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>
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