# Candidate register

The obstruction on natural data decomposes into five factors, and every
candidate attacks exactly one of them. Field correlation stands at 0.656
against the 0.9 that polynomial recovery needs, so the register is
ordered by expected movement per unit of effort rather than by elegance.

Status is one of: **done** (measured, number recorded), **running**,
**queued** (specified, not started), **open** (idea, not specified).

## A. Raise the field correlation

The binding constraint. Everything else is worth less than a tenth of
what this is worth if it moves.

| Candidate | Cost | Status |
|---|---|---|
| Continuous states instead of class codes | low | done: 0.166 to 0.489 |
| Content projection, between-scene over within-scene | low | done: 0.489 to 0.656 |
| Augmentation orbits, four views | medium | done: no gain, 0.6559 either way |
| Larger vision backbone (DINOv2-large, -giant) | medium | queued |
| Multi-scale patch features, several receptive fields | medium | queued |
| Corpus-trained text encoder instead of PPMI vectors | medium | queued |
| Structured phrase encoding: head and modifiers apart | low | queued |
| Intervention responses: occlude a region, re-encode, use the change | high | queued |
| Native multi-view corpora: product photographs, video frames | high | queued |
| Nonlinear content projection (kernel discriminant) | low | open |
| Adaptive segment count with quality filtering | low | open |

The intervention row is the one the closed world argued for hardest and
the one never run on natural data. It needs real forward passes -- masked
re-encoding on both sides -- not cached-feature algebra, which is a
tautology.

## B. Reshape the landscape

Decoys exist because they match second-order statistics. Either raise
them or widen the true basin.

| Candidate | Cost | Status |
|---|---|---|
| Third-order terms, tr(M^3) | low | done: deepens truth and decoys alike |
| Rank-truncation ladder, coarse to sharp | low | done: at chance, all ranks |
| Convex-concave path following (PATH, FAQ family) | medium | queued |
| Entropic doubly-stochastic relaxation | low | done: stalls far above truth |
| Degree normalisation of the fields | low | queued |
| Hubness correction (CSLS) on field similarities | low | queued |
| Sparsified fields, top-k neighbours only | low | queued |
| Functional term: held-out generalisation of the induced map | medium | running |

The rank ladder settles the category. At the natural-data field quality
every rank from four to full returns chance accuracy, as does full-rank
spectral initialisation with refinement -- 0.0039 against 0.0039 chance.
**Landscape reshaping does not substitute for field correlation:** below
the polynomial threshold the deficit is information the algorithm class
cannot use, not basins it cannot find, so smoothing has nothing to
recover. Path following is the canonical landscape-deformation method for
quadratic assignment and remains untried, but this result lowers its
expected value at the current correlation and it should wait until the
correlation moves. The functional term
is landscape reshaping by orthogonal information: a decoy that copies
relational statistics still has to induce a map that generalises.

## C. Search harder

| Candidate | Cost | Status |
|---|---|---|
| Spectral initialisation plus exact refinement | low | done: 53.5% to 95.3% composed |
| Parallel tempering with exact swap deltas | medium | done: equilibrates into the decoy shelf |
| Multiple spectral inits over eta, keep best by energy | low | queued |
| Seeded expansion from high-confidence pairs | medium | queued |
| Belief propagation on the assignment factor graph | high | queued |
| Cluster moves: swap blocks rather than single pairs | low | queued |
| Path relinking and crossover over permutations | medium | open |
| Branch and bound, exact on small blocks | medium | open |
| Sum-of-squares or low-degree hierarchies | high | open |

Accepting exponential cost does not rescue the hard phase at fixed size:
tempering already is exponential-time and fails by equilibrating, not by
running out of budget. Exponential methods pay only where the block is
small enough to finish, which is why C and D are the same programme seen
from two sides.

## D. Shrink the problem

The phase boundary is joint in correlation and population size, so a
corpus need not be matched at once.

| Candidate | Cost | Status |
|---|---|---|
| Largest recoverable N at the natural field quality | low | running |
| Diversity-selected bootstrap subset instead of random | low | queued |
| Hierarchical: cluster first, match within cluster | medium | queued |
| Many small blocks, merge by consistency voting | medium | queued |

Diversity selection is the cheapest idea in the register and possibly
the best value: decoys arise from scenes that resemble each other, and
choosing a mutually dissimilar bootstrap set thins them at no
algorithmic cost. It composes with small blocks by construction, since
if only a few hundred pairs are needed they should be the most
distinguishable few hundred.

## E. Add orthogonal signal

| Candidate | Cost | Status |
|---|---|---|
| Held-out generalisation of the induced map | medium | running |
| Intervention response consistency | high | queued |
| Counting and cardinality, gauge-free by construction | low | queued |
| Higher-order joint tables (colour by shape by count) | low | queued |
| Temporal or sequential structure where corpora carry it | high | open |

## F. Change the formulation

Real corpora are not in bijection, so the permutation formulation is a
scaffold that has to be replaced eventually regardless of performance.

| Candidate | Cost | Status |
|---|---|---|
| Unbalanced coupling: partial mass, unmatched items allowed | medium | queued |
| Many-to-many soft assignment | medium | open |
| Part-level matching across the corpus rather than scene-level | medium | open |
| Joint factor-level and scene-level, alternating | medium | open |

## Order of attempt

Cheap and untried first, because the register's value is in eliminating
branches quickly: diversity selection, degree normalisation, hubness
correction, sparsified fields, multiple spectral inits, structured phrase
encoding, higher-order joint tables. Then the medium tier led by path
following and the functional term. The two expensive rows worth pre-
committing to are intervention responses and a natively multi-view
corpus, because the closed world named both as protocol requirements
rather than optimisations.