World Alignment: unpaired cross-modal correspondence by relational identifiability

Method: scene states are sets of part states; relation fields are built
within each modality and are invariant to how each side labels its own
features; the cross-modal bridge is a coupling searched under an energy
that is a closed-form functional of one matrix; solving is spectral
initialisation followed by exact local refinement.

Evidence: in a procedurally generated closed world, blind recovery of a
hidden image-caption correspondence reaches 95.3% at 256 scenes against
0.39% chance, and the recovered pairs transfer to 200 held-out scenes at
93.0% exact retrieval with random-pair and shuffled-image controls at or
near chance. Cross-modal value correspondence is derived from disjoint
corpora rather than declared. On Visual Genome the field correlation
reaches 0.656 against the 0.9 that polynomial recovery needs, with the
deficit attributed away from segmentation and discretisation.

Protocol: no image-text pair enters any objective, optimiser,
initialisation, or model selection; hidden pairs score orderings only.

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

1 files changed, 179 insertions, 0 deletions

diff --git a/RECOVERY_RESULTS.md b/RECOVERY_RESULTS.md
new file mode 100644
index 0000000..00125df
--- /dev/null
+++ b/RECOVERY_RESULTS.md
@@ -0,0 +1,179 @@
+# Blind recovery on gate-passing states: the search phase
+
+Run date: 2026-07-29
+
+## Question
+
+The basin audit licenses recovery: on content-projected VG states the true
+assignment sits 5--19% below every quench, so a search that reaches the
+bottom of the landscape would recover it. This experiment asks whether
+three search families get there without any pair information. The text
+side enters through a hidden shuffle; hidden truth scores outcomes only.
+
+## Arms
+
+- Parallel tempering: eight-replica Metropolis over permutations with
+  exact closed-form swap deltas and neighbor exchange, 60,000 rounds.
+- Entropic Sinkhorn annealing: soft coupling, barycentric states,
+  temperature and entropy schedules, four restarts.
+- Spectral-band homotopy: the Sinkhorn arm warm-started through
+  progressively finer content bands (4 to full dimensions), a spectral
+  proxy for band-limited continuation on the symmetric group.
+
+## Result: the energy is right and the search is the wall
+
+On all three gate-passing conditions (512 nodes, subset seed 0):
+
+| Condition | Gate margin | Tempering: E over true / accuracy | Sinkhorn | Homotopy |
+|---|---:|---:|---:|---:|
+| VG isolated, k=256 | +7.8--12.0% | +9.4% / 3.1% | +72% / 0.4% | +62% / 0.4% |
+| VG isolated, k=128 | +5.1--10.8% | +10.1% / 2.1% | +100% / 0.2% | +76% / 0.6% |
+| VG in-context, k=128 | +11.6--18.5% | +19.8% / 1.0% | +99% / 0.8% | +78% / 0.0% |
+
+Tempering equilibrates into exactly the quench band of the gate report --
+the glass's low shelf -- and never enters the true funnel; chance is 0.2%,
+so 1--3% accuracy is a population-level shift, not recovery. The soft
+arms stall far higher; their couplings put no verified mass anywhere
+(top-margin-decile precision 0.000 in all runs, mutual-NN precision below
+1.2%). Continuous relaxation loses to discrete annealing everywhere, and
+discrete annealing loses to the landscape.
+
+The picture matches the hard phase of planted-assignment problems: the
+configuration is information-theoretically identifiable (the gate), while
+local dynamics equilibrate into an exponential shelf of wrong states
+whose energies sit just above the planted one. The distance between the
+statistical and the algorithmic threshold is now the project's central
+quantity.
+
+## What moves the algorithmic threshold
+
+Ordered by expected yield, per the tool inventory of
+`STRUCTURE_DESIGN.md`:
+
+1. Constraint density per node: the N-scaling probe below, and at fixed N
+   the view-level recursion and R8 intervention constraints, which
+   sharpen the funnel rather than the search.
+2. Side information: gauge-free unary anchors (counting, mm-space
+   observable invariants) that bias the sampler into the funnel; in
+   planted problems even weak unary priors change the search class.
+3. Message passing: cavity/BP dynamics for planted matching reach partial
+   recovery at densities where annealing fails; this is the queued
+   spin-glass tool.
+
+## N-scaling probe
+
+Tempering-only runs at N=1024 and N=2048 with a fixed 80,000-round budget
+test whether the phase boundary approaches with constraint density. Best
+replica by energy:
+
+| N | E over true | Accuracy | Chance |
+|---:|---:|---:|---:|
+| 512 | +9.4% | 3.1% | 0.20% |
+| 1,024 | +5.2% | 2.5% | 0.10% |
+| 2,048 | +15.0% | 0.05% | 0.05% |
+
+Doubling to 1,024 cuts the energy gap by nearly half at constant budget.
+The under-equilibrated 2,048 point is budget-confounded, so an
+equal-moves-per-node rerun with 320,000 rounds was run:
+
+| N | Moves per node | E over true | Accuracy |
+|---:|---:|---:|---:|
+| 512 | ~117 | +9.4% | 3.1% |
+| 1,024 | ~78 | +5.2% | 2.5% |
+| 2,048 | ~156 | +10.4% | 0.05% |
+
+At matched per-node budget the reachable shelf returns to the same
++9--10% band as N=512, and the small accuracy signal present at 512 and
+1,024 vanishes entirely. One seed per point, so the 1,024 dip may be
+noise; what the curve rules out is the optimistic reading -- **no
+algorithmic threshold is approached by density alone within the tested
+range** (N up to 2,048, 320k rounds). The recovery bottleneck stands, and
+the priority order shifts to the other two levers: gauge-free unary side
+information and cavity-style message passing, with constraint sharpening
+(view-level recursion, interventions) as the structural route.
+
+## Unary anchors, first wave: coarse anchors do not move the search
+
+The C5 battery extracted gauge-free unary anchors with no model passes:
+numeral content, color-lexicon histograms, chromatic fraction, size and
+light words on the text side; hue-band histograms, chromatic fraction,
+luminance, and box areas on the vision side, with the lexicon-to-physics
+maps frozen before evaluation. The combined anchor field has real
+population-level signal at the true pairing (z = 17.2) and no per-pair
+reliability: median truth rank 1,969 of 5,000, top-1%-margin precision
+2%, mutual-NN precision 0.17%.
+
+Injected into the tempering energy as a dense unary field, it strictly
+degrades recovery, monotonically in the weight (best-by-relational-energy
+replica):
+
+| Unary weight | Init | Accuracy | Relational E over true |
+|---:|---|---:|---:|
+| 0 (baseline) | random | 3.1% | +9.4% |
+| 0.5 | random | 1.6% | +30.2% |
+| 2.0 | random | 2.0% | +76.2% |
+| 2.0 | anchor-Hungarian | 1.0% | +82.6% |
+| 8.0 | random | 1.0% | +115.2% |
+
+The mechanism echoes the spectral audit: scene-level color, luminance,
+and size are coarse attributes that already live inside the shared
+topical subspace, so the field adds a noisy copy of information the
+relational energy has, plus its own wrong optima for the sampler to
+compromise with. Side information can move a hard phase only if it is
+per-pair reliable (sparse verified seeds -- there are none to sparsify
+here) or statistically independent of the relational field. Pixel and
+lexicon statistics at node level are neither.
+
+The constructive redirect: fine-grained anchors live at view level, where
+region phrases carry numerals and spatial predicates about individual
+regions, boxes carry geometry, and the candidate set is sixteen rather
+than five hundred. The second anchor wave belongs inside the view-level
+recursion, not the node-level search; message passing with priors remains
+the queued search-side tool.
+
+## Unary anchors, second wave: view-level anchors yield the first seeds
+
+Per-view anchors -- twelve-class color lexicon against crop pixel classes
+(white, black, gray, and brown rules on value and saturation carry most
+of the coverage), size words against within-scene box-area ranks, light
+words against crop luminance, position words against box centers -- give
+the first per-pair-reliable signal of the project:
+
+- frame-free view matching 11.5% against 6.25% chance; 16.9% on the
+  35,484 views whose phrase carries an attribute (2.7x);
+- **seed calibration: 35.6% precision on the top-1%-margin views** (354
+  seeds at 5.7x chance), against 0--4% for every prior high-confidence
+  channel;
+- position words are nearly absent from region phrases (coverage under
+  1.1%), so color and light do the work.
+
+Aggregated to node level (Hungarian value of the per-pair 16-view anchor
+matrix as node affinity, three subset seeds at 512): retrieval R@10
+8.6--9.6% against 1.95% chance (4.4--4.9x, the best frame-free node
+signal to date; graph signatures reached 3.1--3.5x), z = 16--17, median
+rank 113--123 of 512. Node-level seed precision stays at 2--6%: one
+order below seeding threshold. Adding fixed-assignment relational
+agreement trades recall for precision without changing the order.
+
+Pixel and lexicon statistics are now exhausted at both levels. The two
+open moves are the numeral-to-repetition anchor (patch-level extraction,
+untested) and the aggregation everything above points to: a factor graph
+over node and view assignments with anchor unaries and relational
+pairwise terms, solved by message passing, so that 35%-precision view
+seeds, 5x node affinities, and the gate-passing relational energy
+reinforce globally instead of being consumed by independent per-node
+Hungarians.
+
+## Main artifacts
+
+- `artifacts/manifold_gate/recovery_vg_k{128,256}_seed0.json`
+- `artifacts/manifold_gate/recovery_vg_ctx_k128_seed0.json`
+- `artifacts/manifold_gate/recovery_scale_n{1024,2048}.json`
+- `artifacts/manifold_gate/recovery_scale_n2048_long.json`
+- `artifacts/manifold_gate/anchor_battery.json`
+- `artifacts/manifold_gate/anchor_unary.pt`
+- `artifacts/manifold_gate/recovery_unary_lam{0.5,2.0,8.0}.json`
+- `artifacts/manifold_gate/recovery_unary_lam2_unaryinit.json`
+- `artifacts/manifold_gate/view_anchor_battery_v{2,3}.json`
+- `artifacts/manifold_gate/view_anchor_features.pt`
+- `artifacts/manifold_gate/view_anchor_affinity{,_v2}.json`