lin-0214

17.5 Controlled relational recovery

Closed and twisted synthetic databases have similar pairwise relation graphs but different join closure. Ten paired seeds compare flattened PCA, several UMAP and Hodge projections, edge- and triangle-aware geometric transformers, and RADAR. Labels never construct a representation or select a repair. Every method receives exactly the information named in its row: pairwise baselines receive flattened relations, the triangle baseline receives engineered join features, and apex methods receive typed witnesses. A common downstream linear probe scores the resulting coordinates. Accuracy therefore measures information retained under a fixed two-dimensional bottleneck, while incidence, private-edge, anchor, and cocycle audits determine whether the RADAR realization itself is admissible.

Representation	Accuracy	ROC AUC	Interpretation
PCA, flattened relations	\(0.485\pm 0.029\)	\(0.512\pm 0.035\)	homogeneous feature projection
UMAP, \(R\)-Jaccard	\(0.506\pm 0.029\)	\(0.490\pm 0.040\)	pairwise face only
UMAP, triangle features, 2D	\(0.545\pm 0.060\)	\(0.568\pm 0.101\)	higher-order features under the same bottleneck
Pooled Hodge \(L_1\)	\(0.519\pm 0.011\)	\(0.499\pm 0.051\)	fixed typed-complex comparator
GT apex, PCA to 2D	\(\mathbf{0.906\pm 0.014}\)	\(0.905\pm 0.015\)	join-aware apex before repair
RADAR	\(\mathbf{0.906\pm 0.014}\)	\(\mathbf{0.905\pm 0.012}\)	compatible audited realization

Table 17.1 Primary endpoint at mode-1 closure probability \(0.15\). RADAR preserves the apex signal; it does not improve it using hidden labels.

Experiment: RADAR two-dimensional recovery

Against \(R\)-only UMAP, the paired accuracy gain is \(0.400\), with bootstrap interval \([0.380,0.421]\). Across the complete 50-run noise sweep, incidence falls in every run; maximum private-edge violation is \(9.34\times 10^{-9}\), minimum anchor retention is \(99.22\% \), and induced cocycle RMS is at most \(1.66\times 10^{-16}\).

Boundary

The capacity diagnostic is deliberately retained. Triangle-feature UMAP reaches \(0.906\) accuracy in 8D and 16D. The result is therefore an audited two-dimensional recovery under a severe bottleneck, not an information-theoretic separation from every engineered graph representation.