Reconstruction thresholds: scoped graph assurance and distinct tree companions

Selected reading now distinguishes the precise three-community iid-label graph corollary from the regular/Poisson tree statements, the four-state ferromagnetic theorem and the arbitrary bounded-degree tree capacity criterion. Six PDF pages were inspected, with exact extents in the ledger. Formal interfaces select above-threshold three-state reconstruction only; no native certificate or kernel ran.

Detailed review: 1,048 controls, exact parameter rectangles, finite-label null calculations and 54 generated graphs. A conventional spectral baseline substantially outperforms a forced-three-group modularity recipe on selected above-threshold graphs, showing algorithm failure differs from lack of model information. Finite equality scores do not refute an asymptotic claim. Real-model fitting and commercial value remain open.

Opportunity 032 v2 stays low confidence. Source revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb; full source proofs and programs unverified.