Raw Strongly Tree-Child Level-2 Species Networks from Metric Genealogy Laws under the NMSC: Signed pair co-occupancy, rooted graph reconstruction, and demographic fibers

We classify the observable rooted graph and parameters of finite strict temporally separated binary strongly tree-child species networks of level at most two under the independent-lineage Network Multispecies Coalescent (NMSC), with a fixed known surjective taxon sampling map. Equality of complete normalized rooted metric-genealogy laws is equivalent, pointwise on the full positive domain, to an explicit quotient generated by eight move types. The proof recovers current pair co-occupancy from the analytic germ of a finite positive Kingman mixture. Signed jumps reveal every tree event and every pair-exposed reticulation. Descendant clusters, chronology, tree-childness and level-two shortcut completion then reconstruct the retained graph, including arbitrary decorations. Deterministic witnesses and actual three-lineage merger densities recover oriented inheritance weights and the entire finite demographic rate words on exposed populations. Scalar propagation through ancestral reticulations links both incoming arms, including equal rates and inheritance one-half. The indexed collection of all genealogy marginals on at most three sampled lineages suffices for the complete quotient. A reticulation with one descendant sampled lineage retains its two typed first-exposure outlets and route weights, while its internal clock and singly occupied demographic words are unobservable. Sister-transfer direction is invisible exactly when both sides are sample unary; the corresponding reversal transfers the whole exposed donor rate word. Together with presentation suppression, equal-rate composition, unary-corridor replacement, parent exchange and labelled isomorphism, these responses exhaust the quotient. Exact hidden-clock and rank-one nonbridge examples explain essential obstructions. A separate graph argument derives the five cores and twelve minimum strong repairs. The companion study transfers the classification to all indexed ordered locus laws with known boundaries, jointly with the stationary distribution and trace-normalized shared irreducible stationary reversible substitution generator. The reconstruction uses observable current-grouping scalars throughout. Notes: Version 1.0.0 preprint, manuscript dated 5 September 2026. The accompanying files contain the main paper, technical supplement, manuscript sources, and curated reproducibility materials with exact technical checks. The three-lineage result concerns the indexed collection of all marginals on at most three sampled lineages of the same fixed known sampling design; it is a sufficient bound, with no claim of sharpness or recovery from one fixed triple. The sequence corollary uses the companion paper Direct Sequence Laws under the Network Multispecies Coalescent.