Direct Sequence Laws under the Network Multispecies Coalescent: A genealogy equivalence, canonical coalescent-response behavior, and finite determination with an epoch-complexity obstruction

We determine the exact population-level information in multilocus sequence alignments generated by a temporal species network under the independent-lineage Network Multispecies Coalescent (NMSC), without treating estimated gene trees or concordance factors as observed data. With known nonrecombining locus boundaries, contemporaneous labelled sampling, and one shared irreducible stationary reversible finite-state continuous-time Markov chain, the indexed family of length-L locus laws is equivalent to the stationary distribution, the trace-normalized substitution generator, and the mutation-scaled metric-genealogy law. The first step is a compact moment theorem for the random conditional site-pattern tensor. A determinant coordinate then recovers each realized additive gene-tree metric, while a continuum in the ancestral coalescence tail identifies the shared generator, including JC, K2P, K3P, and repeated-eigenvalue GTR strata. We compile every finite temporal NMSC presentation into a finite typed continuous-time response system on partition–occupancy states. Its canonical reachable–observable residual behavior, denoted CCR, is coordinate independent, removes analytically invisible event times, and is unique up to typed linear isomorphism. Equality of CCR is equivalent to equality of normalized metric-genealogy laws. The minimal time-indexed realization has an intrinsic positive cone, but its finite analytic encoding does not supply a stochastic occupancy basis or an NMSC graph. Literal graph recovery is impossible on the strict physical domain: arbitrary sample-unary temporal funnels have identical response in every context. Species trees are nevertheless recovered pointwise with all exposed population rates. For one-reticulation presentations we give an exact finite port-response quotient and a graph theorem on the incidence-separated subclass; the reduced level-1 result additionally assumes scalar-separator faithfulness, and no unconditional level-1 or level-2 graph theorem is claimed. The finite-locus hierarchy has two complementary conclusions. On every fixed finite union of compact semianalytic finite-presentation charts, one finite locus length determines the complete indexed family. The proof uses the Frisch–Siu Noetherian ring of analytic germs along a compact set and remains valid at coalescent–substitution resonances. No determining length can depend only on the number of sampled taxa, reticulations, alphabet size, and sampling design if demographic epoch count is unbounded. For every L we construct two strict two-taxon, zero-reticulation, known-JC demographic histories with identical laws through length L and different length-(L+1) laws. A Vandermonde Jacobian and the implicit-function theorem make the collision exact. We finally separate Borel inversion, finite proposed-model equivalence testing, moment approximation, and statistical consistency. 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 main observation is the indexed family of ordered locus laws with known locus boundaries; the graph conclusions retain the qualifications stated in the paper.