The Ladder of Truncation L2: Generation-Side Identity and Extension-Side Rupture—A Split Adjudication of the Member-Layer Criterion and the Selberg Axiom System

The precise equivalence problem between the member-layer criterion and Selberg axioms S1–S4 is re-adjudicated as a “split closure.” This adjudication doesnot depend on the existence of any natural-boundary witness; its unconditionalpart is carried by an explicit object: W0(s) = (1 − 2−s)ζ(s), whose coefficients are1 at odd integers and 0 at even integers. W0 satisfies the member-layer conditions,S1, S2, and S4, but does not satisfy S3; the proof of the failure of S3 is an elementary zero argument that uses only the standard fact that ζ has no zeros onℜs = 1 and ℜs = 0. Therefore the statement that the member-layer conditionsdo not imply the negation of S3 is unconditional and self-sufficiently verifiable.The universal negation on the S2 side remains conditional: if there exists a witness sequence satisfying bounded multiplicativity, abscissa of absolute convergence1, natural boundary at ℜs = 1, and computable coefficients, then the memberlayer conditions do not imply S2; the existence of the witness is registered as ablocking-level ledger item, blocking only the S2-side proposition and not the overalladjudication. The generation-side correspondence is synthesized from a positiveconstruction and a reverse Euler-product decomposition, characterizing the loadbearing upper limit of the member layer: it carries exactly S1 and S4 and does notcarry extension-side information. The machine-check protocol provides finite-rangeexact arithmetic credentials on the generation side and the member side; naturalboundary and non-continuability have no machine criterion. The net effect of thisadjudication is that the negation of the original equivalence claim changes from“depending on an unverified witness” to unconditional and elementary, and theextension-side debt contracts from the foundation of all conclusions to a specialledger item for the strengthened S2 version.