The Full-Gram First-Return Renewal Law: Two Residuals, a Finite Certificate, and Tail Reconstruction

We derive a full-Gram renewal law for first-return histories generated by unitary evolution. Recording every first-hitting time yields an isometry of the complete history that preserves the full Hermitian Gram matrix of a family of inputs, including all complex cross-correlations. The resulting balance separates finite exits from two positive residuals: amplitude trapped forever within a single excursion and amplitude undergoing infinitely many completed returns. The vanishing of their sum characterizes completeness of finite physical exits. For variable return sections, every positive labelled Gram record, together with prescribed trapping and persistent-tail Grams, admits a realization. For a stationary section of dimension \(rsec\), where the same first-return amplitudes act at every layer, all layer Grams satisfy a common linear recurrence of order at most \(rsec^2\), and their generating functions are rational. The first \(2rsec^2\) Grams determine the entire sequence within this class. A test of length \(rsec\) decides whether the persistent tail vanishes and yields exponential decay; when the hidden residual is zero, it also decides completeness of finite exits. Recorded times define a positive matrix-valued measure. Its waiting-record Gram integrates to the delay moment, while radial readout reconstructs individual layer Grams and their moment hierarchy. The two-residual decomposition remains stable under controlled apparatus limits. A local q³D construction realizes finite exits, hidden trapping, persistent returns, and nonzero complex cross-terms within a single unitary apparatus.