Boundary-Deck Propagation at a Hypothetical First Odd Localized Weil Failure: Carleman Closure, One-Sided Source Transfer, and a Zero-Gap Ancestry Barrier

This paper develops a rigorous interface theory for a hypothetical first failure of odd localized Weil positivity. Assuming the Riemann Hypothesis fails, parity-restricted continuity of the localized Weil ground value is used to identify a first support scale at which the odd localized operator acquires a zero ground state. From this state, the arithmetic and Archimedean contributions are reorganized into a boundary-deck formalism. The main results are: - a first-odd-failure construction with support saturation;- an infinite-rank obstruction to direct bounded congruence with finite-center common-depth source models;- higher prime-log translation-orbit growth together with a two-sided fixed-base metric obstruction;- a positive boundary-deck decomposition for compressed translates;- bounded and injective Carleman-type boundary closure;- a supported-domain lifting theorem for the Archimedean multiplier;- an exact boundary-response identity for the actual first-failure ground state;- a rank-one Loewner lower bound and a one-sided fixed-base source transfer;- a translation-gauge obstruction to universal pointwise response alignment;- an exact full-lattice critical-sampling provenance formula yielding the canonical source coordinates \[ x_j=Lγ_j, y_j=Lδ_j; \]- a finite-rank zero-gap ancestry theorem: exact matching of a prime-log packet to parent zero coordinates forces a bounded-height integer relation among sufficiently many zero ordinates;- a gauge-averaged same-transform coercivity theorem, including a cubic response lower bound inherited from the common-depth source-rigidity framework. A key structural distinction is established between two-sided metric identification, which fails asymptotically for the canonical fixed-base comparison, and one-sided positive-order transfer, which survives. The paper does not prove the Riemann Hypothesis and does not claim a complete realization of the parent bandwidth-one corridor. In particular, exact parent ancestry, derivation of the corridor normalization and depth parameters, whole-gauge-to-support-window concentration, and heterogeneous-depth closure remain open. This Version v1.0 is the public technical freeze of the M1-J interface stage.