Structural Compatibility V: Conditional Lorentzian Principal Structure and Quadratic Mass-Shell Constraint from External Translation-Type Actions

Structural Compatibility V develops the fifth step of the Structural Compatibility series, explicitly on the basis of the broader Finite-Horizon Structures I–VI framework. Starting from the Hermitian-projective operatorial regime established in the preceding Structural Compatibility articles, it addresses the next major question left open by the internal operatorial layer: what external kinematical structure becomes available when a distinguished commuting Hermitian family is reinterpreted as an external translation-type action and supplemented by finite-propagation and covariance requirements. The article begins by extending the inherited complex Hermitian regime toward an external translation setting. It then introduces a finite-propagation requirement motivated by the maintenance perspective and shows that, once this requirement is combined with a local differential admissibility hypothesis at principal level and the existence of a well-posed evolution problem with a single distinguished evolution parameter, the admissible principal propagation structure can no longer remain elliptic or Euclidean in type. Under these added assumptions, the principal structure is forced into a Lorentzian form. On this basis, the paper introduces an explicit Lorentz-covariance hypothesis at state-space level, requiring the translation generators to transform covariantly under a continuous linear action of the proper orthochronous Lorentz group. Within this Lorentz-covariant setting, the canonical quadratic contraction of the translation generators is shown to be, up to normalization, the unique nontrivial quadratic scalar intrinsically available from the external translation structure. The article then derives the sectorial consequence of this invariant. On irreducible sectors of the joint translation–Lorentz action, the quadratic invariant reduces to a scalar multiple of the identity and yields the operatorial mass-shell constraint. The result is therefore a structural theory of external relativistic quadratic kinematics: the paper does not yet provide a first-order propagation law, but it isolates the minimal Lorentzian principal form and the corresponding canonical quadratic invariant that become available once the external translation setting is supplemented by the required additional assumptions. The scope of the article remains deliberately limited. No first-order factorization, no Clifford algebra, no spinorial realization, no gauge coupling, and no quantum field-theoretic completion are assumed or derived. The paper does not claim to provide a full relativistic dynamics, but rather isolates the missing quadratic kinematical layer between the internal Hermitian-projective regime of the earlier Structural Compatibility articles and the later search for a minimal first-order relativistic realization. This article is part of the Ranesis framework, developed by Alexandre Ramakers.