Structural Compatibility VI: First-Order Relativistic Factorisation and Conditional Clifford-Spinorial Realization

Structural Compatibility VI develops the sixth step of the Structural Compatibility series on the basis of the broader Finite-Horizon Structures I–VI framework and of the relativistic quadratic sector established in Structural Compatibility V. Starting from the Lorentz-covariant quadratic mass-shell structure obtained on irreducible external sectors, it addresses the next question left open by that result: under which additional assumptions can this quadratic relativistic layer be realized as a minimal first-order propagation law acting on a single genuine state space? The article begins by clarifying why a purely scalar second-order realization is structurally insufficient for the purposes of the present programme. Although such a realization reproduces the relativistic quadratic dispersion relation, its Cauchy problem closes only on doubled data and therefore does not provide propagation on a single complete state object in the sense required by the maintenance-compatible framework. The relativistic realization problem must therefore be reformulated as a first-order factorisation problem. The paper then introduces a translation-linear factorisation ansatz together with an explicit internal-external tensor decomposition of the propagated state space. Within this setting, the external translation generators act on the inherited relativistic sector, while the coefficient operators act on an internal sector and commute with the external translation action. Under these compatibility assumptions, and under an additional external quadratic non-degeneracy hypothesis ensuring that the external quadratic sector separates symmetric coefficients, the factorisation problem is shown to force the Lorentzian Clifford anticommutation relations. On this basis, the article derives the structural consequences of the factorisation step. A one-dimensional internal realization is excluded, so the propagated state space must carry a genuine multicomponent internal sector. In the complex irreducible case relevant to four-dimensional Lorentzian kinematics, the minimal such realization is spinorial. The resulting first-order propagation law is therefore of Dirac type, not as an independently postulated starting point, but as the minimal admissible realization selected within the stated class of factorisations. The article also shows that this first-order spinorial layer can be coupled back to the Hermitian-projective maintenance framework inherited from Structural Compatibility II–III. Under a maintenance-admissible Hamiltonian realization, and under the invariance of suitable ray-level coherence and horizon descriptors, the associated maintenance-compatible scalar is preserved along the induced projective evolution. The result is thus a conditional compatibility statement linking the quadratic relativistic sector of Structural Compatibility V to a minimal first-order Clifford-spinorial realization and to a compatible maintenance layer. The scope of the article remains deliberately limited. No gauge coupling, no interacting field structure, no second quantization, and no quantum field-theoretic completion are assumed or derived. The paper does not claim to provide a full relativistic theory of matter, but rather isolates the missing first-order structural layer that bridges quadratic relativistic kinematics and spinorial propagation within the Structural Compatibility programme. This article forms part of the Ranesis framework developed by Alexandre Ramakers.