Gauge-Invariant Field Theory of Signal-Worker Interactions: Deriving the Logical Cloning Prohibition from First Principles of Quantum Architectonics

The transition of Quantum Architectonics from a materials-centric discipline to a fundamental ontology of information-matter interaction necessitates a rigorous field-theoretic description. This study bridges that gap by promoting the Signal-Worker ontology to a continuous Gauge-Invariant Quantum Field Theory, where “Workers” are treated as fermionic matter fields and “Signals” as non-Abelian gauge bosons. We construct a $SU(2)$ invariant Lagrangian that recovers the discrete $H_{SW}$ model as a stable infrared fixed point, demonstrating that the lattice architecture of current models is a fundamental emergent reality rather than a mere approximation. The central finding of this research is the first-principles derivation of the Logical Cloning Prohibition (LCI). We demonstrate that the LCI is not merely an architectural heuristic but a symmetry-enforced conservation law—specifically, a Ward identity of the Signal gauge field. Numerical analysis reveals that the LCI scales exponentially with system size ($N$), providing robust topological protection against local decoherence. By identifying the specific anomalous terms responsible for information robustness, we provide a theoretical blueprint for the next generation of topologically protected quantum devices, validating the Signal-Worker ontology as a complete description of quantum information dynamics.