Full Darkon-Euler-Lagrange Equations of Motion and Field Dynamics of the Darkon-Allotey Macro-Wave Formalism in Big Boot Gauge Theory

Abstract

We present a complete variational field theory derivation of the Euler-Lagrange Equationsof Motion (EoM) governing the Darkon-Allotey Macro-Wave Quantum Mechanics framework within Big Boot Gauge Theory (BBGT). By explicitly varying the non-associative Lagrangian density L_ZIP defined over the octonionic manifold OZIP with respect to both the complex order-parameter field Ψ(n)M and its Dirac conjugate Ψ(n)†M, we establish the exact field equations without heuristic approximations. We derive the complete field Euler-Lagrange system, including non-linear octonionic associator forces, Allotey core-hole scattering phase shifts, and non-orientable Mobius topological derivatives. Furthermore, we construct theassociated conserved Noether probability currents Jµ(n) and energy-momentum stress tensor Tµν(n), revealing how non-associative torque anomalies enforce topological stability and scale-inversion decoherence protection across channel force couplings n ∈ {−1, . . . ,6}.