Foundations of Codex Alpha Dynamics: A Variational Derivation of an Informational Extension of Einstein Field Equations

This manuscript presents a minimal variational formulation of the dynamical core of the Codex Alpha theoretical framework. Starting from an explicit action principle involving the spacetime metric, a scalar coherence field, ordinary matter fields, and a coherence-weighted Schwinger functional, the work derives an informational extension of the Einstein field equations. The central result is the field equation G_{μν} + Λg_{μν} = (8πG / c⁴) ⟨ T̂_{μν} ⟩_{∇𝒦}, where the right-hand side is defined as the metric response of a coherence-weighted functional measure rather than introduced phenomenologically. The manuscript is restricted to the effective continuous sector of the Telascura, the regime in which the underlying discrete informational structure of Codex Alpha can be approximated by a smooth Lorentzian manifold. Non-local, retrocausal, and fully discrete nodal effects are explicitly outside the scope of this work and are treated in the broader Codex Alpha 3.0 — Iconoclast Edition. The paper establishes the main assumptions, dimensional consistency, minimality of the scalar coherence extension, absence of Ostrogradsky ghosts in the minimal sector, recovery of General Relativity in the uniform-coherence limit, and falsifiability through possible deviations in cosmological perturbations, gravitational-wave propagation, gravitational lensing, and post-Newtonian regimes. This work is intended as a companion foundational manuscript to Codex Alpha 3.0, isolating and deriving the minimal mathematical core of the effective continuous dynamics. Further information about the Codex Alpha research programme is available at: https://codexalpha.org