Temporal History-Dependent Routing in a Coherence Field: EP-Adjacent Control from First-Principles Derivation

This work develops a first-principles framework for temporal history-dependent routing in non-Hermitian dynamical systems derived from an underlying coherence-based scalar field theory.

Starting from an exact Euler–Lagrange linearization about a stationary background with finite spatial phase gradient, the analysis shows that amplitude–phase mixing arises at linear order. This produces asymmetric, non-normal coupling pathways without invoking gain–loss symmetry, phenomenological parity–time constructions, or imposed exceptional points.

The resulting quadratic field theory is reduced to finite-dimensional generators that retain the essential non-normal structure. A four-mode generator is constructed directly from the field linearization, and a minimal three-mode routing spine is introduced to study temporal switching. Forward and backward switching protocols with identical instantaneous parameters but different temporal ordering are shown to yield distinct terminal states, demonstrating intrinsic history dependence even under strictly lossy dynamics.

The mechanism is then extended to controlled activation through subcritical negative damping, which amplifies history-dependent routing while enforcing bounded evolution. Systematic numerical cartography identifies a finite EP-adjacent corridor characterized by timing-resonant sensitivity, strong transient interference, and pronounced dependence on loss-pattern structure.

Finally, mechanically and electronically realizable surrogate platforms are proposed, including coupled pendula, linear carts, and RLC circuits, enabling direct experimental falsification of the temporal routing predictions.

The results establish temporal ordering as a genuine control parameter in non-normal systems derived from conservative field theories, and position EP-adjacent transient dynamics as a robust route to history-dependent signal routing.