Non-Holomorphic Coupling: Mode Switching Across a Singular Boundary and Self-Sustained Oscillations

We study a non-holomorphic iterative system arising from a derivation error
in the seven-space framework. The original holomorphic coupling connects the
S-space Newton iteration and the U-space entire function iteration through
the U--S relation, with monotonic relaxation dynamics. Replacing ln u with
ln s in the S-space equation renders the iteration non-holomorphic. When
coupled synchronously with the U-space iteration, the dynamics undergo a
qualitative change from monotonic relaxation to self-sustained oscillation.
We define a new special function E(w) = w e^w (e^w - 1) and establish its
zeros, critical points, and asymptotic behaviour. We prove global convergence
of the non-holomorphic iteration and derive the multiplier formula. We prove
rigorously that the partial derivative of V_s with respect to the real part
of u has opposite signs on the two sides of s = 1 (positive for s < 1,
negative for s > 1), so that the response of V_s to u reverses direction
upon crossing s = 1. The orbit repeatedly crosses s = 1, producing a period-4
self-sustained oscillation with amplitude 4.0801. The multiplier of F^4 on
the period-4 orbit is 0.504 < 1, guaranteeing strict attraction. We prove
that period-4 is the minimal closed period crossing s = 1 (crossings must be
paired, P = 2 is numerically excluded, P = 4 is confirmed by closure error
2e-17). We prove that the sign of V_s necessarily alternates on the period-4
orbit, guaranteeing a 50% proportion of s < 1. We prove that divergent
parameters form an isolated set on the real axis (the divergence condition is
a finite-step collision with singularities; F^N is real-analytic and not
identically equal to singular values; its zeros are isolated). Numerically,
divergence is classified into two types: u turning negative and being
absorbed toward u = 0, and u accumulating excessively during |s| < 1 steps
leading to escape. Large-scale parameter scans (u in [0.001, 100], spanning
five orders of magnitude) show that the period-4 cycle exists at almost all
parameters in the tested range, with divergence occurring only at isolated
points. Complex initial values and complex u initial values all lose their
imaginary parts to below 1e-20, converging to the purely real period-4 orbit;
the real axis is a transversely attracting invariant subspace. This work
provides the mechanism of mode switching for non-holomorphic complex
dynamical systems, a mechanism with no counterpart in holomorphic dynamics.