CMT--GEX IV: Numerical Emergence of the Exchange Kernel

We numerically solve the Fredholm exchange flow predicted by the
Conservative Motion Theory (CMT) to demonstrate that the galactic
Giant Exchange Kernel (GEX) profile is not merely a fitting
ansatz, but the analytic fixed point of an information-conserving
Fredholm contraction.

Starting from arbitrary positive kernels $K_{1}(r,r')$ and evolving
them via the CMT flow equation
\[
\partial_{\kappa}K_{\kappa}=-K_{\kappa}\!\circ\!K_{\kappa},
\]
we show that the $\kappa\!\to\!0^{+}$ limit converges to a unique
scale-rigid kernel whose induced velocity profile takes the form
\[
v_{\rm GEX}(r)=v_0(1-e^{-r/r_s}).
\]

Crucially, the emergent $(v_0,r_s)$ satisfy the same relations
previously found from observational fitting:
\[
v_0 = V_{\rm flat},\qquad
r_s \simeq 3 R_{\rm disk}.
\]

The BTFR constructed from the emergent GEX velocities yields an
intrinsic scatter of $0.074$–$0.081$ dex—identical to the observed
SPARC scatter of $0.076$ dex reported in GEX II.
Thus the GEX relationships are not phenomenological but arise as
necessary consequences of the Fredholm contraction.