Proposition: Euler mass schedule and frozen-step channel comparison

Let $f(M)=-\kappa /M^2$, with $|f|\le K$ and Lipschitz constant $L$ on a common positive-mass domain. For step size $h$, let $M_{j+1}=M_j+hf(M_j)$. If the exact solution, numerical iterates, and connecting intervals remain in that domain, then

One may take $K=\kappa /{M_{\mathrm{c}}}^2$ and $L=2\kappa /{M_{\mathrm{c}}}^3$ on $[M_{\mathrm{c}},M_0]$.

Separately, suppose a finite proxy Hamiltonian obeys

Replacing the actual mass path over one step by its initial value changes the complete unitary channel by at most $K_{H}v{h}^2/2$ in half-diamond distance. Multiple compatible step allowances add by telescoping.

Proof. The exact solution's local Euler residual is at most

Thus the global error obeys $e_{j+1}\le (1+Lh)e_j+LK{h}^2/2$. Sum the geometric series and use $(1+Lh{)}^j\le e^{Ljh}$.

Duhamel's formula bounds the propagator difference by

Apply the isometry-to-channel estimate and CPTP contraction at each matched interface.