Rigorous justification of the finite-difference mass correction

Derive a rigorous explanation for the factor $(1+1/n)$ multiplying the scalar-field mass term in the boundary-corrected effective mass used for the discretized radial Hamiltonian.

Background

The numerical data indicate that replacing the mass term $M^2$ by $(1+1/n)M^2$ substantially improves the apparent universality of the single-mode entropy across radial coordinates. The paper identifies this correction as a discretization artifact that may arise from the finite-difference operator.

No derivation is provided for this particular correction, so establishing its origin would clarify the discretization dependence of the effective-mass scaling variable and help distinguish genuine physical universality from numerical effects.