Hartree Fragmentation and Long-Range Pair Networks in Aperiodic Chains

For fermions in a slowly varying aperiodic potential with random interactions $V_{ij}\mathrm{/}\mathrm{\mid }i-j{\mathrm{\mid }}^{\alpha }$, the potential fixes the resonance supply $N_0\propto L^{2-n}\mathrm{/}h$. On crossing $\alpha -2n$, wherever Hartree fragmentation is operative, the resonant-object ensemble reconstructs: turning-point runs fragment into clusters whose intrinsic matching law resolves no logarithmic singularity, while isolated wing bonds survive at finite density and restore a reduced logarithm. The scaling $x_{pair}^{4-2n-\alpha }ln{x}_{pair}\propto h^2\mathrm{/}(Vt0)$ is unchanged. Random coefficient fluctuations sustain the $R^{-\alpha }$ coupling; uniform ones cancel to $R^{-\alpha -2}$.