Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains

In a slowly varying aperiodic potential $h_i=h\cos (2\pi \beta i^n+\phi )$ with random power-law density interactions $V_{ij}\mathrm{/}\mathrm{\mid }i-j{\mathrm{\mid }}^{\alpha }$, the resonant-object ensemble changes across $\alpha =2n$, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling $x_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}^{4-2n-\alpha }\ln x_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}\propto h^2\mathrm{/}(V{t}_0)$ is unchanged~\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics — the supply $N_0\propto L^{2-n}\mathrm{/}h$, the correlated common-phase comb, and the closed-form Hartree variance — proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at $L\le 18$ shows that the order-one forward-branching scale $h_{\mathrm{F}\mathrm{B}}\propto L^{2-n}$ carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged $q\ln (1\mathrm{/}q)$ law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold $V_{\ast }(\alpha )$ and the boundary-healing recursion; and we treat the marginal case $\alpha =2$, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings — $\alpha =1\mathrm{/}2$ (the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate), $\alpha =2n$ (change of the local resonant objects), and $\alpha =2$ (marginality of the long-range shell sum) — none of which is, by itself, a localization transition.