Higher-order disorder correlations control reflected phonon counting statistics

We investigate how structural correlations beyond the power spectrum affect phonon counting in a one-dimensional acoustic stack. An integer-thickness construction cancels the extensive leading backscattering term and leaves a finite-chain reflection amplitude determined by endpoint innovations. For symmetric independent innovations with unit variance and fourth moment ${\mu }_4$, the normalized reflected-power second moment tends to $({\mu }_4+5)\mathrm{/}4$. With vacuum environmental inputs, a two-phonon Fock input therefore gives an ensemble-averaged reflected factorial correlation $g^{(2)}=({\mu }_4+5)\mathrm{/}8$. Two explicitly constructed ensembles with identical mean and complete two-point correlation function yield $g^{(2)}=3\mathrm{/}4$ and $9\mathrm{/}8$, placing their counting variances on opposite sides of the Poisson value. Conditional tensor averaging also gives a general fourth-order resistance-growth coefficient for arbitrary impedance contrast, including a positive higher-scattering contribution not recovered by shifting the lowest-order spectrum. Exact enumeration confirms the fixed-$N$, small-detuning limit and shows that increasing $N$ at fixed detuning produces a crossover governed by accumulated higher-order bulk scattering rather than convergence to the endpoint formula. Further calculations test bandwidth, thickness error, propagation loss and thermal background. The result concerns disorder-averaged pulse statistics; it neither creates nonclassicality from coherent input nor changes the conditional factorial correlation of an individually resolved passive sample.