Discrete Stochastic Cascade Dynamics on Finite Graphs, Part II: Necessity of Species Asymmetry for Structural Hierarchy

The companion paper established a discrete stochastic cascade model with guaranteed termination, emergent structure, and a four-level hierarchy. A foundational question remained open: is the asymmetry between excitation species a necessary condition for this hierarchy, or an incidental modelling choice? We prove that asymmetry is necessary, within the class of models defined in Part I, in a precise theorem-level sense. Symmetric Collapse: when both species share identical self-interaction coefficients and frequencies, the system degenerates into a single-species annihilation process, the Landau free-energy landscape flattens, and the ordered bonding phase ceases to exist. Differential Depletion: the forbidden Y–Y self-interaction channel is independently necessary for bond formation; it creates an asymmetric consumption rate that concentrates the scarcer species at bond sites. Beat Frequency: the frequency mismatch ω_X ≠ ω_Y is independently necessary for sustained cascade dynamics. The cross-channel interaction efficiency is modulated by a phase-interference factor with beat period 2π/|ω_X − ω_Y|, generating a recurrent deterministic drive with explicit amplitude A_beat = (1 − ϑ) sin(|ω_X − ω_Y|/2) γ_1 α_XY ω_X ω_Y X̄Ȳ; at equal frequencies this drive vanishes and only decaying transients remain. An asymmetry index is introduced that reduces the combined deviation from species symmetry to a single scalar. A critical threshold is proved in closed form below which the ripple cannot reach the explosive regime, and a monotone bound on hierarchy depth is established as a function of this index. The results show that the structural richness of the discrete cascade model is not robust to symmetrisation: asymmetry is a necessary structural condition, not merely a parameter choice.