A Sharp Real-Time Pushdown Realizability Dichotomy for Linear Modular Height Guards

The paper develops an exact realizability classification for two independently tagged Dyck-prefix stacks constrained by a linear modular condition on their intermediate heights. It asks precisely when two independent last-in–first-out sources can be represented by a single deterministic real-time pushdown stack without losing their complete ordered contents or the legality of individual source actions. For every modulus (p\geq 2) and coefficients (a,b), it proves a sharp dichotomy: canonical one-stack realization is possible exactly when (a=0), (b=0), or (a+b=0) modulo (p); in every other case, the guarded language is not context-free, ruling out even nondeterministic pushdown recognition. The positive results provide explicit source-faithful encodings and a bounded-repair compilation into strict real time, while the negative results use three-block and periodic-detour obstructions. The classification includes composite moduli and noninvertible coefficients and counts exactly (3p-2) realizable coefficient pairs. Its main novelty is a complete algebraic necessary-and-sufficient criterion connecting modular height geometry, preservation of independent stack contents, and physical real-time realizability within a precisely specified two-source model.