We demonstrate that the persistence of organized structure in open, driven systems requires satisfaction of a precise relationship between three dynamical parameters: feedback delay (tau), phase alignment (Delta_phi), and the balance between dissipation and correction capacity (D:C). When these parameters satisfy the Coherence Propagation Constant (CPC), systems can metabolize exported entropy into persistent coherence. Deviation from the CPC viability band leads to inevitable structural collapse, regardless of energy availability. The CPC is defined as Upsilon = tau * Delta_phi / D:C, where Upsilon must remain within a viability band [Upsilon_min, Upsilon_max] for structure to persist. Below the band, over-correction freezes the system. Above it, under-correction causes decoherence. Core results: - Persistence condition: stable structures require Upsilon within the viability band. This is the post-formation maintenance requirement — the companion to Coherence-Gated Formation, which governs when structure can emerge. - Resonance scaling: within the viability band, resonance obeys R proportional to ln(1/Upsilon). Metabolic efficiency peaks at band center, declining monotonically toward edges. - Thermodynamic constraints: coherence gain cannot exceed entropy export (no-free-lunch bound). Semantic work must satisfy the Landauer-consistent ceiling. Two-ledger accounting separates physical and semantic flows. - Four falsifiable predictions: non-monotonic noise response (formation peaks at optimal noise, not zero noise), phase-mismatch hysteresis, optimal delay windows in any system with controllable feedback, and correction capacity ceiling beyond which persistence fails regardless of other parameters. - Cross-domain applications mapped with specific variable identifications: stellar oscillations (tau = pulsation period, D:C = radiative damping vs thermal driving), chemical oscillations (tau = reaction timescale, D:C = entropy production vs chemical potential), neural synchrony (tau = synaptic delay, D:C = membrane leak vs synaptic drive), institutional governance (tau = policy implementation lag, D:C = corruption vs corrective capacity), working memory (tau = rehearsal loop duration). - Operational tools: Coherence Propagation Mapper (real-time Upsilon tracking), IM-Gauge dashboard (informational metabolism monitoring), Harmonic Governance protocol (correction timing alignment). - Connection to RMAxt gradient equation: CPC is the persistence-specialized form where tau maps to feedback delay, chi to phase alignment, and D:C to the suppression-regeneration balance at equilibrium. The viability band corresponds to the stability condition M > dS/d(rho_c) established in the gradient equation's formal properties. The CPC emerges naturally from the RMAxt framework, which has demonstrated predictive power across 30 orders of magnitude. Where the gradient equation describes formation conditions, CPC governs persistence requirements once formed. Together with the Translation Principle's projection framework, these three perspectives provide a complete mechanistic account of how coherence emerges, stabilizes, and eventually decoheres across scales. Part of the Coherence Institute publication sequence (paired with Coherence-Gated Formation). This paper governs persistence; its companion governs formation.