We develop a globally viable moderated cosmological branch of the finite-capacity latency–erasure program in which late-time cosmic acceleration is sourced by an effective horizon-erasure sector, while the uncontrolled high-redshift behavior of the minimal unsuppressed model is removed by an explicit suppression mechanism. The starting point is the finite-capacity load variable associated with matter-plus-radiation loading on the Hubble scale. In the minimal unsuppressed realization, the effective erasure density grows too aggressively as the matter-plus-radiation loading approaches saturation, making the early-universe behavior phenomenologically problematic. Earlier implicit moderation schemes also lead to global branch tension because the expansion history enters the source sector recursively. We therefore replace the implicit construction by an explicit moderated erasure ansatz built from a bounded load variable , where . The resulting erasure density parameter is chosen to be proportional to , which guarantees finiteness, late-time relevance, and high-redshift suppression. The model yields an explicit closed expression for the normalized expansion history , from which the deceleration parameter and the effective equation-of-state parameter are derived analytically. We analyze the low-redshift and high-redshift limits, derive the normalization condition fixing the amplitude parameter , identify the viability requirements on the moderation parameter , and formulate a first-pass background comparison framework relative to a reference late-time cosmology. The result is a controlled and globally defined cosmological realization of the finite-capacity theory and the second major testable branch of the broader program.