We define a family of cumulative functionals Mₚ (z) = ∫₀ᶻ H (ζ) (1+ζ) ⁻ᵖ dζ over the ΛCDM expansion history and show that the asymptotic behaviour of H (z) during matter domination imposes a strict convergence condition p > 5/2. The smallest integer kernel satisfying this requirement, p = 3, yields a finite, monotonically increasing representation of cosmic expansion in which the matter-dominated era contributes roughly half of the total cumulative memory. Calibration to the Planck 2018 deceleration-to-acceleration transition gives Mcrit = 0. 357, recovering zₐcc = 0. 632 to machine precision. Three independent null tests parameter sweep, limiting-case analysis, and closed-form inversion validation confirm internal consistency to ~10⁻¹². The framework derives entirely from standard ΛCDM and introduces no new physics.
Xaviann Campbell-Stephens (Sat,) studied this question.