When Did Acceleration Begin, and How Sharply? Transition Redshifts and Jerks of the Competing Histories

The deceleration parameter and its derivatives compress an expansion history into a few numbers that can be compared across models without sharing their assumptions: the present deceleration q0, the transition redshift z_t at which deceleration gave way to acceleration, and the jerk j0, which the standard history fixes at exactly unity. This paper computes all three for four histories: the Planck standard, the small-ruler combination, the evolving-dark-energy variant favoured in recent joint fits, and a phantom-leaning contrast variant. The transition redshifts are 0.632 (Planck), 0.761 (small ruler), 0.741 (evolving dark energy), and 0.587 (phantom variant) — the two histories constructed in response to the current tensions both move the transition to 0.74–0.76, close to the median of about 0.74 reported by model-independent compilations, while the standard history sits lower. The present deceleration values are −0.527, −0.598, −0.352, and −0.578 respectively. The sharpest discriminator is the jerk: both cosmological-constant histories return exactly 1, as they must, the phantom variant returns 1.470, and the evolving-dark-energy variant returns −0.205 — a deviation of more than a full unit, of definite sign, which forthcoming cosmographic determinations of j0 at the ±0.3 level would confirm or exclude outright. The maximum separation between the evolving and standard histories' deceleration curves is 0.175, located at the present day, which is why the transition and jerk — present-day and near-present-day quantities — carry the discriminating power. This study is computed from stated assumptions; it is not a measurement.