When does a shuffled deck really become random? If some cards are selected less often than others, the slow cards can retain information that a single overall count misses. This research candidate examines a precise mathematical model. Each step chooses two labels independently and swaps them. Half the labels are slow, and half are fast. Choosing the same label twice leaves the deck unchanged. An existing conjecture predicted the shape of the transition towards randomness. The new argument tests that prediction using one simple event: at least two slow labels remain in their original positions. A label that has never been selected must still be fixed. That observation provides a lower bound on how often the event happens during the shuffle. The proof compares it with how often the same event happens in a uniformly random permutation. At a particular point in the transition, the resulting distance is at least about zero point seven one one. The conjectured formula gives about zero point six eight two. The strict gap rules out that formula. It does not overturn the established cutoff theorem, and it does not supply a replacement profile. The paper also develops a general way to use chosen subsets to bound mixing from below. Exact arithmetic verifies the key numerical inequalities, and a separate implementation within the package replays finite examples. These checks support inspection but do not formally verify the asymptotic proof. The practical lesson is a research direction: subgroup-sensitive diagnostics can reveal information hidden by aggregate statistics. No performance guarantee for other sampling algorithms follows. This is an unrefereed Evidence Press candidate. The full paper, code and evidence are linked on the release page. This AI-generated OpenAI voice summary is communication, not additional evidence.