How abruptly does a biased shuffle become random? Imagine assigning every card a probability, drawing two labels independently, and swapping them. If the same label is drawn twice, nothing happens. This candidate studies probabilities bounded above and below by fixed multiples of one divided by the number of cards. It allows arbitrarily many different probability classes. The proposed theorem says that these shuffles have total variation cutoff: the transition from far from random to close to random occupies a vanishing fraction of the overall mixing time. The mixing time is of order n log n. The proved window upper bound is of order n log log n. These are asymptotic orders, not a formula telling you exactly how many shuffles to perform. The key argument compares a weighted gradient with the symmetric gradient of the uniform shuffle. The comparison cost remains fixed over time. Entropy estimates then give the narrow transition, and a smoothing argument transfers the result back to ordinary discrete steps despite the shrinking chance of doing nothing. The written proof is the main evidence. Exact checks on permutations of two through five labels test several finite identities, but cannot certify the theorem for every size. Earlier work gives sharper results for two equally sized probability classes. This candidate instead addresses general bounded product weights. It does not resolve arbitrary nonproduct swap rates, an optimal window, or a limiting profile. This is an unrefereed Evidence Press candidate dated eight September twenty twenty six. The paper and replay package are linked on the release page. This OpenAI synthetic voice briefing is not additional evidence.