How long does a constrained random shuffle take to forget where it started? Imagine a path made of up and down steps. It must stay above ground and finish back at ground level. Mathematicians study these constrained paths. In the shuffle studied here, each update picks two positions independently at random and swaps their steps. If the swap would break the constraint, nothing changes. This unrefereed proof candidate says that, for paths of length twice n, the worst-start mixing time grows on the order of n times log n. Crucially, rejected proposals still count. The argument looks at heights at two separated positions. Conditioning on either height controls much of the remaining uncertainty. A one-dimensional inequality then controls the information shared by those two heights. The final comparison returns this analysis to the original shuffle, without paying an extra size-dependent factor. A path that first climbs all the way up and then descends supplies the matching lower order. The package includes the analytic argument, exact finite checks and preserved unsuccessful routes. Finite checks do not prove the statement at every size, and internal AI review is not independent mathematical validation. The upper constant is unspecified. This is not a numerical burn-in prescription, an implementation-runtime bound, or a claim to the fastest sampling algorithm. This is an Evidence Press unrefereed candidate release. The full paper, evidence and limitations are linked. The voice is AI-generated using an OpenAI synthetic voice. This audio briefing is not additional research evidence. It is only a communication aid. Thank you for listening.