= Solution
\b[Yes: replacing the observed statistic by a significance label loses information.] The binary positive event has <Bayes factor> $(1-\beta)/\alpha\simeq16$, but the observed-statistic <Bayes factor> varies continuously with $y$. A result just at the threshold has $B_{10}(1.65)\simeq2.74$, while $B_{10}(3)\simeq79.0$. Both would receive the same positive label.
The binary calculation is correct for that deliberately coarsened observation; its problem is treating it as the whole evidence when the statistic is available. This is the point developed in https://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0040168[Goodman and Greenland's response].
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