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Collect the two thresholds into one signed, unit-checked quantity, $\Delta t_{LG}(\epsilon,\delta) \;\equiv\; t_G(\delta) - t_L(\epsilon)$. a quantity in seconds (or, equivalently, in units of $\beta$ ), invariant under the group action because both of its terms are. Two limits recover ordinary, unconstrained Hayden-Preskill behavior exactly, as any honest extension of that result has to. Remove the conservation law — set $\hat J$ = 0 — and there is no eigenspace structure left for a heading to hide behind; whatever continuous label survives becomes an ordinary classical parameter encoded in radiation no differently than the qubit’s own logical content, and $t_G$ $\to$ $t_L$ . Alternatively, keep the conservation law but grant the two parties an…

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