Pretraining doesn’t evade the lower bound: a “pointer” is just a compressed index into a large hypothesis space, and constructing it already requires resolving the same M-way ambiguity during pretraining. The lower bound applies regardless of where the bits are paid.

You can certainly put it in U2 instead (U2 is just a special case of U4 with one auxiliary), but putting it in U4 already ensures it’s suboptimal to preserve the switch & defer yet "kill all humans", because it collapses many future intervention and recovery options simultaneously. In other words, it’s a hard constraint in effect — U4 enforces it as a global irreversibility invariant, whereas U2 is only needed for narrow single-channel invariants like switch reachability.

That's correct, it can be naturally folded into U4 as one of its auxiliary utilities, in the same manner as we do for off-switch preservation.

Thanks! I really appreciate this, and I think your natural-latents framing fits nicely with the Part I point about needing to compress D down to a small set of crisp, structured latents. On the lexicographic point: it's worth noting that even though Theorem 3 writes the full objective as a discounted sum, the safety heads U1-U4 aren’t long-horizon objectives — they’re local one-step tests whose optimal action doesn’t depend on future predictions. For example, U1 is automatically satisfied each round by waiting (and once the human approves the proposed action, the agent simply executes it, thereby engaging U2-U5), and U4 is a one-step reversibility check against an inaction baseline, not a long-run impact estimate. The only head with genuine long-horizon structure is U5, which sits below the safety heads, so discounting never creates optimization pressure on them. This makes the whole scheme intentionally deontic and “natural-latent–friendly”, exactly matching the tractable regime suggested by the large-D lower bounds of Part I.