Conclusion

Conclusion

The information a readout discards is not, by itself, an obstruction. It becomes one against a nominated target, and it does so under an exactly identifiable condition: the correct answer must vary across a single readout fiber. When that condition fails, the coarse description is complete for the question at hand, however much it has forgotten (Theorem 4.4, Example 9.2). When it holds, no deterministic construction from the readout , and no auxiliary quantity computed from the readout , can be correct throughout (Corollary 3.2, Proposition 3.4), and the honest residual is the compatible-answer set, which is the least sound set-valued rule available and is not a probability (Theorem 5.4).

Repair, when required, is then constrained from below, and the lower bound is attained abstractly within the paper’s category of extensions. Any sufficient enrichment must separate every pair of states whose correct answers differ (Theorem 7.1), and hence must distinguish at least the joint $(p,a_Q)$-classes (Corollary 8.4). The joint target image $E_Q=\operatorname{Im}(p,a_Q)$ attains this lower bound exactly as a set-theoretic construction: it is sufficient for $Q$ (Proposition 8.2), it retains no information beyond the original readout value and the target answer, and every sufficient enrichment maps onto it (Theorem 8.3). This characterizes the least distinctions any repair must carry; it does not by itself construct a physically measurable repair, which must independently supply the enriched extension and the answer map that factors $a_Q$. The flat-connection example below does supply one concretely.

This is a statement about the floor, not the ceiling. Other $Q$-sufficient extensions may , and in practice usually do , carry additional information beyond the $(p,a_Q)$-classes: the identity extension $\operatorname{id}_S$ retains the entire source, and any extension adequate for several targets at once will generally be strictly finer than $E_Q$. Nothing forbids this, and Theorem 8.3 does not assert otherwise; it asserts only that whatever else such an extension carries, it maps canonically onto $E_Q$, which is the least it could have carried and still answered $Q$.

The flat $\mathrm{U}(1)$ connections on a circle make all of this concrete and, we think, persuasive. A curvature readout that erases the entire flat moduli space is perfectly adequate for some questions and provably inadequate for the spectrum of the charged covariant Laplacian, where $H_0$ has a zero mode and $H_\pi$ has none. The completion the spectrum demands is not the source and is not even the full holonomy: the spectrum determines the holonomy only up to inversion, and the coarsest sufficient completion is $\mathrm{U}(1)/(z\sim\bar z)$, coordinatized by $\cos\theta$ , strictly coarser than the holonomy that an oriented transport question would require. Two targets, one source, one readout, two different minimal completions. That is the shape of the phenomenon, and the same criterion, applied to the future readout, decides when a projected dynamics can close on itself.