Abstract

Abstract

A surjective, noninjective readout map $p\colon S\to T$ from a reduced source space necessarily discards distinctions among reduced source states. It does not follow that the discarded distinctions matter. Whether they matter is a property of the question being asked, not of the readout alone. This paper makes that dependence precise. A target is specified by a correct-answer map $a_Q\colon S\to A_Q$, and we prove that $Q$ admits an exact readout-only solution if and only if $a_Q$ is constant on every fiber of $p$; consequently a lost source relation obstructs $Q$ precisely when it is active for $Q$, i.e. when the realized answer varies across some readout fiber. When exactness fails, the compatible-answer set $\mathsf{A}_Q(t)=a_Q(p^{-1}(t))$ is shown to be the unique pointwise-least uniformly sound set-valued rule available without further information; it is a set-theoretic, not probabilistic, object. We then characterize what any repair must contain. Every surjective readout extension $e\colon S\to E$ through which $a_Q$ factors must separate every pair of source states with different correct answers, and the joint target image $E_Q:=\operatorname{Im}(p,a_Q)$ is itself such an extension and is universally coarsest: every $Q$-sufficient extension $e$ admits a unique surjection $\varphi_E\colon E\to E_Q$ with $j_Q=\varphi_E\circ e$. The framework is realized concretely for flat $\mathrm{U}(1)$ connections on a circle. Curvature is a deliberately restricted readout that collapses the entire flat moduli space $S_{\mathrm{flat}}\cong\mathrm{U}(1)$ to a point, yet the spectrum of the charged covariant Laplacian $H_\theta=-D_{x}^2$ separates $\theta=0$ from $\theta=\pi$; we prove that $\operatorname{Spec}(H_\theta)=\operatorname{Spec}(H_{\theta'})$, with multiplicities, if and only if $\theta'\equiv\pm\theta \pmod{2\pi}$, so that the coarsest completion sufficient for the spectral target is the inversion quotient coordinatized by $\cos\theta$, strictly coarser than full holonomy. Finally the same criterion is applied to finite-time projected dynamics: an autonomous readout propagator $U_\tau$ with $p\circ\Phi_\tau=U_\tau\circ p$ exists if and only if the future readout is constant on present readout fibers.

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