Paper 02Canonical2026-07-15

Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain

Whether discarded distinctions matter is a property of the question asked, not of the readout alone

DOI: 10.5281/zenodo.21370501 View on Zenodo

Authority role

Makes obstruction target-relative: a question is answerable from the readout exactly when its correct answer never varies within a readout fiber, and every sufficient extension must separate states with different correct answers, with the joint target image as the coarsest such extension.

Summary

Determines exactly when distinctions discarded by a lossy readout matter. This paper proves a target is exactly solvable from the readout if and only if its correct-answer map is constant on every readout fiber; when that fails, the honest residual is the compatible-answer set, the pointwise-least sound set-valued rule, and every sufficient repair must separate states with different answers, with the joint target image as the universally coarsest completion. Realized concretely for flat U(1) connections on a circle, where curvature collapses the whole moduli space yet the charged spectrum needs only holonomy up to inversion.

Notes

Reading notes

A surjective, noninjective readout necessarily discards distinctions. Paper 2 determines exactly when those distinctions matter: the answer is a property of the question asked, not of the readout alone. A target $Q$ is formalized as a correct-answer map $a_Q: S \to A_Q$ on the reduced source.

Theorem: Exact target-solvability criterion

$Q$ 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 $Q$ , meaning that the realized answer varies across some readout fiber. Globally inactive loss never obstructs: that is what makes coarse descriptions viable, not defective.

When exactness fails, no auxiliary quantity computed from the readout can repair it (everything readout-computed is still fiber-constant). The honest residual is the compatible-answer set $\mathsf A_Q(t) = a_Q(p^{-1}(t))$ , the unique pointwise-least uniformly sound set-valued rule. It is a set, not a probability: a probabilistic prediction answers a different target and requires extra input not contained in the readout.

Theorem: Necessary separation and the coarsest completion

Every readout extension sufficient for $Q$ must separate every pair of source states with different correct answers, and the joint target image $E_Q = \mathrm{Im}(p, a_Q)$ is itself sufficient and universally coarsest: every sufficient extension surjects uniquely onto it. This is the floor any repair must contain. It retains target-relevant distinctions rather than reconstructing the source.

The worked example is flat $U(1)$ connections on a circle. Curvature is a deliberately restricted readout that collapses the whole flat moduli space $S_{\mathrm{flat}} \cong U(1)$ to a point. This is perfectly adequate for the flatness target, fatal for the charged-scalar spectral target. The spectrum sees holonomy only up to inversion: the minimal spectral completion is the inversion quotient coordinatized by $\cos\theta$ , strictly coarser than the full holonomy that oriented transport requires. One source, one readout, two targets, two different minimal completions. The same criterion decides finite-time dynamical closure: an autonomous readout propagator exists iff the future readout is constant on present readout fibers.

Cite this paper

Rodgers, Jeremy. (2026). Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain. https://doi.org/10.5281/zenodo.21370501