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

This is Paper II of the seven-paper Shadow Theory foundational series.

A readout can discard genuine source-level distinctions without becoming inadequate for every question. This paper gives an exact criterion for when the lost distinctions matter: a target is solvable from a readout precisely when its correct answer is constant across each readout fiber. If two source states have the same readout but different correct answers, no deterministic construction using that readout alone can answer the target correctly in both cases.

When exact solvability fails, the paper identifies the compatible-answer set as the least uniformly sound set-valued answer available without further information. It then proves a lower bound on every successful extension: any extension sufficient for a target must distinguish every pair of source states with different correct answers. The joint readout-and-answer image is the coarsest sufficient extension in this formal, set-theoretic sense.

The paper also makes an important boundary explicit. This universal minimal object characterizes the least distinctions that any sufficient repair must retain; because it refers to the correct-answer map, it does not itself provide an operational method for extracting answers from the original readout or establish a physically measurable extension. A concrete repair must independently supply and justify the added structure.

The framework is realized for flat U(1) connections on a circle. A curvature-only readout collapses all flat connections, yet the spectrum of the charged covariant Laplacian distinguishes different holonomies. The paper proves that the spectrum determines holonomy only up to inversion, so the minimal completion for this spectral target is strictly coarser than full holonomy. A finite-time projected-dynamics result then gives the exact criterion for when a readout admits an autonomous deterministic propagator.

Paper II builds on Paper I’s source–readout non-equivalence and descent framework. It prepares the later papers on canonical minimal completion, geometric realization, observable quotients and projected dynamics, identifiability, and the RS2 bulk-to-brane physical witness.