Introduction

Introduction

The question left open by exactness

Physical descriptions are routinely obtained by passing from a rich source description to a coarser one. Gauge fixing, symmetry reduction, coarse graining, restriction to a chosen family of observables, and projection onto a subsystem all have the same formal shape: a surjection $$p\colon S\longrightarrow T\tag{1}$$ from a space $S$ of admissible states onto a space $T$ of readout values. The companion paper to the present one, on source–readout non-equivalence [8], established two facts about such maps. First, an exact quotient readout need not be equivalent to the reduced source it was computed from: even when no approximation is committed and no information is thrown away by error, the map $p$ may fail to be injective, and then $T$ simply does not carry all the distinctions that $S$ carries. Second, a source-defined relation $r\colon S\to V$ descends to a relation on $T$ precisely when $r$ is constant on the fibers $F_t:=p^{-1}(t)$.

That is a clean criterion, but by itself it is a statement about the relation $r$, and it is tempting to over-read it. The over-reading runs: since $p$ is noninjective, information is lost; since information is lost, the readout description is defective; since the readout description is defective, it must be repaired by additional variables. Each step of this chain is either false or unsupported. Noninjectivity is a structural fact about $p$, and structural loss is cheap: almost every useful description loses something. The relevant question is never whether $p$ loses distinctions, but whether it loses the distinctions that the question at hand depends on. A thermodynamic readout that forgets molecular positions is not thereby unfit to answer thermodynamic questions.

The present paper is devoted to making that dependence exact. We ask:

Given a nominated mathematical or physical question, when does the information discarded by the readout genuinely obstruct answering it, and, when it does, what must any successful repair retain?

Both halves of this question are answered here by positive mathematics. In particular the impossibility half is not obtained by assuming that no readout-only construction exists. It is derived from the observation that a deterministic construction taking only the readout as input is necessarily of the form $b\circ p$, hence constant on fibers, whereas the correct answer may vary along a fiber. Variation of the correct-answer map on a single fiber is the entire source of the obstruction, and the obstruction is therefore exactly as strong, and no stronger, than that variation.

Targets and correct-answer maps

The technical device that makes the theory target-relative is elementary and, we will argue, unavoidable. A target $Q$ is not a piece of prose. It is a pair, consisting of a set $A_Q$ of admissible answers together with a map $$a_Q\colon S\longrightarrow A_Q\tag{2}$$ assigning to each reduced source state the answer that is in fact correct for that state. Nothing mysterious is intended by “correct”: if the target is “does the charged Laplacian have a zero mode?” then $A_Q=\{\text{yes},\text{no}\}$ and $a_Q$ records the true answer for each connection; if the target is “what is the spectrum, with multiplicities?” then $A_Q$ is a set of multisets and $a_Q$ records the actual spectrum. The values of $a_Q$ may be Boolean, numerical, functional, geometric, spectral, or dynamical; the theory is indifferent.

What the device buys is precision about the object of the obstruction. “Information is lost” is a statement about $p$. “$Q$ cannot be solved from the readout” is a statement about the pair $(p,a_Q)$, and Theorem 3.1 shows the second is equivalent to the failure of $a_Q$ to be fiber-constant. Two different targets, evaluated against the same lossy readout, may receive opposite verdicts, and in Section 9, Section 10, Section 11 we exhibit exactly that: a curvature readout that is perfectly adequate for one class of questions about flat $\mathrm{U}(1)$ connections and provably inadequate for another, together with a completion that is minimal for the second class and strictly insufficient for a third.

Contributions

The paper is organized around a single logical progression, and we summarize it here so that the dependency order is visible from the outset.

1. Section 2 fixes the reduced source space, the readout, and the target, and insists that all statements concern reduced source states , states already quotiented by every declared gauge and coordinate redundancy , so that no apparent “loss” is an artifact of retaining redundant labels.

2. Section 3 proves the exact solvability criterion (Theorem 3.1): $Q$ is exactly solvable from the readout iff $a_Q$ is constant on every fiber of $p$; and derives the obstruction (Corollary 3.2) in the form actually needed, namely that a single fiber on which $a_Q$ takes two values already defeats every deterministic readout-only construction. We show additionally (Proposition 3.4) that auxiliary quantities computed from the readout cannot repair this, because they too factor through $p$.

3. Section 4 separates structural loss from target-active loss. A source relation $r$ is active for $Q$ on a fiber when the realized answers it induces there differ; it is inactive when they agree, even if $r$ itself varies. Activity on one fiber obstructs (Theorem 4.3); global inactivity preserves exact solvability (Theorem 4.4).

4. Section 5 identifies what is still available when exactness fails: the compatible-answer set $\mathsf{A}_Q(t)=a_Q(F_t)$, proved to be uniformly sound and pointwise contained in every uniformly sound set-valued rule (Theorem 5.4). We stress that $\mathfrak{P}(A_Q)$ is the power set and that this is a set-valued, not probabilistic, statement.

5. Section 6, Section 7 define readout extensions and target sufficiency. We then prove the necessary-separation theorem (Theorem 7.1): every $Q$-sufficient extension must separate each pair of source states whose correct answers differ.

6. Section 8 constructs the joint target image $E_Q=\operatorname{Im}(p,a_Q)$, proves it is a $Q$-sufficient extension, and proves the universal factorization (Theorem 8.3) in the correct direction: every $Q$-sufficient extension $E$ maps onto $E_Q$, not conversely. $E_Q$ is therefore the coarsest sufficient completion for the single target $Q$.

7. Section 9, Section 10, Section 11 realize all of this for flat $\mathrm{U}(1)$ connections on $S^1$ with a curvature readout, a charged covariant Laplacian, and its spectrum, culminating in the spectral rigidity result $\operatorname{Spec}(H_\theta)=\operatorname{Spec}(H_{\theta'}) \iff \theta'\equiv\pm\theta$.

8. Section 12 applies the same criterion to autonomous projected dynamics at finite time.

Section 13 delimits scope. The sequel [9] generalizes from a single correct-answer map to families of realization relations; continuous-time and memory-kernel questions belong to [10]. Neither construction is used here, and the results below are self-contained.