Abstract

Abstract

Let $\Omega$ be a space of admissible realization states, let $\simeq_{\mathrm{phys}}$ identify descriptions differing only by declared gauge or coordinate redundancy, and let $\Pi\colon\Omega\to T$ be a readout map invariant under $\simeq_{\mathrm{phys}}$. After passing to the reduced source space $S=\Omega/\!\simeq_{\mathrm{phys}}$, the readout induces a surjection $p\colon S\to T$. We show that $p$ always presents $T$ exactly as the quotient $S/\!\sim_{p}$ of $S$ by equality of readouts, and that this exactness is fully compatible with the failure of $p$ to be equivalent to $S$.

Two distinct obstructions are established. First, a physically invariant source relation $r\colon S\to V$ descends through the readout if and only if it is constant on every fiber of $p$; a relation that varies within a fiber is therefore not a function of readout data. Second, if a group $G$ acts on $S$ and $T$ with $p$ equivariant, a $G$-equivariant section of $p$ exists if and only if every readout stabilizer $G_{t}$ fixes a point of its fiber $F_{t}$; a fiber with no stabilizer-fixed point admits no symmetry-compatible deterministic representative. A finite marked-carrier occupation model and the Hopf fibration $h\colon S^{3}\to S^{2}$ realize these obstructions in the discrete and continuous settings respectively, and the Hopf example additionally separates the equivariant obstruction from the independent topological obstruction furnished by a nonvanishing first Chern class.

The results separate three statements that are frequently conflated: exactness of a quotient presentation, existence of a selected representative, and equivalence with the realization source. The first may hold while the second and third fail.