Paper 01Canonical2026-07-15

Source–Readout Non-Equivalence: Descent and Equivariant Reconstruction Obstructions

An exact readout quotient of a reduced source need not be equivalent to that source

DOI: 10.5281/zenodo.21369967 View on Zenodo

Authority role

Establishes the foundational distinction: after declared redundancy is quotiented out, the readout always presents the source exactly as a quotient — yet a physically invariant relation can fail to descend, and no symmetry-compatible rule need recover the realized state.

Summary

Proves that when gauge and coordinate redundancy are quotiented out first, the public readout always presents the reduced source exactly as the quotient by equality of readouts — and that this exactness is fully compatible with non-equivalence. Two independent obstructions are established: a physically invariant source relation descends through the readout if and only if it is constant on every readout fiber, and a symmetry-compatible deterministic representative exists if and only if every readout stabilizer fixes a point of its fiber. A finite marked-carrier occupation model and the Hopf fibration realize both obstructions, keeping exact presentation, representative selection, and source equivalence rigorously apart.

Notes

Reading notes

The paper's central move is to keep three relations rigorously apart. Start from a space $\Omega$ of admissible realization states, declare the physical equivalence $\simeq_{\mathrm{phys}}$ (gauge, coordinates, relabelling), and pass to the reduced source $S = \Omega/\!\simeq_{\mathrm{phys}}$ before anything else. The invariant readout then induces a surjection $p: S \to T$ .

Exact quotient presentation: $T \cong S/\!\sim_p$ — always holds. The readout labels the readout-equivalence classes without error or omission.
Representative selection: a section of $p$ picks one state per fiber. A section is a convention, not an inverse.
Source equivalence: $p$ bijective. Only in this degenerate case does the readout carry the whole reduced source.

Theorem — Fiber–descent theorem

A physically invariant source relation $r: S \to V$ is a function of readout data alone — factors as $r = \bar r \circ p$ — if and only if $r$ is constant on every fiber of $p$ . A relation that varies within a single fiber descends through no post-processing of the readout whatsoever.

Theorem — Equivariant-section criterion

With a group $G$ acting on $S$ and $T$ and $p$ equivariant, a $G$ -equivariant section of $p$ exists if and only if every readout stabilizer $G_t$ fixes a point of its fiber. A single fiber with empty stabilizer-fixed set excludes every symmetry-compatible deterministic reconstruction rule.

Two worked models realize the obstructions: a finite marked-carrier occupation model (where the permutation group acts transitively on fibers, so no equivariant selector exists — though an equivariant probability-valued lift does, and an ensemble is not a realized state), and the Hopf fibration $h: S^3 \to S^2$ , which also separates the equivariant obstruction from the independent topological obstruction of a nonvanishing first Chern class.

Cite this paper

Rodgers, Jeremy. (2026). Source–Readout Non-Equivalence: Descent and Equivariant Reconstruction Obstructions. https://doi.org/10.5281/zenodo.21369967