Conclusion

Conclusion

An exact quotient is not a source-equivalent representation. The readout $p\colon S\to T$ always presents $T$ faithfully as $S/\!\sim_{p}$, and this fact carries no implication whatever about whether $S$ can be recovered from $T$.

The difference between the two is captured by two independent tests, each with a clean criterion and each with a clean failure mode. A source relation is present at the readout level exactly when it is constant on readout fibers; if it varies within a fiber, no function of readout data reproduces it. A symmetry-compatible deterministic representative exists exactly when every readout stabilizer fixes a point of its fiber; if some stabilizer fixes no point, no equivariant selection rule exists, though invariant ensembles may still exist and set-theoretic sections exist under the relevant choice principle. The finite marked-carrier occupation model realizes the second obstruction discretely, and the Hopf fibration realizes it continuously while simultaneously exhibiting an independent topological obstruction of a different character.

The failure of either test yields a precise source–readout obstruction, relative to the relation or the reconstruction class that the test names. Together they give the first mathematical layer of the framework: the distinction between a realization carrier and the exact but potentially non-equivalent image through which that carrier is read.