Conditional carrier/readout dynamics adapter

This theorem is a conjunction. First, for every small topological state type, and every object consisting of a subset $S$, a continuous trajectory $z:I\times I\to\mathrm{State}$ satisfying $z(t,x)\in S$ everywhere and $z(t,1)=z(t,0)$ for every $t$, and a continuous readout $r:S\to\mathrm{Circle}$, consider the normalized circle loop

The floor-defined winding of this loop at $t=0$, computed from the endpoint of its real exponential-cover lift starting at $0$, equals its winding at $t=1$. Second, for every small topological space $X$ with a simply-connected-space instance, every continuous $r:X\to\mathrm{Circle}$, every $x\in X$, and every path from $x$ back to $x$, the path obtained by applying $r$ is endpoint-preservingly homotopic to the constant path at $r(x)$. The first assertion is conditional on an entire carrier-valued closed trajectory and readout already being supplied; it states no differential equation, existence, uniqueness, carrier invariance, openness, or simply connectedness of the carrier. For an empty state type there can be no square-valued trajectory, so the universal assertion over such segments is vacuous. The second assertion is vacuous when there is no base point $x$, and it concerns every supplied loop but supplies no loop-existence claim. The time and spatial comparisons use the endpoints $0$ and $1$, and the winding sign is inherited from the chosen exponential-cover lift and floor.