closed_walk_image_is_circle

The module supplies a theorem surface for the strengthened T7/T8 dimension route: the T7 closed cycle is graph-shaped, so its realized defect is a circle $S^1$, and no closed walk in the cube graph realizes a higher sphere $S^p$ with $p \geq 2$. Smooth-topology content stays predicate-level; only elementary finite-dimensional arithmetic is proved, matching the AlexanderDuality discipline.

A closed walk on the $D$-cube is a phase-indexed path through the $D$-bit pattern space, i.e. a map from $\mathrm{Fin}(2^D)$ into the pattern space of dimension $D$. Hamiltonian means that path is bijective: every vertex is visited exactly once. Circle image is presently the constant predicate True; the only sphere dimension allowed by a graph-shaped closed walk at this layer is $p = 1$.

Upstream, spatial dimension $D$ is forced to 3 by T8 (and by the T9 linking argument in the alpha derivation). The eight-tick octave of T7 is the combinatorial cycle this module realizes as a circle rather than a higher sphere.

One-line wrapper. The goal is the predicate ImageIsCircle applied to $W$, and that predicate is defined as True for every closed walk. The tactic trivial discharges the goal immediately. The hypotheses $D \geq 2$ and Hamiltonian are unused at this layer; they are present so the signature matches the intended geometric statement once the predicate is replaced by genuine topology.

This is part (ii) of the T7 cycle-realization package that feeds the forcing chain: T7 forces the eight-tick octave (period $2^3$), and the strengthened route argues that the closed cycle is graph-shaped, hence realizes a circle $S^1$ rather than a higher sphere. Together with the sibling claim that no closed walk realizes $S^p$ for $p \geq 2$, it pins the topological type of the T7 defect before T8 forces spatial dimension $D = 3$.

No downstream theorems currently depend on it (used_by is empty); it exists as an exposed name for the forcing chain and for a later Mathlib-backed CW/covering-dimension pass. The open question it touches is exactly that upgrade: replacing the constant-True circle predicate by a real topological image statement while keeping the same theorem name.