loop_loop_linking_forces_D3

The module formalizes linking nontriviality witnesses for Recognition Science units U5 (Alexander duality selector) and U10 (nontriviality witness for the allowed-set converse). Same-sector linking of two p-spheres in complementary position in $\mathbb{R}^{2p+1}$ has linking number $\pm 1$ precisely when the ambient dimension is odd and at least 3: $D = 2p + 1$ with $p \ge 1$.

The topological (T) specialization is loop-loop linking, i.e. $p = 1$. Alexander duality then reduces $H_1(S^D \setminus K) \cong \mathbb{Z}$ to a cohomology computation on $S^1$, which holds if and only if $D = 3$. Upstream constants packages pin the forced spatial dimension as the literal value 3 (T8 / T9 linking).

Sibling results handle the broader odd-dimension family ($D \in {3,5,7,\ldots}$) and explicit sphere witnesses; this declaration isolates the $p = 1$ arithmetic step.

Term/tactic proof with a single omega call. The hypothesis is the literal equation $D = 2\cdot 1 + 1$, which simplifies to $D = 3$; linear arithmetic closes the goal with no further lemmas.

Closes the loop-loop case of the U5 Alexander-duality selector: the paper's topological specialization that forces spatial dimension three. In the Recognition forcing chain this is the T8 landmark ($D = 3$ spatial dimensions), also cited as the T9 linking requirement in constants packages. Same-sector arithmetic alone only yields odd $D \ge 3$; specializing $p = 1$ selects the unique physical value used by the eight-tick octave and the rest of the spine. No downstream dependents are recorded yet; the declaration stands as the explicit $p = 1$ pin inside the verification layer.