witness_p

The module formalizes two paper claims on linking nontriviality. U5 is the Alexander-duality selector: loop-loop linking ($p=1$) forces $D=3$ via $H_1(S^D \setminus K) \cong \mathbb{Z}$. U10 is the converse direction of Proposition 3.5: every odd $D \ge 3$ should support nontrivial same-sector linking, not merely by parity but by an explicit geometric witness.

In $\mathbb{R}^{2p+1}$, two standard $p$-spheres placed in complementary position have linking number $\pm 1$. The integer $p$ is therefore the defect (sphere) dimension dual to ambient $D$. Upstream constants pin the physical spatial dimension at $D=3$ (T8/T9), which recovers the classical loop-loop case $p=1$; the same arithmetic is kept parametric so the full odd ray can be certified.

Pure definition: the body is the natural-number expression $(D-1)/2$ (integer division). No lemmas, tactics, or hypotheses. Downstream facts such as $p \ge 1$ when $D \ge 3$, and the reconstruction $D = 2p+1$ for odd $D$, are proved separately by unfolding and omega/decide.

This is the explicit witness construction demanded by U10 / paper Proposition 3.5. The characterization theorem for the allowed set $A_A$ uses it on the converse arm: every odd $D \ge 3$ admits $p = (D-1)/2 \ge 1$ with $D = 2p+1$. Concrete evaluations (witness_D3 through witness_D9) and the inequality $p \ge 1$ are immediate corollaries. In the forcing chain this sits under T8 ($D=3$ spatial) and the same-sector parity filter: loop-loop specialization $p=1$ recovers $D=3$, while the general odd ray remains available for the topological audit of nontrivial linking.