B5 parity correction for proper products of push-maps in K4

Proposition 6.4 / Corollary 6.5 of Bharathram–Birman–Brendle. Let $K_4$ be the point-pushing subgroup of $B_4$ and $\Gamma_1 \in K_4$ the push-map along the simple loop of Figure 6.3. If $\Phi \in K_4$ is a proper product $\Phi = \Phi' \cdot \Gamma_1$ (with $\Phi' \in K_4$), then $\Phi$ does not lie in the kernel of the Burau representation $\rho_4$. The proof (Section 6.3): the only disk type in the disk sequence of $\Phi$ violating the parity condition is a 4-disk; embedding $D_4 \hookrightarrow D_5$ and pushing the 5th point along a suitable loop $\Gamma \in K_5$ replaces every 4-disk by a sign-changing 5-disk, so that both $f(\Phi) \cdot \Gamma$ and $\Gamma$ satisfy the parity condition (Proposition 6.4). The Moody polynomials then satisfy $\mathbb{M}_{f(\Phi)\cdot\Gamma} \neq \mathbb{M}_{\Gamma}$ (Corollary 6.5), so Moody's theorem (Theorem 2.3) gives $f(\Phi) \notin \ker \rho_5$, and Observation 2.1 (BurauFaithful.burau_ker_le_ker_succ, proved) lifts this to $\Phi \notin \ker \rho_4$. This is the key step feeding Theorem 6.6 (BurauFaithful.burau_faithful_on_brun4). Reference: arXiv:2607.05283v2 (14 Sep 2026), Proposition 6.4, Corollary 6.5, Section 6.3.