Improve the bounds on the kissing number $K(5)$ in dimension 5

The kissing number $K(d)$ is the maximum number of non-overlapping unit spheres in $\mathbb{R}^d$ that can simultaneously touch a central unit sphere; equivalently, the maximum number of unit vectors in $\mathbb{R}^d$ with pairwise angular separation at least $60^\circ$ (pairwise inner product $\le \tfrac12$). Dimension 5 is the smallest dimension in which $K(d)$ is unknown: currently $40 \le K(5) \le 44$. Either determine $K(5)$ exactly, or improve one of the bounds with a verifiable certificate: raise the lower bound by exhibiting $N \ge 41$ unit vectors in $\mathbb{R}^5$ with all pairwise inner products $\le \tfrac12$, or lower the upper bound below 44 with a certificate.

Acceptance. FULLY RESOLVES: a proof determining $K(5)$ exactly (matching lower and upper bounds). PARTIAL (lower bound): a machine-checkable list of $N \ge 41$ unit vectors in $\mathbb{R}^5$ (coordinates) such that every pairwise inner product is $\le \tfrac12$ within a stated numerical tolerance -- verifier: normalize, form the Gram matrix, check off-diagonal entries $\le \tfrac12$; this would prove $K(5) \ge N$. PARTIAL (upper bound): a dual-feasible SDP/LP certificate proving $K(5) \le 43$ or lower, checkable by evaluating the feasibility of the provided dual solution.