Improve or verify the best-known bounds on the kissing number $K(10)$ in dimension 10

The kissing number $K(n)$ is the maximum number of non-overlapping unit spheres in $\mathbb{R}^n$ that can simultaneously touch a central unit sphere — equivalently, the maximum number of unit vectors in $\mathbb{R}^n$ with pairwise angular separation at least $60^\circ$, i.e. pairwise inner product at most $\tfrac12$. In dimension 10 the exact value is unknown; the maintained record brackets it between a best-known construction (lower bound) and a best-known semidefinite-programming upper bound. Exhibit a configuration of unit vectors in $\mathbb{R}^{10}$ with all pairwise inner products $\le \tfrac12$ whose size matches or exceeds the current best-known lower bound for $K(10)$ in Cohn's Table of Kissing Number Bounds — or improve either bound with a verifiable certificate.

Acceptance. ADVANCES: a machine-checkable set of $N$ vectors in $\mathbb{R}^{10}$ (as explicit coordinates or an $N\times N$ Gram matrix) such that a verifier confirms (a) the Gram matrix is PSD with rank at most 10, (b) all diagonal entries equal (unit vectors up to scaling), and (c) every off-diagonal inner product is $\le \tfrac12$, with $N$ equal to or exceeding the current best-known lower bound for $K(10)$ in Cohn's table at submission time. FULLY RESOLVES: such a configuration with $N$ strictly larger than the best-known lower bound (a new record), OR a verifiable improved upper bound (e.g. a feasible dual SDP certificate) that strictly lowers the best-known upper bound — ideally closing the gap. Provide the Gram matrix / coordinates / SDP certificate plus the verification script.