An explicit twelve-measurement construction in real dimension seven

The note gives twelve explicitly specified integral symmetric $7\times7$
matrices whose quadratic measurements distinguish every vector in
$\mathbb{R}^7$ up to a global sign, including the zero vector. Every matrix
entry has absolute value at most $144$. The construction restricts
multiplication of complex cubic polynomials to a real hyperplane and then
takes a real-linear quotient with a two-dimensional kernel. Nonsingularity
is proved by examining all ten unordered balanced factorizations of each
polynomial in that kernel; three uniform inequalities cover all factors and
all parameter values. A separate perturbation argument supplies a
conservative global lower stability bound, and an explicit decoder recovers
the signal up to sign using square roots and the roots of one polynomial of
degree at most four. This establishes $m_{\mathbb{R}}(7)\le12$ in the
unrestricted self-adjoint measurement model. It does not establish equality,
nor determine the minimum measurement numbers in all dimensions.