Quadratic translation germs over parameter stalks

Let $K$ be algebraically closed of characteristic zero and let $E$ be a connected commutative algebraic group with a locally closed projective embedding. There is a projective realization $F\subseteq\mathbf P^N$ and an algebraic group isomorphism $E\to F$, regular in both directions, with the following property.

Every pair $(x,y)\in F\times F$ admits an index $c$ with $Y_c(y)\ne0$. On this parameter chart write $z_j=Y_j/Y_c$, let $I_c$ be the ideal of polynomials vanishing on all its normalized group points, and put

There are forms

homogeneous of degree two in $X$. Choose for each stalk coefficient its fixed localization representative $a/s$, with $s\notin\mathfrak p_y$. Evaluate that coefficient at $y'$ by $a(z(y'))/s(z(y'))$ whenever the pivot is nonzero, with value zero off the pivot chart. Evaluate each form by its finite monomial-support sum, denoting the resulting value by $P_t(X(x');y')$.

For all $(x',y')$ sufficiently near $(x,y)$ in the actual induced multiprojective Zariski topology, the evaluated tuple is nonzero and

The coefficients belong to the parameter stalk at $y$; the assertion does not supply a common principal denominator or a principal neighborhood.

Formalization Note. This is an auxiliary formulation of local quadratic translation data, not a verbatim source theorem. Fraction representatives are the fixed choices in the parameter-stalk definition. Their pointwise evaluation is not asserted to be a ring homomorphism at all points; the displayed polynomial value is explicitly a finite sum. Every chosen denominator is nonzero at $y$. The parent reduction proves that all coefficients spread to one principal chart, that this chart is open in the pair topology, and that degree and projective specialization are preserved. Existence of the reembedding and of these quadratic germs, including comparison with the concrete projective point model, remains Open. Individual zero forms are permitted.