Pointed Section 5 selection with isolated sampled cosets

Statement under review — 1 October 2026. A candidate counterexample suggests that this auxiliary isolation assertion may be stronger than the source addendum requires. Consider $G=\mathbf G_a^2$, $a=(1,0)$, $\Sigma=\{0,a\}$, $T=0$, and the multihomogenization of $P(x,y)=(y-x)(x-1)(x-2)$. With the standard translation charts, the first two chain zero sets should be

The claimed sampled-coset containment and transporter identity would force an irreducible $V$ through $0$ into $Z_2$, hence $V=\{0\}$ and $H=\{0\}$. But $a+H$ lies on the line $x=1$ in both chain loci, obstructing the required isolated-component assertion.

Only the affine two-polynomial zero-set calculation has been checked in Lean. Construction of the exact SectionFiveInput, verification of its chart-dependent ideal-chain loci, and the minimal-prime contradiction still need formalization. This is a review warning, not a verified disproof and not a counterexample to Philippon's addendum. The proposed auxiliary statement should be reviewed before further proofs are built on it.

Let $K$ be a Philippon base field, let $G$ be an embedded product of commutative algebraic groups with $n=\dim G>0$, and let $A$ be an analytic subgroup. Fix Section 5 input: a finite set $\Sigma$ containing $0$, a multihomogeneous nonzero polynomial $P$ of multidegree $D$, contact at least $nT+1$ on $\Sigma(n)$, and the bounded translation atlases. Write $I_r$ for the resulting polynomial-operator ideal chain.

There exist an integer $1\le r\le n$, a closed irreducible subset $V\subseteq G$ containing $0$, and a connected algebraic subgroup $H$ whose carrier is the identity component of the translation stabilizer of $V$, such that, with

every sampled coset $g+H$, $g\in\Sigma$, is incompletely defined both by $J$ and by its order-$T$ differential prolongation $\partial_A^{\le T}J$.

Incomplete definition has the original minimal-prime meaning: the coset is a union of isolated components, rather than merely a subset of the zero locus. This is the pointed geometric selection needed for Philippon's 1987 sampled-translates addendum. It does not assert that $V$ has globally maximal dimension in $Z(I_r)$.

Formalization Note. The chain and translated ideals are the previously published concrete definitions. The identity component is taken in the induced Zariski topology. No multiplicity lower bound, defining-degree bound, or Hilbert inequality is part of this statement. This is an Open geometric lemma; the parent reduction does not prove its existence assertion.