Properties of the recursive Taylor comparison polynomial

Notation. $S=[-3,3]^2$ is the model square; a metric is a smooth positive matrix field $g$; $z$ is a scalar height; $H=\nabla_g^2z$ is its covariant Hessian; $K_g$ is the Gaussian curvature; $E=\det g\,(1-|dz|_g^2)$; $q=H_{xy}/H_{yy}$ is the Hessian quotient; $\kappa(x^2-h(y))$ is the model curvature; the patch box and the central box are the squares $[-1/10,1/10]^2$ and $[-1/5,1/5]^2$; the sheared coordinates are $\xi=x,\ \theta=y+q_0x$, with sheared height $z_s$; $L$ and $v_*$ are the class width and speed of (4.2); $\tilde g_\tau$ is the test metric (4.12) built from $g_*$; a bounded admissible height of bound $M$ is an admissible height with (4.1).

Let $g$ be smooth and positive on an open set $U$, let $I\subseteq\mathbb R$ be open, let $u_0,u_1$ be smooth on $I$ and $a$ real with $(x,a)\in U$ for $x\in I$, and suppose the $(0,0)$ entry of $\nabla_g^2$ of the linear Cauchy polynomial $u_0(x)+u_1(x)(y-a)$ is nonzero at each $(x,a)$. Then for each $N$, the recursively defined Taylor approximation $P_N$ is smooth on the strip $I\times\mathbb R$, has the same six-variable jet as the linear polynomial on the cut $y=a$, and its $Q_g$-residual satisfies

Formalization Note. This is the upstream Lean declaration OAI.SmoothLocal.Taylor.taylorApproximation_properties in lean/OAI/Geometry/IsometricImmersion/Taylor/FiniteTaylorComparison.lean, stated verbatim; the Lean statement is authoritative for the precise hypotheses. It is an intermediate result of OpenAI's formalization.