Linear Cauchy polynomial has the height jet on the initial cut

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 $z$ be smooth on an open set $U$ containing the horizontal segment $\{(x,a):x\in(\ell,r)\}$. Put $u_0(x)=z(x,a)$, $u_1(x)=\partial_yz(x,a)$ and $P(x,y)=u_0(x)+u_1(x)(y-a)$. Then for every $x\in(\ell,r)$ the six-variable jets agree:

This says that the degree-one time polynomial built from the Cauchy data of $z$ on the cut $y=a$ reproduces every argument of the Darboux right-hand side at that cut. It is the base case of the Taylor comparison across the thin strip.

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