Vorticity-Aligned Stretching in Null Coordinates: A Cauchy--Euler Hyperbolic Normal Form

For smooth three-dimensional incompressible Euler flow, Cauchy's vorticity formula identifies pointwise vorticity amplification with the stretch of an infinitesimal material line initially aligned with the vorticity.  This paper packages that standard Lagrangian fact into a two-coordinate reciprocal pair. If $Λ$ is the directional amplification, set $T=Λ$ and $S_{\mathrm{eff}}= Λ^{-1}$.  The half-normalized null transform then gives $U=(T+S_{\mathrm{eff}})/2$, $W=(T-S_{\mathrm{eff}})/2$, and the exact unit-shell identity $U^2-W^2=1$.  The logarithmic coordinate $\phi=\log Λ=-\tfrac12\log c$ evolves by
$\dot\phi=\xi^{\mathsf T}S_u\xi$, where $\xi$ is the vorticity direction and $ S_u$ is the rate-of-strain tensor.  Thus Euler supplies a variable speed and orientation along the fixed hyperbola, while the reduced geometry supplies only the chart.  A separate circulation route gives the same reciprocal area law only for a flux-normal effective cross-section, or under an infinitesimal locally coaxial tube hypothesis.  The construction is a one-scalar reduction of a volume-preserving three-dimensional deformation; it is not a closure of Euler, a relativistic model, a metallic-ratio selection law, or a blow-up criterion.  The contribution is the bounded synthesis and its explicit fence ledger, not a new Euler regularity theorem.