Carrier Birth, Primitive Renewal, and Extension Frustration: A Certified Phase Sequence in a Multivariate Chebyshev Extremal Problem

We study a concrete two-variable uniform extremal problem on the nonconvex semialgebraic aperture $$A = { (x,y) \\in [-1,1]^2 : |2x + y| \\ge \\frac{3}{5} },$$ with evaluation at a fixed exterior target. A first centered conic carrier supports a finite Chebyshev tower and loses an antipodal support pair at the seventh lift. An exact degree-21 second conic carrier then supports a primitive genus-two trace with a strictly positive extremal signature and a certified norm-one ambient extension. Its first Chebyshev composition is not degree-42 extremal. Nevertheless, the same carrier admits a unique new degree-42 primitive trace extremizer — again of reduced genus two — with a target gain exceeding 2.8% over the composed trace. The renewed trace fails a different gate: its optimal degree-42 ambient extension cost is strictly larger than one. A first-jet theorem reduces norm-one extension to a single compatibility condition and identifies it with stationarity of the renewed trace under the carrier modulus. An exact integer-grid / rational-envelope tangent certificate proves the derivative is strictly positive on the complete certified stationary-carrier parameter box, thereby establishing ambient extension frustration. The certified phase sequence is: $$\\text{carrier birth} \\to \\text{composition death} \\to \\text{same-carrier primitive renewal} \\to \\text{ambient extension frustration}$$ We also isolate a stress/Farey mechanism controlling composition lifetime, and a positive old-to-new spectral bridge at the first carrier birth. No third-carrier existence theorem or universal multi-carrier renormalization law is claimed.