Sensitivity Dynamics and Realizable Detector Synthesis in Fixed-Ambient Coupled Fields: Tangent Hyperbolicity, Constrained Completeness, and a Homogeneous de Sitter Benchmark

This paper develops the sensitivity and detector-synthesis layer of the fixed-ambient normal-graph program, connecting admitted nonlinear dynamics to finite, hardware-constrained measurements. Parameter differentiation of the quasilinear Euler–Lagrange system produces an inhomogeneous tangent Cauchy problem with the same principal symbol as the ordinary linearization, so sensitivity evolution inherits the local hyperbolic domain of the background. In a homogeneous small-amplitude de Sitter sector, a closed triangular system governs the probe, leading normal backreaction, and sensitivities to three dimensionless parameters. Small-time expansions establish independence of the corresponding two-channel response profiles in the declared nondegenerate parameter region, and smooth dependence preserves full detector rank on nearby exact coupled backgrounds. Restricting measurements to finite hardware subspaces gives an exact-dual criterion: the projected profile Gram matrix must be positive definite. Nested dense hardware spaces eventually preserve completeness, while the constrained Gram dual uniquely minimizes the norm among exact duals. A reproducible three-bin dimensionless benchmark exhibits rank three under unit whitened noise. Amplitude budgets, positive-only kernels, covariance whitening, robust design, and singular-value margins are treated as separate restrictions rather than automatic consequences of structural rank. The work establishes a conditional mathematical and computational architecture; it does not provide observational calibration, empirical confirmation, global identifiability, or a hardware-budget guarantee.