Six or Seven Readings for Scalar Identification with Separable Nuisance

This mathematical preprint establishes sharp reading counts and stability bounds for identifying a scalar coefficient in a known cosine channel with separable nuisance terms. It proves a complete six-or-seven reading classification, exact optimal sensitivities for both small budgets, and a global sensitivity limit determined by best uniform affine approximation. Explicit constructions distinguish high sensitivity from resistance to marked erasures. The appendices provide reproducible algebraic certificates, an exact threshold for distinguishing two prescribed hypotheses, and phase-calibration bounds valid even at zero amplitude. A separate convex formulation retains a noisy common positive scale and correlated errors. The conclusions are mathematical theorems under explicit assumptions; physical preparation, registration, and normalization require independent justification.