Sharp Cubic Bounds and Critical-Point Transitions for arsinh (x+ λx³ + ηx⁵)

For \( \operatorname{arsinh}(x+\lambda x^3+\eta x^5) \), consider the best cubic bounds with linear term \(x\). When \(\eta=0\), a complete classification of the normalized remainder determines the optimal coefficients on \([0,1]\) for every \(\lambda\geq0\). For positive \(\lambda,\eta\), a parameter-uniform sign-variation argument gives at most two positive critical points. For fixed \(\eta\), the critical-point regimes occur in a fixed order as \(\lambda\) increases. The fold has a unique endpoint height \(\eta_c\). A positive fold threshold exists exactly for \(\eta_*<\eta<\eta_c\). The fold terminates transversely at \(\lambda=0\). Near \((\lambda_*,\eta_*)\), two analytic curves mark the appearance of an interior minimum and its equality with the boundary value. Across the second curve, the minimizing point changes discontinuously and the optimal lower coefficient is continuous but not differentiable. The local sixth-order pattern is classical. The critical-point classification is global for the full composite function. Exact sign bounds also give a rational example.