Before the Spike: Exact Bifurcation Anchors and Stochastic Ceilings at the Edge of Stability

This deposit provides the manuscript and replication materials for an exact dynamical-systems analysis of gradient descent on the classical quartic double-well loss. The discrete update defines a cubic endomorphism whose first period-doubling bifurcation coincides with the Edge of Stability studied in deep learning. Closed-form anchors locate the onset of period two, the onset of period four, the first superstable parameter, and the attractor-merging crisis, all unified by a single formula for the two-cycle multiplier. The map is shown to lie in the Feigenbaum universality class via an explicit trapping interval, a nondegenerate quadratic critical point, and a globally negative Schwarzian derivative. A high-precision Newton protocol on superstable orbits recovers the universal Feigenbaum constants, while additive-noise scaling (Crutchfield κ) explains why practical stochastic training typically remains near period-two Edge of Stability. Learning-rate scheduler design is out of scope; industrial implications such as pre-crisis telemetry and quantization-noise ceilings are discussed only schematically.