Deterministic phase memory I: A minimal bistable gate model with explicit switching thresholds

We formalize the notion of phase memory in deterministic dynamical systemsas the coexistence of multiple asymptotically stable equilibria at a xed control pa-rameter, separated by an unstable invariant set and switchable via nite parametervariation. We present a minimal two-dimensional autonomous model with smoothtanh-gate nonlinearity and establish the following rigorous results: (i) a closed-formformula for the saddlenode switching threshold uSN(β), expressed via a universalidentity linking arctanh and arcosh; (ii) a complete stability classication of allequilibrium branches via the Jacobian spectrum; (iii) global attractiveness of theinvariant box [−1, 1]2 and nonexistence of asymmetric equilibria, conrming thatthe diagonal reduction captures the full dynamics; (iv) a Lyapunov function es-tablishing global convergence to the set of symmetric equilibria; (v) a geometriccharacterization of basins of attraction and the hysteresis mechanism; (vi) a numer-ical verication of all analytical results at β = 2.5. The paper constitutes Part I ofa three-part foundational program on deterministic phase memory. This work is a preprint and has not yet been peer reviewed.