Regime-Aware Interpretation of Fixed-Point Stability Under Finite Precision

Classical fixed-point stability theory provides a complete local characterization of convergence in exact arithmetic through the derivative of the iteration map at the fixed point. In floating-point computation, however, observed iteration behavior may diverge from theoretical predictions due to rounding effects, rapid convergence, and derivative degeneracy near machine resolution limits. This paper develops a regime-aware interpretation of local stability under finite precision. Using infinite tetration as a minimal nonlinear fixed-point system with analytically tractable stability properties, we distinguish between latent dynamics implied by derivative bounds and observed dynamics produced in floating-point arithmetic. We identify three asymptotically distinct numerical regimes—precision-dominated, transient-starved, and degenerate fixed-point regimes—in which trace-based inference becomes unreliable. We prove that derivative-based regime classification remains valid independently of precision-induced sign behavior and demonstrate through a secondary nonlinear example that the three-regime taxonomy reflects structural properties of floating-point iteration in general. The results establish a diagnostic framework for interpreting stability quantities under finite precision and clarify the epistemic limits of trace-based convergence assessment. Version 2 changes: Added Proposition 5.1 with formal proof under the standard floating-point model; added Section 7 with a secondary nonlinear example (xₙ₊₁ = xₙ − λ sin xₙ) validating the three-regime taxonomy; added Section 3 (Related Work); tightened regime definitions using asymptotic notation; expanded reference list from 5 to 8 entries.