Metric vs. Syntactic Saturation: A Fare Algebra Audit of Heron's Method

Classical numerical analysis treats floating-point error as a gradual accumulation of rounding mistakes. The Monkey Business Arithmetic Spectator Pipeline (MBP) models numeric representations as algebraic environments with specific capacities and incapacities. A central question is whether the saturation of a finite environment is a single phenomenon or a bifurcated one. This paper presents an exact-arithmetic audit of Heron's method for the square root of 2, revealing two distinct structural failure modes. Syntactic Saturation occurs when an operation lacks the moveset to construct a value (e.g., nested irrationals), triggering a geometric Apeirogon limit - the point where the finite environment shatters while attempting to simulate a continuous curve. Metric Saturation (a Representational Lock) occurs when a legal rational trajectory falls permanently into a float's rounding interval. The audit shows Heron's method triggers the Metric Lock without reaching the Apeirogon limit. Furthermore, exact auditing reveals a round-to-even tie that splits the single-cast and float-pipeline trajectories by exactly one ULP, quantifying the geometric cost of intermediate rounding.Status: Technical Research Note (v0.5). The exact rational fractions and 1-ULP tie data were generated via a closed computational run (Z.ai GLM) acting as an independent auditor against the public fare-audit engine. The engine and broad test suites are available on GitHub.