Six Birds Protocol Trap: Holonomy Without Entropy Production

Protocol-driven dynamics often conflate route dependence (noncommutativity/holonomy of update steps) with an arrow of time, making time-asymmetry audits vulnerable to scheduling artifacts. We introduce an autonomous lifted-protocol formulation and a path-reversal audit decomposition, operationalize it via a fully explicit finite-state benchmark, and demonstrate a clean null-versus-driven separation between holonomy and directionality. Concretely, we define two reversible Markov kernels K₀, K₁ on three states and compare (i) an externally scheduled stroboscopic composition on X with (ii) an autonomous time-homogeneous lift on Z = X × Φ driven by a clock. Under external scheduling, each step is reversible but the stroboscopic update has strictly positive entropy production EPR ≈ 0.083, with a noncommutativity witness ‖K₀K₁ − K₁K₀‖∞ = 0.1296. Under the autonomous lift with a reversible (unbiased) clock (null control), the lifted chain is reversible and the arrow audit collapses to EPR < 10⁻¹⁷ (equivalently, path-reversal asymmetry vanishes). Turning on clock bias (drive) yields a positive arrow and the measured EPR_lifted matches α · EPR_clock across a bias sweep (Figure 1). This separation reduces false positives when interpreting protocol effects as directionality and provides a reproducible audit template for switching Markov models and algorithmic protocols. We do not claim a general characterization of all nonequilibrium protocols; the conclusions are for finite-state discrete-time kernels and the random-scan autonomous lift studied here. Keywords: protocol holonomy; arrow of time; entropy production; Markov chains; autonomous lift; directionality audit; Six Birds framework