Six Birds Protocol Trap: Holonomy Without Entropy Production

When a system is driven through a fixed sequence of operations, two different things can happen. The outcome can depend on the order in which the operations are applied (route dependence, or holonomy), and the motion can acquire a preferred direction in time (an arrow, measured by entropy production). It is tempting to read the first as evidence of the second. We give a fully explicit example showing why this is a mistake. Two Markov kernels K0, K1 on three states are each reversible with respect to the uniform distribution, yet they do not commute. If an external controller applies K0 and then K1, the resulting two-step chain carries a steady probability current of 27/625 around a cycle and has entropy production rate (27/625)(2 log(2905/1933) + log(1447/475)) ≈ 0.0833. If instead the protocol phase is made part of the state, as a clock variable that is updated at random alongside the system, the combined chain is reversible: its entropy production is zero and its path-reversal divergence vanishes at every horizon, even though the kernels still fail to commute. Any partial observation of this lifted chain also shows no arrow. Biasing the clock restores an arrow, and all of it comes from the clock: the lifted rate equals α EPRclock = αβ b log((1+b)/(1−b)), where α is the clock-update probability, β the clock step rate, and b its bias. Along the way we prove that for every finite stationary Markov chain the path-reversal divergence over T steps equals T times the entropy production rate, including when both are infinite, and show by example that the identity fails without stationarity. The main theorems and the exact values for the example are machine-checked in Lean 4 with Mathlib (265 declarations, using only Lean's standard axioms), with one general case of the entropy decomposition argued by hand. Independent double-precision experiments reproduce the results, and control cases test each hypothesis. The conclusions concern finite-state, discrete-time chains and the random-scan lift studied here. Code, data, and Lean proofs: https://github.com/ioannist/six-birds-protocol-trap