When can maximum entropy production select a branch? Two entropy productions and exact counterexamples

Plain-language summary Driven systems — a fluid heated from below, a chemical mixture kept reacting, a planetary atmosphere — usually have several stable states available and settle into one of them. A long-standing rule of thumb, the maximum entropy production (MEP) principle, guesses that the system will pick whichever state dissipates energy fastest. The rule agrees with observation in some settings and fails in others, and there has been no criterion to say in advance which is which. This paper traces the difficulty to an identification that MEP arguments make without stating it. Two different entropy productions appear in the problem. The first is the dissipation a branch would sustain on its own, and this is the quantity an MEP rule maximizes. The second is the dissipation along the rare paths that carry the system from one branch to another, and it is this one that sets the quasi-potential and therefore fixes which branch is selected. They are distinct large-deviation objects, carrying different units, and relating them requires a dimensionful conversion. The paper isolates that conversion as an explicit closure assumption — the bridge condition — and not a theorem; far from equilibrium it fails. Two thermodynamically consistent finite-state Markov models make the failure exact. They are small enough to solve in closed form, and in both of them the branch that dissipates more is the branch the system avoids. A single-gate cut-balance identity fixes the occupation ratio at every finite parameter value, with no metastability or timescale-separation assumption. The two models fail in opposite ways: in the first, the affinity of the gate connecting the branches reverses the ordering while the model remains MEP-like with respect to its own transition-path entropy production; in the second, an equilibrium gate reverses the same ordering at zero transition-path entropy production, the occupation gap being shaped by the time-symmetric activity profile alone. Where MEP-like behaviour does survive, it survives as a conditional alignment, and the paper states the conditions. Near a supercritical bifurcation the selected branch and the higher-dissipation branch coincide for structural reasons rather than teleological ones. The one route by which the criterion would instead predict anti-MEP selection — a sheared mean flow that suppresses heat transport — is tested directly, by pseudo-spectral direct numerical simulation of Rayleigh–Bénard convection. A released mean flow does suppress the heat flux, transiently by 85% at Ra = 10⁵, but it does not sustain itself: the shear collapses, and the higher-dissipation roll state is selected. The reproduction package (Mendeley Data, doi:10.17632/pgstrtnwj8) The code is not part of this upload: it is deposited at Mendeley Data and linked from this record under "Is supplemented by". The package rebuilds both counterexamples from their generator matrices, with no hard-coded numbers. It verifies local detailed balance on every reversible edge, checks the single-gate cut-balance identity to machine precision, evaluates the driven-ring lemma, and regenerates both figures of the main text. It also reproduces the stochastic Boussinesq illustration of Supplemental Material Sec. S3, together with four independent checks on it: the discrete dissipation returns the analytic conductive value 4.000; it satisfies the exact steady-state identity ∫|∇T|² dA = Lx·Nu; it is grid-converged to 0.5%; and the measured growth rate of the critical mode agrees with linear stability to 0.4% (17.61 against 17.68). A deterministic control run, with the regularizing noise switched off, leaves the reported plateaus unchanged to four digits. The pseudo-spectral DNS of Sec. S7 is archived there as an executed notebook, together with the recorded checkpoint of the high-Rayleigh-number run and a script that re-derives every DNS value quoted in the paper from that checkpoint in about a second, without a GPU. Why it matters MEP is invoked as a working principle in climate science, planetary atmospheres, ecology and materials science, and it has no general derivation. The useful question is therefore not whether it sometimes fails, but which step of the argument is the unjustified one. That step is the bridge condition: the two counterexamples break it from opposite poles of the action decomposition S = C − Σ/2 + B, and one slope criterion collects them into the condition an entropy-production ordering must meet in order to select. A companion paper (doi:10.5281/zenodo.21021547) turns the same decomposition into a quantitative bound on the frenetic component of selection. This version (v3) The title supersedes the one used in v1 and v2, "Frenesy, quasi-potentials, and closure conditions", which remain archived and citable. This version also corrects the discrete dissipation diagnostic of Supplemental Material Sec. S3. Earlier versions evaluated Σ = ∫|∇T|² dA with centred differences that left the two boundary rows at zero, under-estimating it by O(1/Ny): the conductive plateau read 3.76 rather than the analytic 4.00, and the convective plateau was some 15% low. The corrected estimator is the discrete Dirichlet energy over grid edges, which is exact on the conductive profile and satisfies the steady-state identity Σ = Lx·Nu. Every Boussinesq number in Sec. S3 changed accordingly (quench: 3.76 → 4.00 and 10.21 → 12.02, ratio 2.72 → 3.00). The finite-state counterexamples, which carry the central results, are unaffected, as is the DNS of Sec. S7. Preprint; not peer reviewed. The upload contains the main manuscript (30 pp.) and the Supplemental Material (14 pp.). Data and code: Mendeley Data, doi:10.17632/pgstrtnwj8 (all versions).