Source
Remlein, Esposito and Avanzini (2026), LIT-1887, §§V–IX and XI, eqs. 18, 31, 36–44, 49, 57 and Fig. 5, pp. 4–15 of arXiv 2604.14848v2; read in NOTE-1468.
What was actually shown
The regime. Partition a closed network into internal species x, would-be chemostatted species y, and reservoir species Y, with reactions y ⇌ Y faster than the rest by 1/ε and Y more abundant than y by Ω. Taking ε → 0 and then Ω → ∞, the master equation for the slow variables becomes the open network’s, with [y] fixed (eqs. 36–40). Its rates satisfy the open network’s local detailed balance (eq. 49), and its leading-order dissipation equals the closed network’s (eq. 57). With ε → 0 alone the slow equation has the wrong rates and [y] still moves, so time-scale separation is “necessary, but not sufficient” (p. 6). The two limits do not commute.
The window. The reservoir totals change by O(Ω) only after T = O(Ω) on the slow clock (eq. 44). There are three time scales: exchange equilibration, open-network dynamics, reservoir depletion. A network whose emergent open form is detailed-balanced equilibrates within the window. A driven one cannot, and on the ultra-slow scale the closed network “eventually equilibrates” (p. 9).
The check (mine). For x ⇌ y₁, x ⇌ y₂ with fast y_i ⇌ Y_i, [Y₁] = 2Ω, [Y₂] = Ω, a deterministic integration with ε = 10⁻⁴ gives the time at which entropy production falls 1% below its open value as 0.00483 Ω − 0.27 over Ω = 3 × 10³ to 10⁵. The paper’s stochastic fit is 0.0048 Ω + 0.33. With x slaved, the drive y₁ − y₂ decays as e^{−T/Ω} and a 1% fall in (Δ/2) ln((3 + Δ)/(3 − Δ)) needs T/Ω = 0.00483.
What could have failed. The closed network’s slow dynamics could have converged to something other than the open network’s, or its dissipation to a different value. Or the departure time could have scaled otherwise than linearly in Ω.
What this does not say
- That departure always happens at T ~ Ω. It is the onset after which departure can happen; whether it does depends on topology (p. 9).
- That the open description is wrong. Within the window it is exact to leading order.
- That maintenance is a property of the maintained system alone. The window’s upper end is set by the reservoir’s capacity, which belongs to the environment. A maintained organization in the sense of CASE-189 is maintained relative to such a window.
- That this is the only route to an effective constraint. For CLAIM-478 it is one mechanism among several, and it needs two separations, of rate and of abundance, rather than the one the claim names. Like the tracking result of THEORY-1052, it has three time scales.
- Anything outside dilute mass-action chemistry. Buffers are the one laboratory instance the paper gives.