Source
Aubin (1999), LIT-714, read in NOTE-576: the introduction and chapters 1–2 of the mini-course notes. The monograph that founds the theory, LIT-738, is unread.
What was actually shown
These are theorems, not findings, and they hold under their conditions: finite-dimensional states, continuous dynamics with linear growth, and a closed or locally compact K.
- Two separate inputs. Aubin’s notes (LIT-714) define viability relative to a dynamics f and a set K that is given separately. An evolution is viable if it stays in K. K is viable if from every state in it at least one evolution stays in it, and invariant if all do. Viability “depends only on the behavior of f on K”.
- A local test. By Nagumo’s theorem, K is viable exactly when at each of its states the velocity f(x) lies in the contingent cone of K: nowhere does the dynamics force the state out.
- The kernel. When K is not viable, the states of K from which some evolution stays in K for ever form the viability kernel. It is the largest viable subset of K, and from every other state of K every evolution leaves K in finite time. The capture basin of a target is the set of states from which some evolution reaches it in finite time.
What this does not say
- Nothing here compares viability with stability. The notes’ “Stability Properties” (§2.6) concern whether viability survives limits of sets, not stability of an equilibrium or attractor. The claim that a regime can be stable (it returns to its pattern) without being viable (it cannot keep satisfying what it needs) is a correct consequence of the definitions, since nothing ties an attractor to K. But it is a step taken by whoever applies the theory, not something the source states.
- K is not supplied by the theory. Everything depends on specifying the constraint set independently of the dynamics. For a cell or an organization, which constraints count as “required” is the hard question, and viability theory takes the answer as given.
- One evolution suffices. Viability asks whether some evolution can stay in K. With a single deterministic evolution per state, viability and invariance coincide. The interesting cases, controls or set-valued dynamics, are in the unread monograph.
Connections
- Organizational autonomy. Barandiaran, Di Paolo & Rohde (LIT-566) call the network that constitutes an agent “precarious”, and Montévil & Mossio (LIT-582) define organisation by closure of constraints. Reading precariousness as a small kernel, or a state near the kernel’s boundary, is the record’s suggestion; no relation is declared.
- Reef recovery (THEORY-128). Hughes et al.’s reversible phase shift is a case where a degraded state was still inside the capture basin of the coral-dominated one, once herbivory was restored. The parallel is the record’s.
Note of 2026-10-18: an endogenous constraint set, the third time
A6 §4 of the exchange of 2026-10-17 proposed the extension this reading’s title marks as missing. Let an organizational variable z govern the policies and the viability set: ẋ = f(x, u, w; z), ż = g(z, x, u, w), K = K(z), with the joint set 𝒞 = {(x, z) : x ∈ K(z)}, and for a boundary function h the robust controlled-invariance condition sup_u inf_w [∇_x h·f + ∇_z h·g] ≥ 0 at h = 0. A6’s own limit: “This identifies when the organization can keep its state inside its admissible region. It does not yet show who produced the region or why those constraints count as the system’s own.” And: “an engineer could have predetermined the entire update law.”
This is the third time the record meets the constraint set fixed from outside the dynamics: in this reading, in CLAIM-195 (for an argument, K “is fixed by an amendable decision, not discovered”), and now in A6. A6 gives a space in which to ask whose K it is, not an answer, and a counterfactual test: “disable the internal processes that maintain the boundary and measure whether the system loses its capacity to remain in the joint viability set” (QUESTION-086). It stays an open direction. The reading is unchanged. CLAIM-354’s comparison of freedoms inherits the given K.