The claim
Montévil and Mossio (LIT-582, read in NOTE-468) define a constraint by a time symmetry: C acts on a process at a time scale τs(C) at which its relevant aspects are conserved (Definition 1). C1 depends on C2 when C2 constrains, at τs(C2), a process that produces aspects of C1 (Definitions 2-3). A set of constraints realises overall closure if each member “depends directly on at least one other constraint belonging to” the set and at least one other member depends on it (Definition 4, conditions 1-2; strict closure adds that the set cannot be split into two closed sets). Each constraint in a closed set also has a time scale τd(C) at which it is produced or maintained, and the paper reads it off the diagram as the scale of the constraint that produces it (τd(C3) = τ4 = τs(C4)). A dependence is slow when the dependent constraint is conserved longer than the process that makes it (τs − τd > 0) and fast in the opposite case.
Two readings are needed, and the title states them as conditions:
- (A) each member C of the set has a direct generator g(C) in the set with τd(C) = τs(g(C));
- (B) no member is produced at its own conserved scale: τs(C) ≠ τd(C).
Then:
- A closed set has at least two members.
- Under (A) and (B), it has a member with τs − τd > 0 and a member with τs − τd < 0: at least one slow and one fast dependence.
- So a system whose only constraint of its own is a single macroscopic pattern cannot realise closure, under any reading.
Proof
1. Let C be closed and C ∈ C. By Definition 4, condition 1, C depends directly on some other constraint in C, which is distinct from C. So |C| ≥ 2. No reading of (A) or (B) is used.
2. Assume (A) and (B). Choose for each member C a generator g(C) as in (A); by Definition 4 it is another member, so g(C) ≠ C. C is finite, so iterating g from any member eventually revisits one: there is a cycle C₁, …, C_k (k ≥ 2) with g(Cᵢ) = Cᵢ₊₁ and g(C_k) = C₁. Sum the differences around it:
Σᵢ [τs(Cᵢ) − τd(Cᵢ)] = Σᵢ τs(Cᵢ) − Σᵢ τs(g(Cᵢ)) = 0,
because g permutes the cycle’s members, so the two sums have the same terms. By (B) no term is zero. A sum of non-zero terms that equals zero has a positive term and a negative term. ∎
3. Follows from 1: a set with one member is not closed.
A check. For 100,000 random generator maps on 2-8 constraints with τs drawn from {1, …, 5} (Python, seed 0), the 42,328 that satisfy (B) all contain both signs of τs − τd. The generator map was drawn with g(i) uniform over j ≠ i and τd(i) = τs(g(i)), and draws with any τs(i) = τd(i) were dropped.
What this does not say
- Not that a flame fails closure. Part 3 is conditional on the flame having one constraint of its own, which is how Mossio, Saborido and Moreno (2009) describe it (“a unique contribution”) and which no paper checks under the 2015 definitions. A flame whose wick and wax pool turned out to be constraints that it produces, at distinct time scales, would not be touched by this claim.
- Not that closure is differentiation. THEORY-072 reads the 2015 closure as building in the 2009 paper’s organisational differentiation. This claim shows only that closure needs at least two constraints and, under (A) and (B), two time scales. Whether two constraints at distinct scales are “different and localizable patterns … each making a specific contribution” is interpretation.
- Not that the paper’s property holds without (A) and (B). If a constraint can be produced at its own conserved scale, a cycle of such constraints has every difference zero and neither a slow nor a fast dependence. Montévil and Mossio’s Definition 1 makes (B) natural, since a constraint changed at the scale it acts would not be conserved there, but the paper does not state it as a condition.
- Not that closure defines the organism. The paper denies that (footnote 27), and so does THEORY-072.
Note of 2026-10-27: read as the centre of a multiscale account
A13 of the exchange of 2026-10-27 cited this claim correctly and drew its consequence. Around a cycle of constraints C₁ → … → Cₙ → C₁ with conservation time scales sᵢ, the differences sᵢ − sᵢ₊₁ sum to zero, so if none vanishes some are positive and some negative. A13: “A hierarchy of timescales can differentiate organizational roles, but a closed network of dependence cannot be described as a strictly one-directional descent through that hierarchy”, and “Scale separation differentiates processes and permits relative autonomy. Cross-scale dependence integrates those differentiated processes into a larger organization.” It called this “probably the center of the theory” and was careful that “it is not a general theorem that every feedback cycle requires separated timescales”. That is this claim’s content, stated as an interpretation; CLAIM-478, filed from the same exchange, names it among what it does not say. Nothing here changes.