nucleation

CLAIM-516: Where every constraint capacity turns over at a positive rate and is produced only by others, with no basal production and production vanishing as its producers do, capacities whose production graph is acyclic all decay to zero: a capacity that persists needs a productive cycle among its ancestors, and removing a feedback edge set extinguishes every capacity; a cycle is necessary and not sufficient, and a one-member cycle is not closure

3 min read

Properties10
StatusActive
Rolethesis
Warrantdemonstrated
Tagsphilosophy-of-biology, mathematics, complex-systems
Linedistributed-agency, formal-apparatus
GroundsCASE-189 — Two constructions of a maintained organization: two constraints producing each other with saturating production and turnover hold a three-unit coupling, with a box invariant under damage up to 0.4, a saddle-node at damage 0.5 and extinction above it; and a transient encounter that forms a connector enabling that reciprocal production, after which the connector is maintained by what it enables, every single-pathway knockout ends it, and the regime ends in a saddle-node at damage ≈ 0.691 where the coupling's gap is still ≈ 0.989
ComplementsCLAIM-395 — Under Montévil and Mossio's definition of closure, a closed set of constraints has at least two members, and if each constraint is produced at the time scale of a constraint in the set that generates it and not at the scale at which it is itself conserved, the set includes at least one slow and one fast dependence, so a system with a single constraint of its own cannot realise closure
UsesTERM-070 — organizational closure, as closure of constraints: a set of constraints each of which depends directly on another in the set and has another depending on it (Montévil and Mossio 2015), distinct from informational closure and from a flame's maintenance of the conditions of its own combustion
Cited as “illustrates” byCASE-191 — A construction of self-formation: an agent's own frustration at two demands one shared output cannot meet drives production of a sensor and a differentiating channel that then maintain each other; past a threshold of discrepancy (D ≈ 0.645 at unit drive) the capacity forms, outlasts the conflict, meets later conflicts of the same class up to D ≈ 1.047 with no frustration, and is lost at damage 0.5; a capacity upheld from outside is told apart by knockouts afterwards, but one installed by an outside pulse and then self-maintained is told apart only by a knockout during formation
Cited as “rests on” byCLAIM-531 — Generative entrenchment does not by itself give organizational closure: an entrenched core, which later interactions depend on so that perturbing it is catastrophic and perturbing them tolerable, can sit at the root of an acyclic dependency order, and where interactions turn over such an order persists only while something outside supplies its core; an organization that is both entrenched and closed has a core containing a cycle of mutual maintenance, each interaction on which is itself core unless its share of the maintenance is redundant

Graph View

The claim

Let constraint capacities cᵢ ≥ 0 evolve as ċᵢ = Pᵢ(c) − dᵢ(t)cᵢ, with dᵢ(t) ≥ d₀ > 0, Pᵢ continuous, depending only on the capacities that produce cᵢ (its parents in the production graph), and Pᵢ → 0 uniformly as those parents go to zero. There is no basal production.

Acyclic graphs go extinct. Order the capacities topologically. A capacity with no parents has ċᵢ ≤ −d₀cᵢ and decays exponentially. If all of cᵢ’s parents tend to zero, Pᵢ eventually falls below any ε, so lim sup cᵢ ≤ ε/d₀ for every ε, and cᵢ → 0. By induction every capacity tends to zero.

Ancestry. A capacity’s ancestors form a forward-closed set, so the same argument applied to them shows: if none of a capacity’s ancestors lies on a cycle, it tends to zero. A capacity that does not decay has a productive cycle among its ancestors. Removing a feedback edge set, so that the graph becomes acyclic, extinguishes every capacity, provided the knockout leaves each Pᵢ vanishing as its remaining parents do.

What it does not say

  • That a cycle suffices. The two-constraint model of CASE-189 has a cycle and goes extinct from small seeds and above a damage threshold.
  • That the cycle is closure. A self-loop, a capacity producing itself, satisfies the necessary condition. Closure in TERM-070’s sense, and in CLAIM-395’s, needs at least two mutually dependent constraints, and CASE-161 shows a self-dependent production rule does not regenerate a zeroed rule. The theorem’s necessary condition is weaker than closure.
  • Anything about systems with basal production or an outside supply. An exogenous source keeps an acyclic graph alive; that is the externally maintained case.
  • That real chemistry decays to zero. The hypothesis of no basal production and decay to zero is the limit of infinite affinity, with irreversible decay. Under local detailed balance the decay c ⇌ W has a back reaction, and with any waste present the capacities of an acyclic graph relax to (d′/d)[W], not to zero. The proof and the warrant stand for the stated hypotheses.
  • That the production cycle is a stoichiometric cycle. The graph is one of catalytic influence. Catalysts cancel out of the stoichiometric matrix, so a cycle of mutual catalytic production is not a cycle in the reaction network’s kernel; steady currents run on fuel-to-waste cycles (THEORY-1248). Mossio, Montévil and Longo’s closure likewise needs another member, and their hydrological cycle is a cycle of processes that is not closure (LIT-1888).

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