nucleation

CLAIM-517: A successor regulator keeps a viable set positively invariant if, at every boundary context, its response distribution is within γ/L in total variation of a regulator that certifies inward margin γ, the dynamics being L-Lipschitz in that distribution; an average distillation loss certifies this only through coverage of the boundary contexts, so fidelity measured in ordinary operation certifies nothing about viability

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Properties7
StatusActive
Rolethesis
Warrantdemonstrated
Tagscomplex-systems, mathematics, information-theory
Linedistributed-agency, formal-apparatus
GroundsTHEORY-134 — Whether a system can persist is a question about a constraint set fixed independently of its dynamics: a set is viable when from each of its states some evolution can stay in it, and when it is not, its viability kernel is the largest part that is, from the rest of which every evolution leaves in finite time
ComplementsCLAIM-503 — Over finitely many states and finite outcome sets, one experiment is ε-deficient relative to another in every decision problem with losses in [0, 1] exactly when a single Markov kernel brings it within ε in total variation at every state, so the deficiency computed as a linear programme is the least such ε, CLAIM-407 — A coarse-graining exactly informationally closed under a process's ordinary dynamics need not preserve its responses to an intervention on the mechanism: for a declared intervention distribution ρ, independent of the state, the deficiency Γ = I(S; Y^(I) | T(S), I) can be positive, as in the hidden governance bit, where it is ρ(h = 1) bits; and Γ = 0 exactly when, for every intervention ρ charges, the response kernels are constant on T's fibres over the states of positive probability

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The claim

Let a physical state x evolve as ẋ = F(x, p(· | s)), where p is a regulator’s response distribution in context s, F is Lipschitz in x and L-Lipschitz in p under total variation. Let K be a closed set regular enough for Nagumo’s theorem, and suppose a reference regulator p* gives n·F(x, p*) ≤ −γ < 0 at every boundary point x with outward normal n, in every context the boundary meets. Then any successor p with TV(p(· | s), p*(· | s)) < γ/L at those contexts gives n·F(x, p) ≤ −γ + L·TV < 0, so K stays positively invariant (THEORY-134).

A distillation loss is an average. If the successor was trained to forward KL ε under a context distribution μ, Pinsker and Jensen give E_μ TV ≤ √(ε/2), which bounds nothing at any one context. With finitely many boundary contexts each sampled with probability at least μ_min, TV ≤ √(ε/(2μ_min)) at each, so ε < 2μ_min(γ/L)² suffices. Under a deployment distribution ν with dν/dμ ≤ M, E_ν TV ≤ √(Mε/2).

A16: “A distillation procedure cannot certify the preservation of capacities under interventions that its transmission regime never samples.” The contexts at the boundary of viability are those that ordinary operation rarely visits, so a successor faithful on ordinary operation can be arbitrarily wrong exactly where viability is decided. This is the coverage form of CLAIM-407: closure under ordinary dynamics does not fix responses to intervention.

What it does not say

  • That distillation preserves viability in practice. The condition is sufficient, and it asks for coverage that ordinary training does not supply.
  • That fidelity is lineage. A successor meeting the bound is a faithful student; whether it continues the predecessor is CLAIM-329’s question.

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