Outer dynamics with an uncoupled fixed center

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The outer process can depend on its previous value and input, while the center’s update simply copies its old value. The center therefore stays constant even if the outer process is complex or changing. This explicitly demonstrates that surrounding activity cannot affect a stored component without a causal update path.

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Original context

7.1 Proposition 4: no coupling means no center change

Suppose the outer process evolves arbitrarily while the center obeys

ℓt+1=Fℓt,ut,  ct+1=ct.

By induction, ct=c0 for every update. Increasing the number of outer iterations does not change this conclusion. The audit ran 10,000 outer-state updates while preserving the center exactly at two.

For the center to change, the model needs a causal connection, for example ct+1=ct+Gct,ℓt. Physical spatial proximity is not itself a mathematical coupling term. If electromagnetic effects are intended, their field strengths, circuit geometry, material properties, and coupling must be specified; a diagram of information “around” a number supplies none of those quantities.

7.2 Proposition 5: repeated positive input does not always accumulate without bound

{ℓ}_{t+1}=F\left({ℓ}_{t},{u}_{t}\right),  {c}_{t+1}={c}_{t}.