4 Three instances
With \(N = 3\), one contact updates the count \(c \in \mathbb {N}\) by
For all \(n, c \in \mathbb {N}\), if \(c \le N\) then \(f^{n}(c) \le N\).
Induction on \(n\), using that one step never exceeds \(N\).
The model is coarse: hands and the rule, against surfaces and microbes. It assumes the rule is followed.
A set \(S\) is organized when \(\exists \, a, b \in S\) with \(a \neq b\) and \(\operatorname {Bonded}(a, b)\). Two systems with disjoint compositions and at least one bond between them compose by union.
\(\operatorname {Org}(C_{\mathrm{HTTP}} \cup C_{\mathrm{TCP}})\), where the bond is the HTTP client handing data to the TCP sender.
Composition of two systems with a cross bond is organized.
Each layer is modeled as closed, with no network environment. This is composition of two systems, not composition of arrows.
The state is the inter-block time \(s \in \mathbb {Z}\) in minutes, with set point \(10\). The feedback law is
\(\varphi (10) = 10\).
At the set point the error is zero and the correction leaves the state unchanged.
This is an idealized one-step model of the 2016-block retarget. As written, \(\varphi \) returns \(10\) from any state, so the result concerns the homeostat’s structure, not Bitcoin’s dynamics. The real rule adjusts difficulty multiplicatively over a 2016-block window.