The construction
A7 §4: “Suppose repair has a cooperative, self-amplifying component”:
ḃ = μb + κb² − b³ + d, κ > 0,
where b ≥ 0 “measures boundary-maintenance activity, μ changes the linear ability to sustain it, and κb² models positive feedback among repair processes. The cubic term saturates growth.” Take d = 0.
Write ḃ = f(b) = b·g(b) with g(b) = μ + κb − b².
- Equilibria. b = 0, and the roots of b² − κb − μ = 0, b± = (κ ± √(κ² + 4μ))/2, real when μ ≥ −κ²/4. They collide at μ_fold = −κ²/4.
- Stability. f′(0) = μ, so b = 0 is stable for μ < 0. At b±, g = 0, so f′(b±) = b±·g′(b±) = b±(κ − 2b±) = ∓ b±√(κ² + 4μ). The upper branch b₊ is stable, the lower b₋ unstable.
- Bistable range. For −κ²/4 < μ < 0 both roots are positive (their sum is κ, their product −μ > 0). Then 0 and b₊ are both stable, and b₋ is the threshold between them. For μ > 0, b₋ < 0 lies outside b ≥ 0, b = 0 is unstable, and b₊ is the only stable state. For μ < −κ²/4 only b = 0 remains.
- Hysteresis. Lower μ from the maintained state: b follows b₊ until μ passes −κ²/4, then collapses to 0. Raise μ back to any value in (−κ²/4, 0): b = 0 is still stable and b stays there. Maintenance returns only when μ passes 0, or when something outside the dynamics lifts b above b₋(μ) = (κ − √(κ² + 4μ))/2.
- Slowing at the fold. The recovery rate at b₊ is |f′(b₊)| = b₊√(κ² + 4μ), which goes to 0 as μ → −κ²/4.
A7’s chart values for κ = 1 check: (b₊, b₋) = (0.724, 0.276) at μ = −0.2, (0.816, 0.184) at μ = −0.15, and b₊ = 1.048 at μ = 0.05 and 1.171 at μ = 0.2. The derivation was redone by hand and the values numerically for this filing.
What it can show
A7’s four consequences, each now a property of the normal form:
- “Boundary maintenance can have a threshold. An organization may require a minimum existing stock of maintenance capacity to sustain itself.” The stock is b₋.
- “Recovery can depend on history. Two systems with the same current external conditions may differ because one has retained sufficient repair organization and the other has lost it.” In the bistable range, μ does not fix the state.
- “Restoration can cost more than preservation. Once the maintained state collapses, returning the external parameter to its previous value may not regenerate maintenance without an additional intervention.” The additional intervention is μ past 0 or a lift above b₋.
- “Critical slowing can warn of approaching failure. Near the fold, local recovery rates approach zero, though noise, rate-dependent effects, and nonstationarity can make empirical early-warning signals unreliable.”
And A7’s use of it: “A will-organization can change what gets done without changing the underlying capacity to govern. In a deeper transition, it can lose or regain that capacity itself.” That is CLAIM-099 (maintaining a regime of action is not maintaining the capacity to govern) and CLAIM-025 (the two fail separately), with a mechanism for the loss of the second.
It is also the kind of model CLAIM-330 says it lacks: CLAIM-330 does not say “That the bounded nonlinear system has two captured attractors, or how far it hysteretically holds one.” This case is a different variable, the maintenance of a boundary, not the meaning–authority loop. It shows what a nonlinear half would look like. It does not supply CLAIM-330’s.
What it cannot show
- That any boundary obeys it. A7: “This is a proposed minimal model, not an empirical fit.” Nothing measures μ or κ.
- That an abrupt collapse shows a fold. THEORY-136: jumps also come from a stepwise driver or a simple threshold, and only hysteresis, initial-state dependence or a lasting shift after a temporary disturbance comes close to showing alternative stable states. A field test of this case is the hysteresis loop, not the collapse.
- That slowing will warn. On r/place, a boundary of attack and repair, THEORY-214 finds several distinct precursors rather than one generic warning signal, and many overthrows with no warning.
- Who maintains b. The equation does not say whether repair is the organization’s own or supplied from outside. That is QUESTION-086’s question, and A7 §1’s caveat: “writing G and H into an equation does not establish self-maintenance.”