The construction
A8 §4: “Suppose w represents the balance of authority between two action-generating regimes. Let h represent an endorsement favoring one of them. A minimal nonlinear model is
ẇ = μw − w³ + h.
Here μ controls the stability structure, and h tilts the competition toward the endorsed regime.” A12’s Proposition 3 is the same equation in z. Here h is a higher-order volition in Frankfurt’s sense, an attitude that one regime should be effective. The record’s endorsement (TERM-053) is a different thing, an effect of one process on another’s share of influence.
Equilibria satisfy h = φ(w) = w³ − μw, and f′(w) = μ − 3w².
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μ ≤ 0. φ is monotone, so there is one equilibrium for each h, and it is stable.
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μ > 0. φ has a local maximum at w = −√(μ/3) and a local minimum at w = +√(μ/3), of value ±(2μ/3)√(μ/3), so
h_c = 2(μ/3)^{3/2} = 2μ
For |h| < h_c there are three equilibria. The outer two have |w| > √(μ/3), so f′ < 0, and they are stable. The middle one is unstable. At |h| = h_c an outer equilibrium meets the middle one and both vanish, a saddle-node.
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Hysteresis. Start in the competing regime, w < 0, and raise h. For 0 < h < h_c the state moves continuously along the lower branch and stays there. Past h_c the branch is gone and w jumps to the upper one. Lower h back to 0 and w stays positive. Returning to the first regime needs h < −h_c.
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Structure against strength. h_c falls to 0 as μ → 0. A change in μ, the structure of the competition, can make a small h decisive. A8: “With stronger endorsement, or with structural changes that alter μ, the competing attractor may disappear”.
The threshold was checked by hand and numerically for this filing.
What it can show
- Efficacy without transition. A8: “An endorsement can be sincere and causally efficacious while remaining too weak to shift the system out of its current attractor basin. It may change the local balance without producing a behavioral transition.” For 0 < h < h_c the volition moves w, and the regime is unchanged.
- Changing the conditions, not the attitude. Lowering μ with h fixed can switch the regime. That is CLAIM-122’s form: changing the conditions of efficacy changes behaviour while the competing attitudes stay fixed. It is CASE-013’s writer blocking the websites.
- Persistence against revision. A12, Proposition 3: “An enduring commitment can coexist with high costs of revision.” Once in the upper regime the agent stays there for every h > −h_c. That is CLAIM-025’s separation, and the revisability dimension of CLAIM-354.
What it cannot show
- That motivation follows this equation. A8: “This is not a claim that human motivation literally follows a quartic potential.” A12: “application to human commitments remains interpretive.”
- Will as an attractor. TERM-003 defined will as a higher-order attractor and was superseded by will-organization (TERM-023). Here the attractor is a state of the balance of authority, and the will-organization is what sets μ and passes h into w. The case does not restore the old definition.
- That h is the agent’s own. A8 §3: “we could accidentally redefine endorsement as whichever signal successfully changes behavior. That would collapse a normative notion into a causal one.” The model says what an input does, not what makes it authoritative (THEORY-029).
Note of 2026-10-21: the normal form at zero tilt, read as a rule
A19 of the exchange of 2026-10-20 used ḋ = (α − κ)d − βd³ for the authority balance between two organizations, with κ a stabilizing institutional constraint. That is this case’s equation at h = 0 with μ = α − κ: weakening the constraint past κ = α makes the equal allocation unstable (checked). It is a second appearance of the normal form, and a new reading of μ as the strength of a rule. It is filed as CLAIM-408, which this case supports.
At h = 0 the pitchfork is supercritical, so in κ there is no hysteresis: restoring κ > α makes the equal allocation again the only equilibrium, and a tipped allocation decays back to it. This case’s hysteresis is in the tilt h at fixed μ > 0, and it is what A19 did not draw: an allocation tipped by a tilt past h_c is not restored by removing the tilt. This case’s h is what A19 called “explicit symmetry breaking”. CLAIM-408’s check also ran this case’s form with a small tilt through a weakening, restoring and re-weakening of the constraint: restoring it brought w near zero, and weakening it again selected the side the tilt favours, not the side the state had held before.
Arthur’s model, read the same day through its 1983 working paper (NOTE-1052), has the same shape in another setting: a planner can tilt the adoption process at the outset, and once one technology has locked in, small changes in returns have no effect and only large ones swing the market (THEORY-863). That matches this case’s tilt h before symmetry breaks and its hysteresis in h after; Arthur’s lock-in is absorption in accumulated adoptions, not a pitchfork, so the match is of shape only.