capabilityCost
capabilityCost defines the recognition cost of a rising power relative to an incumbent as the J-cost evaluated on their capability ratio. International relations modelers applying Recognition Science to power transitions cite it to locate the zero-cost equilibrium at parity. The definition is a direct one-line application of the J-cost function to the ratio.
claimThe capability cost for rising capabilities $r$ and incumbent capabilities $i$ is $J(r/i)$, where $J$ is the recognition cost function satisfying $J(1)=0$ and $J(x)=(x+x^{-1})/2-1$.
background
The module adapts Organski-Lemke power transition theory to Recognition Science by identifying the parity threshold with the zero set of J-cost. J-cost itself is imported from the Cost module and equals the function $J(x)=(x+x^{-1})/2-1$, which vanishes at unity and rises symmetrically. The local setting states that conflict likelihood peaks when the ratio crosses parity because the cost surface reaches a local minimum there; the war window is the open interval $(1/φ,φ)$ inside which $J(r)≤J(φ)≈0.118$.
proof idea
This is a one-line definition that applies Jcost directly to the ratio of the two real arguments.
why it matters in Recognition Science
The definition supplies the cost function used by capabilityCost_at_parity (which recovers zero at equal capabilities), capabilityCost_nonneg, capabilityCost_at_phi_boundary, and the PowerTransitionCert structure. It realizes the module's core claim that the parity band coincides with the J-cost zero set, connecting to T5 J-uniqueness and the phi-ladder constants. The module's falsifier is any COW dataset analysis showing great-power conflict onset does not cluster inside $(1/φ,φ)$.
scope and limits
- Does not compute numerical war probabilities.
- Does not incorporate alliance structures or third-party states.
- Does not model economic growth rates or technological change.
- Does not address nuclear or deterrence effects.
formal statement (Lean)
43def capabilityCost (rising incumbent : ℝ) : ℝ :=
proof body
Definition body.
44 Jcost (rising / incumbent)
45