warWindowLow
The lower bound of the war window in this Recognition Science model of power transitions is set to the reciprocal of the golden ratio. Researchers extending Organski's power transition theory with J-cost dynamics would reference this constant when specifying the interval of elevated conflict risk. It is introduced via a direct definition that imports the golden ratio from the constants module.
claimThe lower edge of the war window is the real number $phi^{-1}$, where $phi$ is the golden ratio satisfying $phi = 1 + phi^{-1}$.
background
The module applies Recognition Science to international relations by identifying the J-cost zero set with capability parity. The J-cost function satisfies J(1) = 0, and the war window is the band around parity where J(r) remains small, specifically the interval (phi inverse, phi). This definition introduces the lower bound as phi inverse. The module doc explains that the parity band is the J-cost zero set, making the system unstable at zero cost. Phi is the self-similar fixed point from the foundational forcing chain.
proof idea
The declaration is a one-line definition assigning the lower war window bound directly to the reciprocal of phi, using the imported phi from Constants.
why it matters in Recognition Science
It provides the concrete lower bound required by the PowerTransitionCert structure, which asserts zero cost at parity, non-negative costs, and the ordering of the war window bounds. This completes the quantitative specification of the war window predicted by the J-cost model of power transition theory. It connects to the phi fixed point in the core framework. The open falsifier is empirical disconfirmation via the Correlates of War dataset showing no clustering of conflicts in (0.618, 1.618).
scope and limits
- Does not derive the J-cost function or its properties.
- Does not prove the existence of the war window in real conflicts.
- Does not specify the upper bound of the war window.
- Does not handle cases where capability ratios fall outside the defined interval.
formal statement (Lean)
55def warWindowLow : ℝ := phi⁻¹