mobilityTransitionCost
The definition sets the mobility transition cost between achieved status a and expected status e to the J-cost of their ratio. Sociologists modeling the Recognition Science five-stratum hierarchy forced by configDim D = 5 would cite this when quantifying upward or downward mobility steps. It is supplied as a direct definition that invokes Jcost on the status ratio.
claimThe mobility transition cost between achieved status $a$ and expected status $e$ is defined as $J(a/e)$, where $J$ is the J-cost function.
background
The module derives social stratification from configDim D = 5, yielding five canonical strata that follow the same template as Köppen zones, Big Five personality factors, and other natural five-fold divisions. The J-cost function, imported from the Cost module, assigns to a transition with ratio r the value J(r) = (r + r^{-1})/2 - 1. This definition specializes that measure to the ratio of achieved to expected status for mobility between strata.
proof idea
The declaration is a one-line definition that applies the Jcost function directly to the ratio of the two input parameters.
why it matters in Recognition Science
This supplies the cost function required by the SocialStratificationCert structure, which encodes stratum_count = 5 together with zero self-transition cost and nonnegative mobility costs for all positive status pairs. It realizes the module claim that stratum transitions cost J(φ^k) for rung k. The framework landmark is the forcing of five strata from the same template that produces D = 3 spatial dimensions and the eight-tick octave elsewhere in Recognition Science. The module falsifier is any comparative survey on ten or more societies that finds a modal stratum count reliably different from 5.
scope and limits
- Does not establish the value of configDim or the stratum count.
- Does not prove nonnegativity or zero self-cost; those are separate theorems.
- Does not incorporate cultural or historical variations in stratification.
- Does not specify the explicit formula or numerical value of J or phi.
formal statement (Lean)
49def mobilityTransitionCost (achieved_status expected_status : ℝ) : ℝ :=
proof body
Definition body.
50 Jcost (achieved_status / expected_status)
51