mobilityTransitionCost_at_stratum
plain-language theorem explainer
The theorem establishes that the J-cost of a mobility transition vanishes when achieved status equals expected status. Sociologists working within the Recognition Science framework cite it to anchor the zero-cost fixed point in the five-stratum model forced by configDim D = 5. The proof is a one-line wrapper that unfolds the definition, cancels the ratio via div_self, and invokes the Jcost unit lemma.
Claim. For any nonzero real number $s$, the J-cost of the mobility transition from status $s$ to status $s$ equals zero: $J(s/s) = 0$.
background
The J-cost function is defined in the Cost module as $J(x) = (x-1)^2/(2x)$, with the unit lemma stating $J(1) = 0$. The mobilityTransitionCost definition is the J-cost of the ratio of achieved status to expected status. This places the zero-cost case at the fixed point of the cost function on the phi-ladder of strata.
proof idea
The proof is a one-line wrapper. It unfolds mobilityTransitionCost to expose Jcost(s/s), rewrites the ratio to 1 using div_self under the hypothesis s ≠ 0, and applies the lemma Jcost_unit0 to obtain zero.
why it matters
The result supplies the cost_at_stratum field of the SocialStratificationCert structure, which certifies the five-stratum prediction. It closes the stationary case for the J-cost on stratum transitions, consistent with the module's derivation of five layers from configDim D = 5 and the same template used for Köppen zones and Big Five factors. The parent cert definition assembles this with the stratum count and nonnegativity results.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.