measureTheoryCert
plain-language theorem explainer
The definition constructs a certificate confirming that exactly five canonical measures exist in the Recognition Science framework and that the J-cost function is non-negative for all positive real arguments. Mathematicians formalizing the integration and probability foundations of RS would reference this to establish compatibility with classical measure theory. It assembles the certificate by direct assignment from a counting result on canonical measures and a non-negativity lemma for J-cost.
Claim. The measure theory certificate asserts that the cardinality of the set of canonical measures is 5 and that the J-cost function satisfies $J(r) ≥ 0$ for every real $r > 0$.
background
In the module on Measure Theory from RS, measure theory underpins probability and integration, with the J-cost function serving as a central measure-theoretic object. Five canonical measures (Lebesgue, counting, Dirac, Hausdorff, Borel) correspond to configuration dimension 5. The J-cost is measurable as a continuous function of positive r. Upstream results include the theorem that the cardinality of CanonicalMeasure is 5, proved by decision, and the non-negativity of J-cost (IC-005.1: J-cost is non-negative, with unique minimum at x=1).
proof idea
The definition is a one-line wrapper that populates the five_measures field using the canonicalMeasureCount theorem and the jcost_nonneg field using the jcost_measurable theorem, which proceeds by case analysis on whether r equals 1.
why it matters
This definition completes the measure-theoretic certification in the RS framework, linking the five measures to configDim D=5 and ensuring J-cost non-negativity for entropy and tachyon-free conditions. It anchors the module's claim of zero sorry or axiom in the Lean status. It touches the open question of deriving full classical measure theory from the J-cost functional equation.
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