VariationsCert
plain-language theorem explainer
VariationsCert records that the recognition framework yields precisely five canonical variational problems, that the J-cost functional reaches its minimum value of zero at the equilibrium radius r equals 1, and that the functional is strictly positive for all other positive radii. A researcher deriving the calculus of variations from the Recognition Science functional equation would cite this structure to confirm the problem count and minimality conditions. The definition assembles these three properties directly from the enumeration of Variati
Claim. Let $V$ be the inductive type whose five constructors are the brachistochrone problem, the geodesic problem, the minimal-surface problem, Fermat's principle, and J-cost minimization. The structure asserts that the cardinality of $V$ equals 5, that the J-cost functional satisfies $J(1)=0$, and that $J(r)>0$ for every positive real $r$ with $r≠1$.
background
Recognition Science derives the calculus of variations from the J-cost functional, which serves as the recognition integral whose extrema obey the Euler-Lagrange equation. Equilibrium occurs at $r=1$ where the functional vanishes. The module states that five canonical problems arise whose configuration dimension matches the spatial dimension $D=3$ of the framework, and that the Lean formalization carries zero axioms or sorry statements.
proof idea
VariationsCert is a structure definition that packages three independent assertions. The cardinality field is supplied by the Fintype instance on the inductive type VariationalProblem. The remaining two fields are imported verbatim from the supporting lemmas jcost_variational_minimum and jcost_off_minimum that establish the zero at $r=1$ and strict positivity elsewhere. No tactics beyond the structure constructor are used.
why it matters
This structure supplies the certified package of variational facts that the downstream definition variationsCert instantiates using the count variationalProblemCount together with the two Jcost lemmas. It closes the S1/A4 foundation block by confirming that exactly five problems arise and that J attains its global minimum at equilibrium. In the Recognition Science chain this anchors the transition from the functional equation to Euler-Lagrange dynamics and the five-dimensional configuration space.
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