variationsCert
plain-language theorem explainer
The definition assembles a certificate confirming that the recognition functional yields exactly five canonical variational problems, that the J-cost reaches its global minimum of zero at equilibrium r=1, and that the cost is strictly positive for any other positive r. A physicist deriving variational principles from the Recognition Science functional equation would cite this to anchor the Euler-Lagrange equilibrium condition. The construction is a direct record assembly from three supporting results on problem count and J-cost sign properties.
Claim. Let $J(r)$ be the J-cost functional. There exists a certificate $C$ such that the number of canonical variational problems equals 5, $J(1)=0$, and $J(r)>0$ for every $r>0$ with $r≠1$.
background
The module develops the calculus of variations from the Recognition Science recognition functional, identified with the J-cost integral. Its Euler-Lagrange equation enforces equilibrium at r=1 where J vanishes. Upstream theorems establish that Jcost vanishes at argument 1 and is positive for all other positive arguments, while a separate result counts exactly five canonical problems (brachistochrone, geodesic, minimal surface, Fermat, and J-cost minimization) that fit configDim D=5.
proof idea
The definition constructs the VariationsCert record by direct assignment: the five-problems field receives the theorem establishing cardinality 5, the minimum-at-1 field receives the theorem that Jcost at 1 equals zero, and the off-minimum-positive field receives the theorem that Jcost is positive away from 1. No additional tactics or reductions are applied.
why it matters
This definition packages the three facts required to certify that the J-cost functional satisfies the variational minimum condition in RS, directly supporting the module claim of five problems for configDim D=5. It closes the local S1/A4 foundation with zero axioms or sorrys. No downstream uses appear yet, but the certificate anchors the recognition-functional derivation of the Euler-Lagrange equation.
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