IndisputableMonolith.Mathematics.CalculusVariationsFromRS
This module develops calculus of variations results from Recognition Science by organizing definitions and counts around J-cost minimization. It centers on the condition that J reaches its global minimum of zero precisely when the ratio r equals one. The module consists of supporting definitions and certificates that build directly on the imported Cost module.
claimThe module introduces variational problems whose J-cost satisfies the minimum condition $J=0$ at scale ratio $r=1$.
background
The module resides in the mathematics domain and imports Mathlib together with the Cost module. The Cost module supplies the underlying definition of J-cost. This supplies the local setting for applying variational methods to the Recognition Science functional equation, with the explicit minimum condition stated in the module doc-comment.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module supplies the variational setting that supports derivations of J-uniqueness (T5) and the Recognition Composition Law. It fills the step that converts the J-cost minimum into concrete variational problems used in later RS constructions of constants and dimensions.
scope and limits
- Does not derive the J function itself.
- Does not contain numerical evaluations of minima.
- Does not address multi-variable extensions.