IndisputableMonolith.Mathematics.CalculusVariationsFromRS
This module introduces the variational calculus built on the J-cost function from Recognition Science, with the explicit minimum J=0 attained at scale factor r=1. Researchers deriving equations of motion from the Recognition Composition Law would cite these objects when moving from cost functions to extremal principles. The module is a collection of definitions and supporting lemmas that rest directly on the imported Cost module.
claimThe variational problem whose J-cost attains its global minimum value of zero precisely when the scale factor equals one; auxiliary objects count the admissible problems and certify the extremal property.
background
The module resides in the Mathematics domain and imports IndisputableMonolith.Cost, whose J-cost is the function J(x)=(x+x^{-1})/2-1 that vanishes at the fixed point x=1. It defines VariationalProblem together with its cardinality, the explicit minimum statement, an off-minimum comparison, and a certificate type VariationsCert. The local setting is the translation of the Recognition Composition Law into a variational language that later supports physical derivations.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the variational minimum principle that later results in the Recognition Science mirror rely upon when converting the J-uniqueness and phi fixed-point statements into equations of motion. It directly implements the claim that J-cost minimum occurs at r=1, closing one step between the forcing chain (T5) and downstream physical applications.
scope and limits
- Does not derive Euler-Lagrange equations.
- Does not treat time-dependent or field-theoretic extensions.
- Does not prove uniqueness of the r=1 minimizer beyond the stated J-cost.
- Does not connect the variational objects to specific physical constants or dimensions.