IndisputableMonolith.Foundation.RHatFromJCostGradient
The module introduces the midpoint map as the linear contraction x maps to (x+1)/2 with fixed point at 1, supplying the discrete step that decreases J-cost. Researchers deriving the recognition hat from cost minimization in the Recognition Science chain would cite it. The argument rests on algebraic verification of the contraction property together with fixed-point and Lyapunov lemmas.
claimThe midpoint map $m(x) = (x + 1)/2$ is a linear contraction toward the fixed point 1 that decreases the J-cost.
background
Recognition Science derives physics from the J-cost functional whose level sets encode deviation from self-similarity. The midpoint map supplies a canonical discrete gradient step on that landscape. The Constants module fixes the RS time quantum at one tick; the Cost module supplies the J-cost definition and its composition law.
proof idea
This is a definition module. It contains the midpoint map definition together with sibling lemmas that verify the fixed point at 1 and the strict decrease of J-cost under iteration.
why it matters in Recognition Science
The contraction supplies the mechanism that certifies RHat emergence from the J-cost gradient. It feeds the RHatEmergenceCert and rHatEmergenceCert siblings and closes the step from J-uniqueness to the recognition structure in the forcing chain.
scope and limits
- Does not establish convergence rates beyond the linear contraction factor 1/2.
- Does not treat continuous-time gradient flow or higher-dimensional maps.
- Does not derive numerical values for physical constants.