IndisputableMonolith.Foundation.Inequalities
The Inequalities module supplies the AM-GM relation for any positive real and its reciprocal, establishing that their sum is at least two. Researchers deriving the golden ratio from J-cost minimization in Recognition Science cite it to anchor non-negativity of the cost function. Proofs rely on direct algebraic comparison or standard Mathlib quadratic-mean lemmas.
claimFor all $x > 0$, $x + x^{-1} > 2$ with equality only at $x = 1$.
background
The module belongs to the Foundation layer and imports the Constants module, which fixes the RS-native time quantum at one tick. It assembles lemmas that convert the arithmetic-geometric mean into statements about the J-cost function, including its non-negativity, strict positivity away from unity, and minimum location. These results prepare the ground for the subsequent phi-emergence argument by guaranteeing that the cost functional is bounded below by zero.
proof idea
The module organizes a short chain of sibling lemmas. The core statement follows from the standard AM-GM inequality applied to $x$ and $1/x$, proved either by direct expansion of $(x - 1/x)^2$ or by invoking Mathlib's quadratic-mean comparison. Companion lemmas handle the equality case, the strict inequality for $x$ not equal to one, and evaluations at the golden-ratio fixed point.
why it matters in Recognition Science
This module feeds the PhiEmergence development, which extracts the golden ratio as the unique minimizer of J-cost. It supplies the non-negativity step required before the self-similar fixed-point argument can proceed, corresponding to the early segment of the T0-T8 forcing chain where the cost functional first acquires a well-defined minimum.
scope and limits
- Does not treat inequalities on complex numbers or matrices.
- Does not compute numerical values of the golden ratio or related constants.
- Does not address multi-variable or higher-dimensional extensions.
- Does not connect the inequality to measurement or experimental data.