IndisputableMonolith.Foundation.Inequalities
Elementary inequalities that underwrite Recognition Science cost functionals. The module records reciprocal AM-GM (x + 1/x ≥ 2 for x > 0), nonnegativity and unique minimum of the J-cost, and basic algebraic facts for the golden ratio φ. Anyone proving J ≥ 0 or evaluating J on the φ-ladder cites it. Proofs are short algebra or thin Mathlib wrappers.
claimFor all $x > 0$, $x + x^{-1} \ge 2$, with equality if and only if $x = 1$. Hence $J(x) = (x + x^{-1})/2 - 1 \ge 0$, uniquely minimized at $x = 1$. Also $\varphi > 1$, $\varphi^{-1} = \varphi - 1$, $\varphi^2 = \varphi + 1$, and $J(\varphi)$ evaluates to a fixed positive constant.
background
Recognition Science forces a unique cost $J$ from the Recognition Composition Law; the first analytic demand is that $J$ be nonnegative with a unique minimum at the multiplicative identity. This Foundation module supplies those elementary real inequalities and the companion identities for the self-similar fixed point $\varphi$.
The engine is reciprocal AM-GM: the arithmetic mean of $x$ and $x^{-1}$ is at least their geometric mean $1$. Subtracting 1 produces the standard formula $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$). The module also records $\varphi > 1$, the relations $\varphi^{-1} = \varphi - 1$ and $\varphi^2 = \varphi + 1$, and the evaluation of $J$ at $\varphi$. It imports only Mathlib and the Constants module (RS time quantum $\tau_0$).
proof idea
A compact lemma library rather than a single theorem. Reciprocal AM-GM is obtained from Mathlib's two-term AM-GM (or by expanding $(\sqrt{x} - x^{-1/2})^2 \ge 0$ and clearing denominators). Nonnegativity, the unique minimum at 1, and strict positivity off 1 are immediate rewrites of that inequality. The $\varphi$ facts are algebraic consequences of the minimal polynomial $t^2 - t - 1 = 0$; $J(\varphi)$ is direct substitution. No deep analysis is required.
why it matters in Recognition Science
These are the first concrete positivity statements supporting the forcing chain. T5 (J-uniqueness) and the identification of $J$ with $\cosh(\log x) - 1$ rest on $J \ge 0$ with equality solely at 1. Every later use of the cost (mass ladder, eight-tick octave, $\alpha$-band estimates) inherits the sign of $J$ from the lemmas collected here. The module currently shows no external used-by edges, yet it is the natural import for any argument that needs $J$-nonnegativity or $\varphi$-arithmetic inside the monolith.
scope and limits
- Does not prove the Recognition Composition Law or force the form of J.
- Does not derive φ from self-similarity; only records its algebraic identities.
- Does not treat operator-valued or higher-rank costs.
- Does not contain the full forcing chain T0–T8.
- Does not bound physical constants such as α or masses.