IndisputableMonolith.Foundation.Inequalities
Foundation.Inequalities supplies the reciprocal AM-GM inequality x + 1/x ≥ 2 for x > 0 that forces J-cost non-negativity. Researchers deriving φ from J-cost minimization in the Recognition framework cite these results. The proofs consist of direct algebraic expansions or calls to Mathlib quadratic-mean lemmas.
claim$\forall x > 0,\ x + x^{-1} \geq 2$
background
The module imports Constants, which fixes the RS time quantum τ₀ at one tick. It collects inequalities on positive reals that establish J(x) = (x + x^{-1})/2 - 1 ≥ 0 together with ordering facts on the golden ratio φ. These facts are required by the downstream PhiEmergence module to obtain φ as the unique minimizer of J-cost.
proof idea
The central inequality is obtained either by expanding (√x - 1/√x)² ≥ 0 or by direct appeal to Mathlib's add_div_two_ge_sqrt_mul_self_of_sq_le_sq. The remaining declarations are one-line algebraic consequences that establish non-negativity of J and the basic ordering properties of φ.
why it matters in Recognition Science
The module is imported by PhiEmergence, whose doc-comment states it derives the golden ratio from J-cost minimization. It supplies the inequality explicitly identified in the module doc-comment as the one that forces J-cost ≥ 0, a prerequisite for the self-similar fixed point (T5-T6) in the forcing chain.
scope and limits
- Does not treat x ≤ 0.
- Does not derive numerical approximations to φ or α.
- Does not connect the inequality to physical constants G or ħ.
- Does not address equality cases or higher-order inequalities.