IndisputableMonolith.Foundation.Inequalities
This module supplies the AM-GM inequality for x and 1/x that forces J-cost non-negativity throughout Recognition Science. Workers on the phi-emergence step or the forcing chain cite it to anchor minimization arguments. Proofs are short algebraic reductions or direct calls to Mathlib inequality lemmas.
claimFor all $x > 0$, $x + x^{-1} \geq 2$. This inequality implies $J$-cost $\geq 0$ where $J(x) = (x + x^{-1})/2 - 1$.
background
The module lives in the Foundation layer and imports Constants, where the RS time quantum is fixed as $ au_0 = 1$ tick. Its central object is the reciprocal AM-GM inequality, documented as the fact that forces $J$-cost $\geq 0$. Sibling declarations include am_gm_reciprocal, J_formula_nonneg, phi_pos, phi_gt_one, and J_cost_phi, which together establish positivity and ordering on the phi-ladder.
proof idea
The module assembles a family of inequalities. Each is obtained either by direct algebraic expansion or by invoking Mathlib's add_div_two_ge_sqrt_mul_self_of_sq_le_sq. No deep tactic chains appear; the structure is a collection of short, self-contained verifications.
why it matters in Recognition Science
The module supplies the inequality layer required by IndisputableMonolith.Foundation.PhiEmergence, whose doc-comment states it derives the golden ratio from J-cost minimization. It therefore underpins T5 J-uniqueness and T6 phi-forcing in the unified chain.
scope and limits
- Does not treat negative or zero arguments.
- Does not derive numerical values for constants.
- Does not address the eight-tick octave or spatial dimension.
- Does not connect to mass formulas or Berry thresholds.