J_nonneg
plain-language theorem explainer
J(x) is nonnegative for every positive real x. Derivations of baseline rung integers and mass formulas cite this to keep recognition costs well-behaved on the phi-ladder. The proof unfolds the definition, cancels the product x * x^{-1} to 1, expands (x - x^{-1})^2, and applies nlinarith to recover the AM-GM bound x + x^{-1} >= 2.
Claim. For all real $x > 0$, $J(x) = (1/2)(x + x^{-1}) - 1$ satisfies $J(x) >= 0$.
background
J is the recognition cost functional J(x) = (1/2)(x + x^{-1}) - 1, defined for all reals but used only for x > 0. This module upgrades boundary assumptions to derived status by extracting rung integers, octave offset, and generation ordering from the combinatorics of the 3-cube Q3, with every integer tracing directly to D = 3. The upstream structure for records the meta-realization axioms required by the self-reference content in the forcing chain.
proof idea
Unfold J. From 0 < x obtain x ≠ 0 and x * x^{-1} = 1. Expand (x - x^{-1})^2 = x^2 - 2 + x^{-1}^2 and apply sq_nonneg to reach x^2 + x^{-1}^2 >= 2. Use nlinarith on the squares of (x + x^{-1}) and (x - x^{-1}) to conclude x + x^{-1} >= 2, then finish with linarith.
why it matters
This nonnegativity result supports the non-triviality item (B-20) in the baseline table: J(1) = 0 implies x = 1 is unobservable. It anchors the J-uniqueness step (T5) of the unified forcing chain and guarantees nonnegative costs in the mass formula yardstick * phi^(rung - 8 + gap(Z)). No downstream theorems are listed, so the lemma functions as a local foundation for the remaining rung derivations in the module.
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