phiSpacing_jcost_pos
plain-language theorem explainer
The declaration establishes that the J-cost of the golden ratio is strictly positive. Researchers modeling ALEXIS Experiment A on pulse spacing would cite this to confirm that φ-spaced intervals carry a nonzero per-step cost under Recognition Science. The proof is a one-line application of the general positivity lemma for Jcost at any positive argument not equal to one.
Claim. $0 < Jcost(φ)$, where $Jcost$ is the J-cost function and $φ$ is the golden ratio.
background
In the Phi vs Uniform Pulse Spacing module the J-cost quantifies deviation from equilibrium for successive interval ratios in pulse trains. The golden ratio arises as the self-similar fixed point forced by the Recognition Science chain (T5–T6). The key upstream lemma states that Jcost(x) > 0 for every x > 0 with x ≠ 1, proved by rewriting Jcost as a square divided by a positive denominator and invoking positivity of squares away from zero.
proof idea
One-line wrapper that applies the lemma Jcost_pos_of_ne_one to the golden ratio, supplying the facts that phi is positive and phi ≠ 1.
why it matters
The result is invoked directly by experiment_a_prediction_holds to discharge the per-step positivity clause in the RS prediction for ALEXIS Experiment A. It instantiates the T5 J-uniqueness landmark, showing that φ supplies the minimal positive drive among geometric ratios greater than one. The module doc notes that this positivity distinguishes φ-spacing from uniform intervals once boundary conditions are imposed.
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