catalyticBarrierRatio_pos
plain-language theorem explainer
Positivity of the catalytic barrier ratio follows directly from the golden ratio exceeding 1.5, confirming a positive scaling factor that reduces the modeled activation energy in the Haber-Bosch process. Chemists working with Recognition Science predictions for industrial catalysis cite this to anchor the J-cost reduction from 230 kJ/mol uncatalyzed to roughly 27 kJ/mol on iron. The proof is a one-line wrapper that unfolds the definition and applies linear arithmetic to the phi lower bound.
Claim. $0 < phi - 3/2$, where $phi = (1 + sqrt(5))/2$ is the golden ratio and the left side is the catalytic barrier ratio $J(phi)$ that scales the uncatalyzed activation energy downward.
background
The module derives Haber-Bosch parameters from the phi-ladder: optimal temperature ratio equals phi, pressure ratio equals phi squared, and the catalytic barrier ratio is defined as phi minus 3/2 to approximate J(phi) times the uncatalyzed barrier of 230 kJ/mol, yielding the observed 27 kJ/mol on Fe. This sits inside the structural theorem block that predicts industrial conditions (400-500 C, 150-300 atm) without external axioms. The upstream lemma phi_gt_onePointFive states the tighter lower bound phi > 1.5 because sqrt(5) > 2.
proof idea
The term proof unfolds catalyticBarrierRatio to the expression phi - 3/2, then invokes linarith on the hypothesis supplied by phi_gt_onePointFive. No further rewriting or case analysis is required.
why it matters
The result supplies the barrier_pos field inside the HaberBoschCert definition, completing the structural theorem for the phi-ladder model of the Haber-Bosch process. It realizes the module's third RS prediction that E_a^RS equals J(phi) times the homogeneous barrier, tying directly to the J-cost and phi fixed-point landmarks. No open scaffolding remains for this claim.
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