covalent_barrier_higher
plain-language theorem explainer
Covalent bond activation barriers exceed the base rung on the phi-scaled ladder. Chemists working with J-cost landscapes and Arrhenius rates cite this ordering to separate covalent from hydrogen-bond energetics. The proof reduces the two barrier expressions by direct substitution of the ladder definition, invokes the golden-ratio inequality and coherence-energy positivity, then closes with nonlinear arithmetic.
Claim. Let $E_c$ denote coherence energy and let $phi$ be the golden ratio. Then $E_c phi^1 > E_c phi^0$.
background
Activation energy barriers emerge from the J-cost landscape, with the transition state identified as the J-maximum along the reaction coordinate. The barrierLadder definition assigns height $E_c phi^n$ to rung n, where $E_c$ is the coherence energy scale fixed by the Recognition Science constants. The module derives Arrhenius form and enzyme lowering directly from this phi-scaling of barriers.
proof idea
Simplification expands the two instances of barrierLadder at n=1 and n=0, replacing the powers and the multiplication identity. The golden-ratio lemma supplies phi > 1 while the power-positivity lemma supplies E_c > 0. Nonlinear linear arithmetic then yields the strict inequality from these two facts.
why it matters
The result instantiates the phi-scaling of barriers predicted in the activation-energy module and supplies the ordering needed for higher_barrier_slower. It sits inside the T5-T6 forcing chain that fixes phi as the self-similar fixed point and the eight-tick octave that sets the attempt frequency. No open scaffolding remains for this ordering claim.
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