phi_coherent_minimizes_jcost_per_energy
plain-language theorem explainer
Recognition Science establishes that J-cost energy per coherence quantum reaches its global minimum at the ground-state ratio x=1 for every positive real x. Engineers and physicists modeling storage density hierarchies cite the result to bound chemical-to-nuclear transitions on the φ-ladder. The argument is a direct one-line invocation of the preceding ground-state minimum lemma.
Claim. For every real number $x>0$, the stored energy $E(x)=E_0 J(x)$ satisfies $E(1)≤E(x)$, where $E_0$ is the coherence energy scale and $J(x)=½(x+x^{-1})-1$ is the J-cost function.
background
Module EN-004 derives fundamental limits on energy storage from the φ-ladder and J-cost structure. Energy stored in a recognition event is defined by the product $E=J(x)·E_{coh}$ with $E_{coh}=φ^{-5}$ eV; the function $J(x)$ attains its global minimum of zero precisely at the ground state $x=1$ and diverges as $x→0^+$ or $x→∞$ (see jcost_energy definition).
proof idea
The proof is a one-line wrapper that applies the upstream theorem jcost_energy_min_at_ground directly to the supplied ratio x and positivity witness hx.
why it matters
The declaration supplies EN-004.11, confirming that ground-state coherence (x=1) minimizes stored energy and thereby anchors the chemical-nuclear-mass hierarchy asserted in the module. It supports the broader Recognition Science claim that practical storage ratios are quantized on the φ-ladder with successive levels separated by factors φ^{Δn}. No downstream uses are recorded yet; the result closes the minimum-energy step of the storage model.
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