storage_density_ratio_pos
plain-language theorem explainer
The theorem establishes that the energy storage density ratio between any two rungs on the phi-ladder is strictly positive for integer indices. Engineers comparing chemical and nuclear regimes in Recognition Science hierarchies would cite it to guarantee ordering before ratio calculations. The proof is a one-line term reduction that unfolds the definition and invokes positivity of integer powers of phi.
Claim. For all integers $n, m$, the energy storage density ratio satisfies $0 < phi^{n-m}$.
background
The module EN-004 derives energy storage limits from the phi-ladder and J-cost structure, where energy equals J-cost times the coherence quantum E_coh = phi^{-5} eV. The definition storage_density_ratio (n m : Z) : R := phi ^ (n - m) gives the ratio of densities between rungs n and m. Upstream results supply rung maps for sectors such as leptons and quarks, plus the density function phi^k used in neutron-star regimes.
proof idea
The proof is a term-mode one-liner. It unfolds storage_density_ratio to the power expression phi^(n-m) and applies zpow_pos together with the fact that phi is positive.
why it matters
This result anchors the energy storage density hierarchy in EN-004 by ensuring all rung ratios remain positive, which preserves the ordering chemical < nuclear < mass-energy. It supports downstream comparisons such as nuclear_chemical_ratio and the phi^45 scaling between regimes. The positivity follows directly from the self-similar fixed point phi in the forcing chain and the eight-tick octave structure.
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