phi_rung_energy_ratio
plain-language theorem explainer
Consecutive energies on the phi-ladder satisfy E(n+1)/E(n) = phi exactly. Researchers deriving scaling relations in energy storage hierarchies cite this when comparing chemical and nuclear regimes. The proof is a direct algebraic reduction that substitutes the exponential definition and cancels common factors using positivity lemmas.
Claim. Let $E(n) := E_0 · ϕ^n$ denote the energy stored per recognition event at rung $n$ of the ϕ-ladder, where $E_0 > 0$ is the base coherence energy. Then $E(n+1)/E(n) = ϕ$ for every integer $n$.
background
In the Recognition Science framework, energy storage is quantized on the phi-ladder. The function phi_rung_energy assigns to each integer rung n the value E_coh_storage times phi to the power n, where E_coh_storage is the base coherence energy shown positive by the upstream theorem E_coh_storage_pos. This construction appears in the module on optimal energy storage density, which derives limits from the J-cost function J(x) = (x + 1/x)/2 - 1 and the coherence quantum E_coh = phi^{-5} eV. The result relies on the non-vanishing of phi from Constants.phi_ne_zero and the definition of rung energy.
proof idea
The proof unfolds the definition of phi_rung_energy to expose the common E_coh_storage factor and the powers phi^{n+1} and phi^n. It rewrites the numerator power using zpow_add_one₀ instantiated at phi_ne_zero. Positivity of phi^n is recorded to justify the field simplification, after which field_simp cancels the common terms and the positivity hypotheses to yield exactly phi.
why it matters
This theorem completes the EN-004.3 step in the energy storage density hierarchy and is referenced in the en004_certificate that certifies the full derivation. It directly supports the prediction that consecutive rungs differ by the golden ratio phi, which underpins the scaling between chemical (rung 0) and nuclear (higher rungs) storage limits. In the broader framework it reinforces the phi-ladder structure and the energy hierarchy chemical < nuclear < mass-energy.
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