phi_rung_energy
plain-language theorem explainer
Energy stored per recognition event on rung n of the φ-ladder is given by scaling the coherence quantum E_coh by φ to the integer power n. Physicists deriving storage density limits from Recognition Science cite this scaling to establish the hierarchy between chemical and nuclear energies. The definition directly multiplies E_coh_storage by phi^n with no additional computation.
Claim. The energy at rung $n$ of the φ-ladder is $E_0 φ^n$, where the coherence quantum is $E_0 = φ^{-5}$.
background
In the module on optimal energy storage density, Recognition Science posits that energy equals J-cost times the coherence quantum E_coh = φ^{-5} in eV units. The φ-ladder organizes energy scales by integer exponents of φ, with chemical storage at rung 0 and nuclear at rung 45. This definition provides the explicit scaling for any rung n, building on the coherence energy E_coh_storage defined as φ^{-5}.
proof idea
This is a direct definition that multiplies the precomputed coherence energy E_coh_storage by the power phi^n. No lemmas or tactics are applied beyond the built-in exponentiation.
why it matters
This definition supplies the energy scale for each rung in the φ-ladder, directly feeding the theorems on energy hierarchy such as energy_storage_density_hierarchy and nuclear_exceeds_chemical. It realizes the EN-004 claim that energy storage follows the φ-ladder with ratios of φ between levels, consistent with the Recognition Science prediction of nuclear-to-chemical ratio approximately 10^9. It touches the open question of matching observed energy densities to the exact rung assignments.
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