nuclear_exceeds_chemical
plain-language theorem explainer
Nuclear energy storage exceeds chemical energy storage by the factor phi to the forty-five in the Recognition Science phi-ladder model. Researchers deriving energy hierarchies from the coherence quantum and rung structure would cite the result. The term proof unfolds the rung definitions, simplifies the zero and forty-five cases to a multiplication inequality, and closes using positivity of the coherence energy together with one less than phi.
Claim. $E_chemical < E_nuclear$, where $E_chemical = E_coh$ at rung zero and $E_nuclear = E_coh · phi^{45}$ at rung forty-five, with $E_coh > 0$ and $phi > 1$.
background
The Energy Storage Density Structure module places energy scales on the phi-ladder via the function phi_rung_energy n, which returns the coherence energy E_coh scaled by phi to the power n. The sibling definition chemical_rung fixes the integer zero so that E_chemical equals E_coh, while nuclear_rung fixes the integer forty-five so that E_nuclear equals E_coh times phi to the forty-five. The upstream lemma one_lt_phi supplies 1 < phi and E_coh_storage_pos supplies 0 < E_coh_storage, both required for the comparison.
proof idea
The term proof unfolds E_chemical, E_nuclear, phi_rung_energy, chemical_rung and nuclear_rung. Simplification with zpow_zero and mul_one reduces the goal to E_coh < E_coh · phi^{45}. It applies lt_mul_of_one_lt_right to the positivity theorem E_coh_storage_pos and finishes with one_lt_zpow₀ instantiated at one_lt_phi together with a norm_num computation confirming phi^{45} > 1.
why it matters
The theorem supplies the final inequality required by the downstream rs_energy_storage_hierarchy result, which asserts positivity, monotonicity on the ladder, and the chemical-nuclear comparison in one package. It realizes the EN-004.4 claim inside the module documentation that nuclear energy exceeds chemical energy by phi^{45} approximately 10^9. In the Recognition Science framework it instantiates the phi-ladder energy hierarchy derived from the self-similar fixed point phi and contributes to the EN-004 certificate confirming derived status for the storage density structure.
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