E_coh_storage_pos
plain-language theorem explainer
The theorem establishes that the RS coherence energy quantum defined as phi to the power of negative five is strictly positive. Engineers and physicists deriving storage density hierarchies from the phi-ladder cite it as the base positivity fact that propagates to all J-cost energy comparisons. The proof is a one-line wrapper that unfolds the definition and applies the positivity of integer powers of a positive base.
Claim. The coherence energy quantum satisfies $0 < phi^{-5}$.
background
In the EN-004 module on Optimal Energy Storage Density, energy takes the form E = J(x) · E_coh with E_coh = phi^{-5} eV and J(x) = ½(x + x^{-1}) - 1 the J-cost function whose minimum is zero at the ground state x = 1. The sibling definition supplies E_coh_storage := phi ^ (-5 : ℤ) as the base quantum for the phi-ladder energy structure. Upstream results from Constants.RSNativeUnits establish the native energy type while PhiForcingDerived and SpectralEmergence fix phi as the self-similar fixed point.
proof idea
The proof is a one-line wrapper. It unfolds E_coh_storage to phi ^ (-5 : ℤ) and invokes zpow_pos using the positivity of phi.
why it matters
This anchors the EN-004 hierarchy theorems including energy_storage_density_hierarchy (which compares phi-rung energies across mechanical, chemical and nuclear scales) and the family of jcost_energy_* results that propagate non-negativity and ground-state minimality. It fills the EN-004.1 slot in the Recognition Science derivation of storage limits from the phi-ladder and J-cost structure. In the broader framework it supplies the positivity of the coherence quantum tied to hbar = phi^{-5} in RS units, enabling downstream comparisons such as nuclear_exceeds_chemical and the eight-tick octave structure.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.