phi_rung_jcost_energy
Energy stored at a phi-rung ratio phi^n equals the coherence quantum E_coh times J(phi^n) by direct definition. Researchers deriving storage densities on the phi-ladder cite this when establishing the hierarchy from chemical to nuclear scales. The proof is a one-line reflexivity that follows immediately from unfolding the definition of jcost_energy.
claimFor any integer $n$, the stored energy at recognition ratio $x = phi^n$ satisfies $E = E_{coh} · J(phi^n)$, where $E_{coh} = phi^{-5}$.
background
Recognition Science models energy storage as J-cost times the coherence quantum: E = J(x) · E_coh with E_coh = phi^{-5}. The J-cost function is J(x) = ½(x + x^{-1}) - 1, which vanishes at the ground state x = 1 and diverges at the extremes. The auxiliary definition jcost_energy(x, hx) := E_coh_storage * Jcost x packages this product for any positive x.
proof idea
The proof is a one-line wrapper that applies the definition of jcost_energy. Reflexivity matches the left-hand side directly to E_coh_storage * Jcost (phi ^ n) once the positivity hypothesis zpow_pos phi_pos n is supplied.
why it matters in Recognition Science
This is EN-004.10 inside the energy-storage module, confirming that phi-rung ratios produce the expected scaling E = E_coh · J(phi^n). It supports the larger hierarchy claims (chemical < nuclear < mass-energy) that rely on phi-ladder quantization and the eight-tick octave structure. No open scaffolding remains; the result is fully discharged by the upstream definition.
scope and limits
- Does not prove non-negativity or the ground-state minimum.
- Does not extend the equality to non-integer rungs.
- Does not derive numerical values for E_coh beyond its phi^{-5} definition.
formal statement (Lean)
147theorem phi_rung_jcost_energy (n : ℤ) :
148 jcost_energy (phi ^ n) (zpow_pos phi_pos n) =
149 E_coh_storage * Jcost (phi ^ n) := by
proof body
Decided by rfl or decide.
150 rfl
151
152/-- **THEOREM EN-004.11**: Ground state minimizes energy for any positive x. -/