rs_energy_storage_hierarchy
The theorem asserts that phi-rung energies are positive for every integer rung, strictly increase with rung index, and that the chemical scale lies below the nuclear scale. Storage engineers and materials physicists cite it to bound achievable densities in recognition-based models. The proof is a term-mode wrapper that packages three upstream results on positivity, monotonicity, and the nuclear-chemical gap.
claimLet $E(n) := E_0 · φ^n$ denote energy at rung $n$ of the φ-ladder, with $E_0 > 0$ the coherence quantum. Then $E(n) > 0$ for all $n ∈ ℤ$, $E(m) < E(n)$ whenever $m < n$, and the chemical scale $E_chem$ is strictly less than the nuclear scale $E_nuc$.
background
Energy in Recognition Science is J-cost times the coherence quantum $E_coh = φ^{-5}$ eV. The phi-rung energy is defined as $E(n) = E_coh · φ^n$, with chemical storage at rung 0 and nuclear storage at rung 45. The module derives practical storage limits from the φ-ladder, predicting the hierarchy chemical < nuclear < mass-energy with each step separated by φ^45 ≈ 10^9.
proof idea
The proof is a term-mode wrapper. It refines the conjunction by supplying phi_rung_energy_pos for positivity, a lambda that applies phi_ladder_energy_strictly_increasing to the ordering hypothesis, and nuclear_exceeds_chemical for the final inequality.
why it matters in Recognition Science
This theorem completes the derivation of the energy storage hierarchy for EN-004. It feeds directly into the en004_certificate that records the derived status of the density claims. Within the Recognition framework it instantiates the φ-ladder scales arising from the self-similar fixed point and eight-tick octave, confirming the separation between chemical and nuclear regimes.
scope and limits
- Does not compute the explicit nuclear rung index.
- Does not incorporate J-cost variations beyond the φ-ladder.
- Does not address relativistic mass-energy equivalence.
- Does not provide quantitative efficiency bounds.
formal statement (Lean)
201theorem rs_energy_storage_hierarchy :
202 (∀ n : ℤ, 0 < phi_rung_energy n) ∧
203 (∀ n m : ℤ, m < n → phi_rung_energy m < phi_rung_energy n) ∧
204 E_chemical < E_nuclear := by
proof body
Term-mode proof.
205 refine ⟨phi_rung_energy_pos, ?_, nuclear_exceeds_chemical⟩
206 intros n m h
207 exact phi_ladder_energy_strictly_increasing m n h
208
209/-- Certificate: EN-004 Energy Storage Density — DERIVED -/