theorem
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phi_ladder_energy_strictly_increasing
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IndisputableMonolith.Engineering.EnergyStorageDensityStructure on GitHub at line 191.
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188/-! ## §V. Maximum Theoretical Density -/
189
190/-- The φ-ladder energy is strictly increasing: higher rungs always give more energy. -/
191theorem phi_ladder_energy_strictly_increasing (n m : ℤ) (h : n < m) :
192 phi_rung_energy n < phi_rung_energy m := by
193 unfold phi_rung_energy
194 apply mul_lt_mul_of_pos_left _ E_coh_storage_pos
195 exact zpow_lt_zpow_right₀ one_lt_phi h
196
197/-! ## §VI. Fundamental Bound Summary -/
198
199/-- The RS energy storage hierarchy theorem.
200 Derives three distinct energy scales from the φ-ladder structure. -/
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
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 -/
210def en004_certificate : String :=
211 "═══════════════════════════════════════════════════════════\n" ++
212 " EN-004: ENERGY STORAGE DENSITY — STATUS: DERIVED\n" ++
213 "═══════════════════════════════════════════════════════════\n" ++
214 "✓ E_coh_storage_pos: E_coh = φ^(-5) > 0\n" ++
215 "✓ phi_rung_energy_ratio: E(n+1)/E(n) = φ\n" ++
216 "✓ nuclear_exceeds_chemical: E_nuclear > E_chemical\n" ++
217 "✓ nuclear_chemical_ratio_gt_one: 1 < φ^45 (nuclear/chemical ratio > 1)\n" ++
218 "✓ jcost_energy_nonneg: J-cost energy ≥ 0\n" ++
219 "✓ jcost_energy_zero_iff_ground: J(x) = 0 ↔ x = 1\n" ++
220 "✓ jcost_energy_min_at_ground: x=1 is energy minimum\n" ++
221 "✓ energy_storage_density_hierarchy: mechanical < chemical < nuclear\n" ++