energy_storage_density_hierarchy
plain-language theorem explainer
Energies at rungs -10, 0 and 45 on the phi-ladder satisfy the strict ordering E(-10) < E(0) < E(45), corresponding to mechanical below chemical below nuclear storage regimes. Engineers modeling recognition-derived energy systems cite the result when bounding density hierarchies. The term-mode proof unfolds the rung definition, factors out the positive coherence energy, and reduces both inequalities to the monotonicity of phi-powers via one_lt_phi.
Claim. Let $E(n) := E_0 phi^n$ denote the stored energy per recognition event at rung $n$ of the phi-ladder, where $E_0 > 0$ is the coherence quantum. Then $E(-10) < E(0) land E(0) < E(45)$.
background
In the Recognition Science engineering module, energy storage is quantized on the phi-ladder. The definition phi_rung_energy(n) sets the energy at integer rung n to E_coh_storage multiplied by phi^n, with E_coh_storage positive by the upstream theorem E_coh_storage_pos. The underlying J-cost function J(x) = (x + x^{-1})/2 - 1 supplies the cost structure, minimum at the ground state and diverging at extremes, while the module document states practical limits at chemical (one coherence quantum) and nuclear (phi^45 scaling) regimes.
proof idea
The term proof opens with constructor to split the conjunction. Each inequality unfolds phi_rung_energy to expose the common positive factor E_coh_storage, applies mul_lt_mul_of_pos_left using E_coh_storage_pos, and reduces to zpow_lt_zpow_right₀ one_lt_phi together with a norm_num witness on the exponent difference.
why it matters
The declaration supplies EN-004.14, the three-level energy hierarchy mechanical < chemical < nuclear on the phi-ladder. It is referenced by the en004_certificate that certifies the full EN-004 derivation. The result supports the module claim that energy ratios are governed by integer powers of phi and closes the rung-ordering step required for the maximum theoretical density section.
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