storage_density_ratio
Energy density ratios between rungs n and m on the phi-ladder are defined as phi raised to the integer difference n minus m. Physicists and engineers working on RS-derived storage hierarchies cite this when comparing chemical and nuclear energy densities. It is introduced as a direct one-line definition that encodes the exponential scaling without additional proof steps.
claimThe ratio of energy storage densities at integer rungs $n$ and $m$ on the phi-ladder is given by $phi^{n-m}$.
background
The module derives fundamental limits on energy storage per unit mass from the phi-ladder and J-cost structure. Energy takes the form J-cost times the coherence quantum, with E_coh equal to phi to the power of negative five electronvolts. The upstream density definition from NeutronStarCrustalRegimesFromRS sets density at rung k to phi^k, supplying the exponential base used here.
proof idea
This is a one-line definition that directly sets the ratio equal to the phi-exponentiation of the rung difference.
why it matters in Recognition Science
The definition supports the positive-ratio and higher-rung-denser theorems in the same module. It realizes the EN-004 claim that storage densities follow phi-ladder scaling, producing the predicted nuclear-to-chemical ratio of approximately phi^45. It connects to the Recognition Science framework in which phi is the self-similar fixed point and energy hierarchies are quantized on the ladder.
scope and limits
- Does not incorporate the J-cost function into the ratio itself.
- Does not compute absolute energy values or specify units.
- Does not extend to non-integer rungs or continuous parameter tuning.
- Does not address volume versus mass density distinctions.
Lean usage
theorem storage_density_ratio_pos (n m : ℤ) : 0 < storage_density_ratio n m := by unfold storage_density_ratio; exact zpow_pos phi_pos _formal statement (Lean)
162def storage_density_ratio (n m : ℤ) : ℝ := phi ^ (n - m)
proof body
Definition body.
163
164/-- **THEOREM EN-004.12**: Storage density ratios are positive. -/