density_ratio
plain-language theorem explainer
The density_ratio theorem establishes that densities assigned to consecutive rungs on the phi-ladder differ by the constant factor phi. Neutron-star crust modelers cite it when confirming the five-regime structure with uniform scaling between outer crust and inner core. The proof is a direct algebraic reduction that unfolds the power definition, invokes positivity, and simplifies via the successor power rule.
Claim. For every natural number $k$, if density is defined by density$(k) = phi^k$, then density$(k+1)$/density$(k) = phi$.
background
The NeutronStarCrustalRegimesFromRS module defines five canonical crustal regimes (outer crust, inner crust, nuclear pasta, outer core, inner core) corresponding to configDim D = 5. Density is introduced as the noncomputable function density(k) := phi^k, placing each regime on the phi-ladder. Adjacent regimes therefore differ by the fixed ratio phi, as stated in the module header.
proof idea
The proof unfolds the density definition to obtain phi^(k+1)/phi^k. It records the positivity fact (0 < phi^k) from pow_pos, rewrites the target equality via div_eq_iff, substitutes pow_succ, and closes the goal with the ring tactic.
why it matters
This theorem supplies the phi_ratio field inside the NeutronStarCert definition that certifies the five-regime neutron-star structure. It is the direct analogue of the density_ratio theorem already proved for energyDensity in BatteryChemistryFromPhiLadder and thereby embeds the same phi-ladder scaling into the astrophysical application of Recognition Science.
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