scale_ratio
plain-language theorem explainer
The result shows that the scale function increases by the factor phi at each successive natural number rung. Researchers certifying large-scale structure in Recognition Science cosmology cite this when assembling the five-regime certificate. The proof is a short algebraic reduction that unfolds the exponential definition and cancels powers using field properties.
Claim. For every natural number $k$, the ratio of the scale at rung $k+1$ to the scale at rung $k$ equals the golden ratio phi, where the scale at rung $k$ is defined as phi raised to the power $k$.
background
The scale function assigns to each natural number $k$ the value phi raised to the power $k$. This places successive comoving lengths on the phi-ladder. In the Large-Scale Structure from RS module, this supports five regimes: CMB acoustic scale, baryon acoustic oscillation, galaxy clusters, filamentary structure, and cosmic voids, each one rung up the ladder in comoving length. The upstream definition of scale supplies the explicit exponential form used for the ratio.
proof idea
Unfold the scale definition to obtain phi to the power of $k+1$ over phi to the power of $k$. Invoke positivity of phi to the $k$ to justify the division rewrite. Apply the successor power rule and simplify the resulting expression with ring arithmetic.
why it matters
This theorem supplies the phi_ratio field in the largeScaleStructureCert definition, which certifies the five LSS regimes. It implements the self-similar fixed point property of phi from the forcing chain. The result closes the algebraic step for the overall structure certificate with no remaining hypotheses.
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