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def definition def or abbrev high

scale

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scale(k) returns phi^k for each natural number k, supplying the explicit length scales for the five canonical LSS regimes on the phi-ladder. Cosmologists and researchers deriving resonant lengths in coherence devices or musical intervals cite this definition. It is introduced as a direct one-line assignment with no lemmas or computation.

claimThe scale at rung $k$ is defined by $s(k) = phi^k$ for each natural number $k$, where $phi$ is the golden ratio fixed point.

background

The LargeScaleStructureFromRS module defines five canonical LSS regimes (CMB acoustic scale, baryon acoustic oscillation, galaxy clusters, filamentary structure, cosmic voids) with each regime occupying one rung on the phi-ladder in comoving length. The phi-ladder is generated by successive powers of the self-similar fixed point phi forced in the unified forcing chain. This definition supplies the explicit real-valued length for rung k.

proof idea

The declaration is a direct definition that sets scale(k) equal to phi raised to the power k, with no lemmas applied and no tactics used.

why it matters in Recognition Science

This definition supplies the phi-ladder scales that underpin resonant scale definitions in applied coherence technology and the Fibonacci relations in musical scales. It realizes the D=5 LSS regimes in the module and connects to the phi fixed point from the forcing chain. Downstream results such as ResonantScale and scale_fibonacci depend on it to link cosmology to aesthetics and device design.

scope and limits

formal statement (Lean)

  29noncomputable def scale (k : ℕ) : ℝ := phi ^ k

proof body

Definition body.

  30

used by (40)

From the project-wide theorem graph. These declarations reference this one in their body.

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