stepRatio_invariant_under_r0
plain-language theorem explainer
The step ratio for a log-spiral path is unchanged when the base radius is scaled by any nonzero real factor. Propulsion modelers working with φ-scaled rotors cite the result to eliminate absolute size from pitch calculations. The proof is a one-line wrapper that applies the closed-form step-ratio expression to both radii and simplifies.
Claim. For nonzero reals $c$ and $r_0$, and any parameters $P$, the step ratio satisfies stepRatio$(c r_0, θ, Δθ, P) =$ stepRatio$(r_0, θ, Δθ, P)$.
background
The Flight.Geometry module supplies the purely geometric layer for spiral-field propulsion, built on the Recognition Science constant φ. It defines log-spiral rotor paths whose radial growth is governed by a pitch parameter kappa drawn from SpiralField.Params. The upstream lemma stepRatio_logSpiral_closed_form supplies the explicit formula stepRatio = φ^(kappa · Δθ / (2π)) once r0 ≠ 0, showing that the ratio depends only on angular step and pitch.
proof idea
The proof is a one-line wrapper that invokes stepRatio_logSpiral_closed_form once for the scaled radius c·r0 and once for the original r0, then applies simp to equate the two identical closed-form expressions.
why it matters
The lemma is cited inside the master theorem tesla_turbine_master to establish that step ratio depends only on pitch and angle, not base radius. It thereby completes one of the six geometric properties required for the φ-spiral engine model and aligns with the self-similar scaling fixed point in the T0–T8 forcing chain.
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