spiral_pitch_one_is_phi
plain-language theorem explainer
The per-turn compression ratio of a logarithmic spiral with pitch parameter 1 equals the golden ratio phi. Turbine designers seeking the minimum non-trivial stable compression step per revolution would cite this result. The proof is a one-line simplification that substitutes the pitch value into the multiplier definition.
Claim. For pitch parameter $k=1$, the per-turn compression ratio of the logarithmic spiral equals $phi^1$.
background
In the Recognition Science treatment of the Tesla turbine, fluid paths are logarithmic spirals whose pitch parameter k sets the compression per revolution. The per-turn multiplier encodes the scaling $phi^k$, where phi is the self-similar fixed point forced in the T0-T8 chain. This theorem specializes the general multiplier to the integer case k=1, which the module identifies as the minimum non-trivial stable step. The local setting follows the RS decipherment of Tesla's 1913 patent, where disc spacing, spiral pitch, and flow rate are optimized under phi-scaling to minimize J-cost of the velocity profile. Upstream results supply the multiplier definition from SpiralField and the phi-power conventions from PhiLadderLattice and AnnularCost, where the stiffness constant is defined as $(log phi)^2$.
proof idea
The proof is a one-line wrapper that applies the definition of the per-turn multiplier. Simplification substitutes the record {kappa := 1} directly into the exponential scaling formula, yielding phi to the first power.
why it matters
This declaration supplies the second enumerated key result in the Tesla turbine decipherment, confirming that integer pitch 1 produces the minimal stable compression phi. It feeds the master certificate for the phi-spiral engine and connects to the phi-ladder constructions in NumberTheory. The result aligns with T6, where phi is forced as the self-similar fixed point, and with the eight-tick octave periodicity that governs stable compression steps. No open scaffolding remains for this specific claim.
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