 140This captures the “discrete pitch families” idea: per-turn growth is quantized in multiplicative
 141steps of `φ^{d}` for `d : ℤ`.
 142-/
 143lemma kappa_discreteness (P : SpiralField.Params) (d : ℤ) :
 144    SpiralField.perTurnMultiplier { kappa := P.kappa + d }
 145      = SpiralField.perTurnMultiplier P * Real.rpow IndisputableMonolith.Constants.phi (d : ℝ) := by
 146  -- `φ > 0` so we can use `Real.rpow_add`.
 147  have hφpos : 0 < IndisputableMonolith.Constants.phi := IndisputableMonolith.Constants.phi_pos
 148  -- Expand both sides and use `rpow_add` on the exponent `(κ + d)`.
 149  simp [SpiralField.perTurnMultiplier, Real.rpow_add, hφpos, Int.cast_add]
 150
 151end SpiralLemmas
 152
 153end Geometry
 154end Flight
 155end IndisputableMonolith

papers checked against this theorem

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