single_turn_Jcost
plain-language theorem explainer
The J-cost for a single-turn Tesla turbine stage at compression ratio φ equals (φ + φ^{-1})/2 - 1. Engineers optimizing φ-scaled disc turbines cite this to fix the baseline cost for κ=1 steps on the ladder. The proof is a one-line term simplification that unfolds the Jcost abbreviation directly to its Cost definition.
Claim. $J(φ) = (φ + φ^{-1})/2 - 1$, where $J$ is the J-cost function on positive reals.
background
Jcost is an abbreviation for the J-cost function imported from the Cost module, where the cost of any recognition event equals J applied to its state. The module models a Tesla turbine as a φ-spiral engine whose disc spacing, spiral pitch, and flow follow φ-scaling to minimize J-cost; for κ=1 the per-turn compression ratio is exactly φ. Upstream, the cost of a multiplicative recognizer is the derived cost of its comparator, and the cost of an observer forcing event is likewise its J-cost.
proof idea
One-line term proof that applies simp only to unfold Jcost and Cost.Jcost to the explicit formula (φ + φ^{-1})/2 - 1.
why it matters
This supplies the explicit value for the minimum non-trivial cost on the φ-ladder at κ=1, anchoring the single-turn case in the Tesla turbine model. It supports sibling results on optimal disc spacing and spiral pitch that rely on the same J-cost minimization. The result aligns with T5 J-uniqueness and the forcing of φ as the self-similar fixed point.
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