phi_spacing_Jcost
plain-language theorem explainer
At the φ-optimal disc spacing the J-cost of the velocity ratio across the gap equals Jcost(φ). Fluid dynamicists and Recognition Science modelers cite it to confirm minimum momentum-transfer strain in Tesla-turbine gaps. The proof reduces directly to the phi_disc_spacing_optimal lemma by rewriting the gap parameter.
Claim. Let $g = 2δ√φ$ with $δ > 0$. Then the J-cost of the velocity ratio across the gap equals the J-cost of $φ$.
background
In the Tesla turbine model the velocity ratio is defined as $(g/(2δ))^2$ for parabolic flow between plates separated by gap $g$ with boundary-layer thickness $δ$. The J-cost function, taken from the PhiForcingDerived structure, measures interaction strain on the φ-ladder. The upstream phi_disc_spacing_optimal theorem states that the ratio $g/(2δ)$ equals $2√φ$ precisely when the velocity ratio equals φ, which minimizes the cost.
proof idea
The proof is a one-line wrapper that rewrites the velocity ratio using the phi_disc_spacing_optimal lemma, after which equality to Jcost(φ) holds by the definition of optimal spacing.
why it matters
This result places the minimum non-trivial J-cost at the φ-optimal spacing inside the Flight module, supporting the master certificate for the Tesla turbine. It aligns with the Recognition Science φ-ladder where J(φ) ≈ 0.118 is the baseline interaction cost and with the RCL composition law. The declaration closes the single-gap case in the φ-forcing chain for fluid devices.
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