no_compression_zero_cost
J-cost vanishes at unit compression ratio. Turbine modelers would cite this to set the zero baseline for an uncompressed flow stage in phi-spiral geometries. The proof is a direct one-line application of the unit-cost lemma for the J function.
claim$J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ is the cost function induced by the multiplicative recognizer.
background
In the Tesla turbine setting, fluid follows a logarithmic spiral between parallel discs with gap governed by phi scaling. The J-cost function, defined via the Cost module as the derived cost of a multiplicative recognizer comparator, quantifies recognition cost of a compression ratio x. Upstream lemma Jcost_unit0 states J(1) = 0 directly from the squared-ratio expression.
proof idea
One-line wrapper that applies the Jcost_unit0 lemma from the Cost module.
why it matters in Recognition Science
This anchors cost accounting for the turbine stack by confirming the trivial case has zero cost, feeding into phi_disc_spacing_optimal and spiral_pitch_one_is_phi results. It aligns with T5 J-uniqueness and non-negativity of cost in ObserverForcing. No open questions are touched.
scope and limits
- Does not address positive compression ratios or their costs.
- Does not derive the explicit form of J beyond the unit case.
- Does not connect to specific turbine parameters like disc count or spacing.
formal statement (Lean)
194theorem no_compression_zero_cost :
195 Jcost 1 = 0 := Cost.Jcost_unit0
proof body
Term-mode proof.
196
197/-- J(φ) > 0: any non-trivial compression incurs positive cost. -/