Materials Electroplating3 From Jcost

A cost function for plating

Electroplating is the process of coating an object with metal by passing an electric current through a solution containing that metal's ions. The efficiency of the process is the fraction of the electrical charge that actually deposits metal, rather than being wasted on side reactions like producing hydrogen gas. For most industrial plating baths, that fraction lands between 80 and 99 percent, depending on the metal, the bath chemistry, and the current density.

In the Recognition Science framework, the module materials electroplating3 from jcost defines a cost function for this process. The framework's central object is a cost, a measure of how far a ratio of two quantities is from being equal, with the property that the cost is zero when the two quantities match. Here, the module defines the cost of a plating run as J(m / e), where m is the mass of metal actually deposited and e is the mass that the charge would deposit at 100 percent efficiency. The function J is the framework's forced cost function, proved unique by the recognition chain.

The machine-checked library of formal theorems proves three general facts about this definition. First, when the actual deposited mass equals the ideal mass, the cost is exactly zero. Second, for positive inputs, the cost is never negative. Third, the quantity phi minus 3/2 is positive, where phi is the golden ratio. These are the only theorems in the module, and they are stated for any real numbers m and e; the module does not prove anything specific to electroplating, because it never defines m and e in terms of the physics of a plating bath.

A research note attached to the module goes further. It claims that the framework's canonical optimal plating efficiency is 88.2 percent, computed as 1 minus J(phi), and that this sits comfortably within the 80 to 95 percent range typical of real plating. That note is a research aspiration, not a theorem. The note itself says so: the paragraph records where the idea was meant to go, and what would make the module a genuine theorem about electroplating is a definition of m and e in the subject's own terms, which the module does not provide.

What the module actually establishes is a template. It shows that the framework's cost function, applied to a ratio of two positive quantities, always behaves sensibly: zero at equality, nonnegative elsewhere. That is a small but solid foundation. The step from that template to a real statement about plating efficiency remains open, awaiting a physical definition of the two masses.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module proves no fact specific to electroplating; its theorems are about the cost function applied to any positive ratio. The 88.2 percent efficiency figure is a research note, not a proved result. The module does not define m and e in terms of plating physics, so it does not connect to measured efficiencies.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/Electroplating3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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