Chemistry Electrolysis3 From Jcost

Electrolysis overpotential

Electrolysis of water splits H₂O into hydrogen and oxygen using electricity. The theoretical minimum voltage is 1.23 volts, but real cells need 1.8 to 2.0 volts. The difference, 0.6 to 0.8 volts, is called the overpotential. It comes from kinetic barriers at the electrodes, not from thermodynamics.

In Recognition Science (RS), the framework models this overpotential through its cost function J(x) = (x + 1/x)/2 - 1. The framework's machine-checked library of formal theorems proves three general facts about J applied to a ratio m/e. First, J(m/e) equals zero when m equals e. Second, J(m/e) is nonnegative for positive inputs. Third, the constant phi - 3/2 is positive, where phi is the golden ratio. These are general properties of the cost function itself, not specific to electrolysis.

The module named Electrolysis3_FromJCost defines domainCost as Jcost(m/e) and bundles those three facts into a certificate structure. The research note attached to the module records a numerical match: J(phi)^(-1)/14 = 8.47/14 = 0.605 V, which sits inside the measured overpotential range of 0.6 to 0.8 V. But the Lean code proves nothing about water. The definition of domainCost contains no reference to volts, electrodes, or water chemistry. The note is a research intention, not a theorem.

What the module actually establishes is a template. It shows that the cost function has the right shape for a threshold phenomenon: zero at the ideal ratio, positive away from it. The connection to real electrolysis would require defining m and e in electrochemical terms, such as an activation energy ratio. That step is not in the code. The measured overpotential agreement is a numerical observation, not a derived result.

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]

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

What this page does not claim

The module does not prove that real water electrolysis requires 0.605 volts of overpotential. No electrochemical definition of m and e exists in the machine-checked code. The numerical agreement with measured overpotential is not a theorem.

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/Chemistry/Electrolysis3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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