Chemistry Photocatalysis Efficiency2 From Jcost

A template, not a result

Photocatalysis uses light to drive chemical reactions, and solar fuel production aims to split water into hydrogen using sunlight. Practical solar-to-hydrogen efficiencies sit in the single digits, typically 1 to 10 percent. The module named PhotocatalysisEfficiency2FromJCost appears to address this topic, but its formal content stays at the level of a template.

The file defines a cost function domainCost m e = Jcost (m / e), where Jcost is a specific mathematical function from the Recognition Science framework. The Lean code proves three general facts: the cost is zero when the two inputs are equal, it is never negative for positive inputs, and a constant called canonicalThreshold, equal to phi minus 1.5, is positive. These statements hold for any positive real numbers m and e; they say nothing about photons, electrons, or chemical bonds.

In Recognition Science, the framework models recognition events through a ledger, a discrete record of such events, and assigns a forced cost to each recognition. The cost function J(x) = (x + 1/x)/2 - 1 is proved unique under five conditions. The photocatalysis module reuses this cost as a template: it sets m and e as inputs but never defines what they mean for a chemical system. The docstring notes the intended direction: efficiency as J(phi)^(1/2) * phi, giving about 0.557, or J(phi) roughly 11.8 percent, consistent with best reported solar-to-hydrogen values. That paragraph is a research note, not a theorem.

The machine-checked library of formal theorems, which verifies each proof, confirms only the template facts. To turn this into a statement about photocatalysis, one would need a definition of m and e in chemical terms, such as photon flux and electron transfer rate. Without that, the module remains a scaffold: the mathematics is sound, but the subject matter is absent.

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

This module does not prove any efficiency value for a real photocatalyst. The framework does not derive the 1-10 percent solar-to-hydrogen range from first principles.

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/PhotocatalysisEfficiency2FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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