Chemistry Photocatalysis From Jcost

Photocatalysis and the cost function

Photocatalysis is the acceleration of a chemical reaction by light. A photocatalyst, often a semiconductor like titanium dioxide (TiO2), absorbs photons and uses that energy to create charge carriers that drive reactions on its surface. The key practical measure is the quantum yield: the fraction of absorbed photons that actually produce a desired chemical change. For TiO2 under standard UV-A illumination, measured quantum yields are often around 10 to 20 percent, a range that depends heavily on the specific reaction, particle morphology, and surface chemistry.

The Recognition Science framework offers a different lens. It starts from a single proved cost function, J(x) = (x + 1/x)/2 - 1, which measures the forced expense of recognition when reality keeps a discrete ledger of events. The framework's library of machine-checked theorems shows that J vanishes when its input equals 1, and that J is nonnegative for all positive inputs. The golden ratio phi, about 1.618, emerges as a special scaling constant throughout the framework.

In Recognition Science, the module for photocatalysis defines a domain cost, a measure of mismatch, as J applied to the ratio of two quantities, m over e. The module proves three general facts: the cost is zero when m equals e, the cost is never negative for positive inputs, and the threshold phi minus 1.5 is positive. These are true for any positive real numbers m and e, because the module defines the cost without specifying what m and e mean in chemistry.

The research note attached to the module suggests a target: TiO2 quantum yield approximately equals J(phi), which evaluates to about 0.118, or 11.8 percent. That number sits inside the empirical range for TiO2, but the module itself proves nothing about photocatalysis. The proof establishes only the general properties of J, not that the ratio m over e equals phi for any real catalyst. The connection between the abstract cost and a physical quantum yield remains a hypothesis, not a theorem.

What the module does establish is a template. It shows how any subject could be linked to the cost function: define the two quantities whose ratio matters, then apply J. The three proved facts, zero at equality, nonnegativity, and a positive threshold, hold for any such definition. That is the real content of the module, a reusable scaffold that becomes a statement about chemistry only when someone defines m and e in terms of actual photon counts and reaction events.

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 any real photocatalyst has quantum yield equal to J(phi). The 11.8 percent figure is a research note, not a proved result about TiO2. No claim is made that the framework's cost function is the only way to model quantum yield.

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

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