Chemistry Photosynthesis2 From Jcost

The photosynthesis module

Photosynthesis is the process by which plants, algae, and some bacteria convert light energy into chemical energy. The first step takes place in a protein complex called photosystem II, which captures photons and uses their energy to split water molecules. A standard measure of this step's efficiency is its maximum quantum yield: the fraction of absorbed photons that successfully drive the water-splitting reaction. In a dark-adapted state, the measured maximum quantum yield for photosystem II is about 0.88, meaning roughly 88 percent of absorbed photons do useful chemical work.

In Recognition Science, the framework models this efficiency through a cost function, a rule that assigns a penalty to any mismatch between two quantities. The framework's central result proves that any cost function satisfying five plain conditions must take the form J(x) = (x + 1/x)/2 - 1. The module named chemistry photosynthesis2 from jcost defines its domain cost as J applied to the ratio of two quantities, m over e. The intended physical picture is that m and e are two energy or electron counts whose balance determines the quantum yield.

The machine-checked library of formal theorems proves exactly three general facts about this cost function, and nothing specific to photosynthesis. First, the cost vanishes when m equals e, meaning a perfect balance carries no penalty. Second, the cost is never negative for positive inputs, so the penalty never rewards imbalance. Third, the golden ratio phi minus 3/2 is positive, a small algebraic fact about the number phi. The module packages these three facts into a certificate structure, but the certificate contains no definition of m or e in terms of photons, electrons, or any physical quantity.

The research note attached to the module records the intended connection: the photosystem II maximum quantum yield of 0.88 matches the expression 1 - J(phi), which evaluates to about 0.882. That numerical agreement is a research aspiration, not a proved result. The formal theorems in the module would become theorems about photosynthesis only if m and e were defined in the subject's own terms, for instance as a count of absorbed photons and a count of electrons transferred. Until such a definition exists, the module proves facts about the cost function J, not about plants.

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 does not prove that photosystem II has quantum yield 0.88. The module does not define m and e in terms of photons or electrons. The numerical match 1 - J(phi) ≈ 0.882 is a research note, 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/Photosynthesis2FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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