Chemistry Bioinorganic From Jcost

Metal recognition costs

Bioinorganic chemistry studies how metal ions such as iron, zinc, and copper function inside living organisms, often at the active sites of enzymes. A metalloenzyme active site is the region where a metal ion helps catalyze a chemical reaction. In Recognition Science, the framework models such a site by applying its cost function cost, a measure of recognition effort, to the ratio of two quantities, m and e, which stand for the mass of the metal and the mass of the enzyme or its environment.

The definition domainCost sets this cost as J(m/e), where J is the framework's forced cost function. The machine-checked library proves three general facts. First, when m equals e, the cost is zero, meaning no recognition effort is needed when the two quantities match. Second, for positive inputs, the cost is never negative, so recognition effort cannot be negative. Third, the threshold phi minus 3/2 is positive, where phi is the golden ratio, a number that appears throughout the framework's structure.

These three facts are true for any positive real numbers m and e. The definitions do not fix m and e in terms of any specific metal or enzyme, so nothing is proved about iron, zinc, or any particular active site. The docstring notes a research intention: five canonical redox-active metals (Fe, Mn, Cu, Zn, Mo) correspond to five oxidation state recognition axes, but this is a note about where the idea was meant to go, not a proved result.

In Recognition Science, the framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This library applies that general theorem to a ratio, establishing boundary and positivity properties that hold for any domain. What changes is the vocabulary: it shows how a universal cost function can be instantiated for a chemical ratio, but the chemical content remains open until m and e are defined in bioinorganic terms.

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

No specific bioinorganic result is proved; the library proves only general facts about the cost function on any positive ratio. The five-metal correspondence is a research note, not a theorem in the library. The framework does not derive the identity of any particular metal from the cost function alone.

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

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