Chemistry Reactive Oxygen From Jcost

Reactive oxygen and the cost function

Reactive oxygen species (ROS) are chemically reactive molecules containing oxygen, such as superoxide, that form as byproducts of cellular respiration. In mitochondria, a small percentage of the oxygen consumed is incompletely reduced and becomes superoxide. The empirical range commonly cited for this electron leak is about 1 to 5 percent of total oxygen consumption.

The Recognition Science framework models this leak using its cost function. The framework's cost, a measure of the forced price of recognition events, is defined as J(x) = (x + 1/x)/2 - 1. In the module for reactive oxygen, the domain cost is set to J(m/e), where m and e are real numbers. The research note in the module proposes that the ROS fraction equals J(phi) times an electron leak fraction, which computes to roughly 0.118 times 0.1, or about 1.18 percent. That figure lands inside the 1 to 5 percent empirical range.

What the module actually proves is narrower. The machine-checked library of formal theorems establishes three general facts about the cost function: it vanishes when the two inputs are equal, it is nonnegative for positive inputs, and the quantity phi minus 3/2 is positive. These are properties of the cost function itself, not of reactive oxygen. The definition of domainCost as J(m/e) contains no reference to mitochondria, superoxide, or any chemical quantity.

In plain language, the module proves that the cost function has certain basic mathematical properties, and it records an intended application to ROS production that is not yet a theorem. The connection would become rigorous only if m and e were defined in terms of actual chemical or biological quantities. Until then, the ROS fraction calculation remains a research note, shared verbatim with many other modules that use the same template.

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 1.18 percent is the actual ROS production rate. The module does not define m and e in chemical terms. The module does not establish a mechanism linking recognition cost to mitochondrial biology.

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

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