Chemistry Radioactive Precursor From Jcost

The labeling yield claim

Radiopharmaceutical chemistry has a practical threshold: a synthesis that labels less than about 90% of its precursor is usually not worth purifying for clinical use. Good syntheses clear 95%. The question this page examines is whether a mathematical cost function from Recognition Science can say anything about that yield. The framework defines a cost, a forced penalty for any mismatch between two quantities, and the construction under discussion applies it to the ratio of two masses, m over e.

The classical facts come first. Labeling yield is the fraction of precursor that ends up attached to the radioactive isotope. A yield of 98.6% means that out of every thousand precursor molecules, about fourteen fail to label. That failure rate matters because unlabeled precursor can compete with the labeled product in the body, diluting the image or the therapy. The threshold of 90% is a rule of thumb from radiopharmacy practice, not a law of nature; it exists because purification becomes harder and more expensive below it.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 measures the penalty for a ratio x differing from 1. At x = 1, the cost is zero; away from 1, it grows. The construction defines its domain cost as J(m/e), the cost of the mass ratio. The golden ratio phi appears because J(phi) is small: phi is the self-similar scaling that makes the cost minimal among nontrivial ratios. Squaring J(phi) gives about 0.014, or 1.4%, which the research note interprets as a labeling failure floor. Subtracting from 1 gives 98.6%, above the 95% good-synthesis mark.

The machine-checked library of formal theorems proves three general facts about this cost function, and nothing specific to radiochemistry. It proves that the cost vanishes when m equals e, that the cost is nonnegative for positive inputs, and that phi minus 3/2 is positive. The structure RadioYieldCert packages these three facts, and the library proves such a certificate exists. What the library does not prove is that m and e refer to precursor and isotope masses in a synthesis. The definition of domainCost uses the ratio m/e without connecting either symbol to a chemical quantity.

The honest verdict: the 98.6% figure is a research note, a suggestion about where the idea was meant to go, not a derived result. The formal theorems hold for any positive real numbers m and e. To turn this into a chemistry theorem, someone would need to define m as the mass of the labeled product and e as the mass of the precursor, and prove that the recognition cost between them behaves as the framework claims. Until then, the construction establishes a general property of a cost function, and the radiopharmaceutical application remains a hypothesis with a named falsifier: a synthesis that labels below 98.6% without violating the cost axioms would refute it.

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]

theorem cert_inhabited : Nonempty RadioYieldCert := ⟨cert⟩

What this page does not claim

The 98.6% labeling yield is not a proved theorem; it is a research note. The construction does not define m and e as chemical masses. The framework does not derive the 90% clinical threshold; that is external practice.

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/RadioactivePrecursorFromJCost.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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