Chemistry Diels Alder3 From Jcost

What the module proves

The Diels-Alder reaction is a classic organic chemistry transformation: a diene and a dienophile join in a single step to form a six-membered ring. Chemists have long observed that one of the two possible products, the "endo" one, is usually favored, with typical endo selectivities around 80 to 99 percent. The module named Diels_Alder3_FromJCost was intended to connect this selectivity to the Recognition Science framework's cost function, but the file itself does not make that connection.

The module defines cost, a measure of the price of a recognition event, as a function of a ratio of two real numbers, written J(m/e). It then proves three general facts about this cost function: it is zero when the two numbers are equal, it is never negative when both numbers are positive, and a certain threshold involving the golden ratio is positive. These are properties of the cost function itself, not of any chemical system.

What the module does not do is define what m and e mean for a Diels-Alder reaction. The file's own documentation states that the Lean code proves nothing specific to this subject, because the cost is defined without reference to any chemistry. The research note in the file suggests that the endo fraction might equal 1 minus the cost at the golden ratio, giving 88.2 percent, which falls in the observed 80 to 95 percent range, but this is an idea for future work, not a proved result.

In Recognition Science, the framework models recognition events with a forced cost function, and its library contains many modules that apply this cost to different subjects. This particular module, however, is a template: the same three theorems appear verbatim in 2383 sibling modules, because they all use the same definition of cost without subject-specific meaning. The honest summary is that this file establishes three general properties of the cost function and nothing about Diels-Alder chemistry.

What a reader can take away is a clear distinction between a mathematical framework and its application. The cost function has well-defined formal properties, but applying it to a real chemical reaction requires a separate step: defining the quantities m and e in terms of the reaction's own features. That step is missing here, so the module is a placeholder, not a result about selectivity.

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

The module does not prove any Diels-Alder selectivity result. The 88.2 percent figure is a research note, not a theorem. The module does not define m and e in chemical terms.

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/Diels_Alder3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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