Chemistry Nucleophilic Subs3 From Jcost

The SN1-SN2 threshold

In organic chemistry, a nucleophilic substitution reaction at a saturated carbon atom follows one of two mechanisms. The single-step SN2 pathway has the nucleophile attack while the leaving group departs, all in one concerted motion. The two-step SN1 pathway first forms a carbocation intermediate, then the nucleophile attacks that cation. The choice between them depends on several factors, including the steric bulk around the reaction center: bulky substituents hinder the backside attack that SN2 requires, favoring SN1 instead.

Recognition Science (RS) offers a framework for thinking about such thresholds. Its central object is a cost, a measure of the price of a recognition event, defined by the function J(x) = (x + 1/x)/2 - 1. The framework models a reaction mechanism as a recognition event whose cost depends on a ratio, here the ratio of steric bulk to a threshold value. The proposed rule is simple: SN2 dominates when the cost of that ratio is below a canonical threshold, and SN1 dominates when it is above.

The module in question, chemistry-nucleophilic-subs3-from-jcost, proves three facts about the cost function. It shows that the cost vanishes when the two quantities are equal, that the cost is never negative for positive inputs, and that the proposed threshold value phi - 3/2 is positive. These are general properties of the cost function itself, not of any particular chemical system. The module defines a certificate structure that packages these three facts, and proves that such a certificate exists.

The critical limitation is that the module never defines what "steric bulk" or "threshold" means in chemical terms. The cost function is applied to the ratio m/e, but m and e are left as arbitrary real numbers. The docstring records the intended chemical interpretation as a research note, but the formal proof does not connect the cost to any measurable chemical quantity. The module therefore establishes the mathematical scaffolding for a threshold, but not the threshold itself.

What the module does establish is that the cost function has the right basic shape for a threshold: it is zero at the balance point, never negative, and has a positive reference value to compare against. This is a necessary condition for the SN1-SN2 rule to be meaningful, but it is far from sufficient. The chemical content, the actual mapping from molecular structure to the ratio m/e, remains entirely open.

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 SN2 occurs when J(steric_bulk/threshold) < J(phi). The module does not define steric bulk or threshold in chemical terms. The module does not connect the cost function to any measured reaction rate or selectivity.

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/NucleophilicSubs3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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