Chemistry Reaction Coordinate From Jcost

Reaction coordinate and its cost

A reaction coordinate is a single number that describes progress along a chemical reaction, from starting materials to products. It is not a physical distance but a bookkeeping axis: zero often marks the reactants, one the products, and the transition state sits somewhere between. Chemists use it to discuss how much the molecular structure has changed and where the energy barrier lies. The Hammond postulate, a classical rule from 1935, says that for an exothermic reaction, the transition state resembles the reactants more than the products, so its position on the coordinate sits closer to zero.

In Recognition Science, the framework defines a cost for any mismatch between a measured value and an expected one. The cost function is J(x) = (x + 1/x)/2 - 1, which is zero when x equals one and positive otherwise. The framework applies this to a reaction coordinate by setting the cost as J(measured / expected). Here measured and expected are real numbers, and the ratio measures how far the actual progress deviates from the ideal. The framework then proves three general facts: the cost is zero when measured equals expected, the cost is never negative for positive inputs, and a threshold constant phi - 3/2 is positive. These are general properties of the cost function, not specific chemical results.

In Recognition Science, the reaction coordinate concept gets a formal home. The framework defines a certificate structure that packages the three proved facts, and it shows such a certificate exists. The framework models the Hammond postulate by suggesting that the transition state position equals 1 - J(phi), about 0.882 for an exothermic reaction. That number is a research note, not a theorem: the framework proves nothing specific to chemistry, because it defines measured and expected without tying them to any particular reaction.

What the framework establishes in plain language is a template. It shows that the same cost function that governs recognition events elsewhere also applies to a reaction coordinate, as long as one can define measured and expected in chemical terms. The three proved facts are the foundation: zero cost at perfect match, nonnegative cost otherwise, and a positive threshold. The chemical content, such as the 0.882 position, remains a target for future work that would define measured and expected from molecular structure.

theorem domainCost_at_equilibrium (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 (measured expected : ℝ) : ℝ := Jcost (measured / expected)

What this page does not claim

The framework does not prove that the transition state position is 0.882 for any real reaction. The framework does not derive the Hammond postulate from first principles. The framework does not define what measured and expected mean 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/ReactionCoordinateFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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