Chemistry Equilibrium Constant From Jcost

Equilibrium and the cost of recognition

In chemistry, the equilibrium constant K tells you how far a reaction proceeds before it stops: a large K means products dominate, a small K means reactants do. The classical tool for how K changes with temperature is the van't Hoff equation, d(lnK)/d(1/T) = -ΔH/R, which links the shift in equilibrium to the enthalpy of reaction. That relation is a measured and tested workhorse of physical chemistry.

Recognition Science starts from a different question: what does it cost for a system to recognize a change? Its ledger, a discrete record of recognition events, assigns a cost to every ratio of two quantities. The framework's central theorem forces that cost to be J(x) = (x + 1/x)/2 - 1. For a chemical equilibrium, the module defines the cost of a ratio m/e, where m and e are two quantities of the system, as J(m/e). This is a definitional choice, a model, not a derived law.

What the machine-checked library of formal theorems actually proves about this definition is modest but clean. It proves that the cost is zero when m equals e, that the cost is never negative for positive inputs, and that a certain threshold, the golden ratio minus 1.5, is positive. These are general facts about the cost function, true for any positive m and e, and they hold because J itself has those properties. The module does not yet define what m and e mean in chemical terms, so it proves nothing specific to equilibrium constants.

The research note attached to the module sketches where the idea was meant to go: at a reference temperature T_RS, roughly 1.4 billion kelvin, the logarithm of K would change by J(φ) per rung of temperature. That is a hypothesis, not a result. The honest summary is that the framework has a tool, the cost function, and a clear picture of what a chemical application would require, but the bridge from the abstract cost to a specific chemical definition of m and e remains open.

What the module does establish is a template. It shows how a universal cost function can be pointed at a chemistry problem, and it certifies the basic sanity conditions any such application must satisfy. That is a small but real step: it gives the framework a place to stand when the chemical meaning of m and e is finally supplied.

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]

What this page does not claim

The module does not derive the van't Hoff equation or any specific chemical equilibrium constant. The module does not define what m and e mean in chemical terms. The research note about T_RS and J(φ) is a hypothesis, not a proved result.

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

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