Chemistry Osmosis3 From Jcost

Osmotic pressure from a cost function

Osmotic pressure is the pressure that must be applied to a solution to stop the flow of solvent across a semipermeable membrane. The classical law, due to Jacobus van 't Hoff in 1887, states that the osmotic pressure Π equals iMRT, where i is the number of particles per formula unit, M is the molar concentration, R is the gas constant, and T is the absolute temperature. A solution is iso-osmotic when it has the same osmotic pressure as another solution, meaning no net solvent flow occurs.

In Recognition Science, the framework models osmotic equilibrium through its cost function, a measure of the forced price of a recognition event. The module Osmosis3_FromJCost defines a cost for a ratio of concentrations, J(m/e), where m is a molarity and e is a reference molarity. The framework's proved theorem states that this cost vanishes when m equals e, and is nonnegative for all positive inputs. This matches the physical idea that an iso-osmotic pair has zero osmotic cost.

The module proves three general facts about this cost function: it is zero when the two concentrations are equal, it never goes negative for positive concentrations, and the golden-ratio-derived constant φ − 3/2 is positive. These are all true statements about the cost function, and they are checked by the machine-checked library of formal theorems. The module also bundles these into a certificate structure, a formal package that records the proofs together.

What the module does not do is connect these facts to any specific chemical system. The cost is defined as J(m/e) without any reference to a particular solute, membrane, or experimental setup. The docstring notes that the intended application to van 't Hoff's law, where the osmotic pressure would be RT · J(M/M_iso) · M_total, is a research note, not a theorem. To make this a theorem about osmosis, one would need a definition of m and e in the subject's own terms, which the module does not provide.

The practical upshot is clear: the framework has a general mathematical structure that could in principle describe osmotic equilibrium, but the bridge from that structure to actual chemistry remains open. The module is a template, one of many identical siblings, that shows what a cost-based formulation would look like. It establishes the formal properties of the cost function, not the physics of osmosis.

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 cert_inhabited : Nonempty Osmosis3Cert := ⟨cert⟩

What this page does not claim

The module proves van 't Hoff's law or any specific osmotic pressure formula. The cost function J(m/e) is derived from chemical principles rather than chosen as a model. The constant φ − 3/2 has a demonstrated physical meaning in osmosis.

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/Osmosis3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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