Chemistry Henry Law3 From Jcost

Henry's law and the cost ledger

Henry's law is a classical chemistry rule: at constant temperature, the amount of gas dissolved in a liquid is proportional to the partial pressure of that gas above the liquid. The proportionality constant, the Henry's law constant, is a practical number. It tells a brewer how much carbon dioxide stays in beer, a diver how nitrogen moves into blood, and an oceanographer how oxygen enters seawater. Measured values span a wide range, from about 10^-5 to 10^5 atmospheres, depending on the gas and the solvent.

The framework Recognition Science starts from a different object: a ledger, a discrete record of recognition events. It proves that any cost function for recognition, obeying five plain conditions such as symmetry and a forced composition law, must equal J(x) = (x + 1/x)/2 - 1. This J is the framework's universal cost. The module named Henry_Law3_FromJCost applies J to a ratio of two quantities, m over e, where m and e are meant to stand for a molecular property and an environmental property in the chemistry setting.

The Lean module proves three general facts about this application. First, when m equals e, the cost is zero: J(1) = 0. Second, for positive inputs, the cost is never negative. Third, the golden ratio phi is greater than 3/2, so the quantity phi - 3/2 is positive. The module packages these three facts into a structure called HenryLaw3Cert and proves that structure is inhabited, meaning the facts are consistent.

The honest summary is that the module proves nothing specific to Henry's law. The definition of domainCost as J(m/e) makes no reference to chemistry; the same body of text is shared verbatim with 2383 sibling modules. What would make this a theorem about Henry's law is a definition of m and e in chemistry's own terms, such as a molecular volume and an environmental pressure. The research note records where the idea was meant to go, not a result that arrived.

What the module does establish is a template: if a chemistry model defines m and e so that their ratio is positive, then the universal cost J applies cleanly, giving zero at equality and nonnegative cost elsewhere. The numerical note in the module observes that the measured range of Henry's law constants, 10^-5 to 10^5 atm, spans roughly phi^30, and that oxygen in water at about 7.6e4 atm sits near phi^24. That is an empirical observation, not a proved consequence of the framework.

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

This module does not derive Henry's law constants from first principles. The phi-power numerical agreement is not a proved consequence of the framework. The module does not establish any specific chemistry result beyond the general properties of J.

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/Henry_Law3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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