Chemistry Henry Law3 From Jcost Henry Law3 Cert

What the certificate proves

Henry's law describes how much gas dissolves in a liquid at a given pressure. Its constants, which measure solubility, range from about 10^-5 to 10^5 atmospheres across different gases and solvents. A recognition cost, a number that measures the price of matching one quantity to another, can be built from the ratio of two such constants. The declaration HenryLaw3Cert is a machine-checked certificate: a packaged list of three proved facts about that cost function.

The three facts are general properties of the cost function J(x) = (x + 1/x)/2 - 1, evaluated at the ratio m/e. First, when m equals e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the number phi - 3/2, where phi is the golden ratio, is positive. These are proved in the machine-checked library of formal theorems. They hold for any positive real numbers m and e; they do not mention any specific gas or liquid.

The certificate does not establish that Henry's law itself is true, nor that any particular constant equals a phi-power. The research note attached to the module records a hope: that the range of Henry's constants, roughly phi^30, might match the framework's scaling. That note is not a result. The module defines domainCost as Jcost (m / e) without saying what m and e mean. The certificate would become a theorem about Henry's law only if m and e were defined in terms of actual solubility measurements.

In plain terms, the certificate is a small, correct piece of mathematics about a cost function. It proves that the cost is zero at equality, nonnegative for positive inputs, and that a certain threshold is positive. It does not prove any chemistry. The gap between the mathematics and the subject is explicit: the definition of m and e is missing. That gap is what the research note acknowledges, and it is why the certificate, as it stands, is a template rather than a discovery.

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

Henry's law itself is not proved by this certificate. No specific Henry's law constant is derived from the framework. The phi-scaling note is a research idea, not a measured or 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/Henry_Law3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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