Chemistry Vapor Pressure From Phi Ladder Vapor Pressure Cert

What the certificate proves

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form at a given temperature. At 20°C, mercury exerts about 0.0016 mbar, water about 23 mbar, and acetone about 233 mbar. The ratio of acetone to water is roughly 10, which is close to the fifth power of the golden ratio φ ≈ 1.618. That numerical coincidence motivates a research note in the framework's library, but it is not what the formal certificate proves.

The declaration VaporPressureCert, a machine-checked certificate in the framework's library of formal theorems, packages three general facts about a cost function Jcost, a measure of recognition cost. For positive numbers m and e, the certificate proves that the cost vanishes when m equals e, that the cost is never negative, and that the golden-ratio threshold φ − 3/2 is positive. These are abstract properties of the cost function applied to a ratio m/e; they hold for any positive real numbers, not just for vapor pressures.

What the certificate does not do is connect those facts to vapor pressure. The module defines domainCost as Jcost (m / e) without ever defining m and e in chemical terms. Nothing in the Lean code says m is a vapor pressure or e is a reference pressure. The docstring records the intended research direction: that each φ-rung might correspond to a factor of about 1.618 in vapor pressure. But that remains a research note, not a proved result.

The three proved facts are real and machine-checked, but they are generic. The same certificate could apply to any ratio of positive quantities. The chemical interpretation, the φ^5 approximation for acetone over water, and the idea of a φ-ladder in vapor pressures are all unformalized hypotheses. The framework's own documentation states that the module proves nothing specific to this subject.

For a reader, the practical takeaway is to distinguish the formal skeleton from the empirical dream. The certificate guarantees a clean mathematical structure: zero cost at equality, nonnegative cost elsewhere, and a positive threshold at the golden ratio. Whether that structure describes vapor pressure is a separate question, one the certificate leaves open.

noncomputable def cert : VaporPressureCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

structure VaporPressureCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The certificate does not prove that vapor pressures follow a φ-ladder. The certificate does not define vapor pressure or any chemical quantity. The φ^5 approximation for acetone over water is a research note, 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/VaporPressureFromPhiLadder.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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