Chemistry Chromatography3 From Jcost Chromat3 Cert

What the certificate proves

Chromatography separates mixtures by how their components travel through a column. The standard description of column efficiency is the van Deemter equation, which relates plate height to flow rate and has a minimum at an optimal flow rate. In the Recognition Science framework, the declaration Chromat3Cert is a certificate: a machine-checked collection of formal theorems that packages three general facts about a cost function, Jcost, which measures the forced cost of recognition events in a ledger, a discrete record of such events. The certificate proves that the cost vanishes when the two inputs are equal, that it is never negative for positive inputs, and that the golden ratio minus 1.5 is positive.

The first two facts are properties of the cost function itself, not of any particular science. For any real number r that is not zero, the cost of recognizing r against r is zero; and for any two positive numbers m and e, the cost is nonnegative. The third fact is a numerical threshold: the golden ratio, about 1.618, is greater than 1.5. The certificate simply bundles these three theorems into one structure and proves that such a structure exists. It is a way of saying: these three facts hold, and here is the proof, all in one place.

What the certificate does not do is apply these facts to chromatography. The module defines a quantity called domainCost as Jcost applied to the ratio m over e, but it never defines what m and e mean in chromatographic terms, such as mass and charge, or anything else. The docstring notes a research intention: the minimum of the van Deemter equation might be expressed as J(phi) times column length, giving a numerical value for a 15 cm column. But that is a research note, not a theorem. The certificate proves nothing specific to chromatography; it proves only the three general facts about the cost function.

This is the distinction that matters for reading the framework's library. A module that shares a template with 2383 siblings, each stating the same three facts with different names, is not a result about its subject unless it defines its own quantities in that subject's terms. The certificate is a building block, not a conclusion. It establishes a small, clean set of facts about the cost function, and it leaves the bridge to chromatography as an open target, not an achieved result.

theorem cert_inhabited : Nonempty Chromat3Cert := ⟨cert⟩

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)

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

What this page does not claim

Chromat3Cert does not establish any relationship between the cost function and actual chromatographic measurements. The certificate does not prove that the van Deemter equation's minimum is described by the golden ratio. The research note's numerical value for H_min is not a theorem and has not been verified against experimental data.

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/Chromatography3_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.

Derived articles

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