Chemistry Chromatography3 From Jcost

A formal placeholder

Chromatography is a laboratory technique for separating the components of a mixture. A sample is carried by a moving fluid through a column packed with a stationary material; different components travel at different speeds and emerge at different times. The efficiency of a column is often summarized by the van Deemter equation, which relates the height equivalent to a theoretical plate (HETP) to the flow rate of the mobile phase. That equation has a minimum: an optimal flow rate that gives the smallest plate height and therefore the best separation.

In the Recognition Science framework, the object chromatography3 from jcost is a formal construction in a machine-checked library of theorems. The name suggests an intended connection to chromatography, and a research note in the construction records a plan: the minimum plate height should equal the framework's cost function J evaluated at the golden ratio, times the column length. For a 15 cm column, that would give a minimum plate height of 1.77 cm, which is far larger than the 0.01 to 0.2 mm typical of real high-performance liquid chromatography columns. The note is an aspiration, not a result.

What the construction actually proves is much narrower. It defines a cost function on a ratio of two real numbers, m and e, as J(m/e), where J is the framework's forced cost function. It then proves three general facts: the cost is zero when m equals e, the cost is nonnegative for positive inputs, and a certain threshold involving the golden ratio is positive. These are consequences of the general properties of J, not of any chromatography-specific assumption.

The construction's own documentation is explicit: it proves nothing specific to chromatography, because the quantities m and e are not defined in chromatographic terms. The construction is a template, shared verbatim with 2383 sibling constructions. To turn it into a theorem about chromatography, someone would need to define m and e using the physics of a chromatographic column. Until then, the construction is a formal placeholder, not a derivation.

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

The construction does not establish the van Deemter equation or any chromatographic result. The research note's numerical prediction (H_min = 1.77 cm for a 15 cm column) is not a proved statement.

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)

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Derived articles

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