Chemistry Molecular Recognition From Jcost Host Guest Cert

What the certificate proves

In chemistry, host-guest recognition is the process by which a larger molecule, the host, selectively binds a smaller one, the guest, through non-covalent interactions. Crown ethers and beta-cyclodextrins are classic hosts, with binding affinities, measured as association constants K_a, typically ranging from 10^3 to 10^9 per molar. The Recognition Science framework proposes that this binding energy can be modeled by a cost function, a measure of the effort required for recognition, applied to the ratio of two masses.

The machine-checked library of formal theorems contains a declaration called HostGuestCert. This certificate establishes exactly three general facts about the cost function J(x) = (x + 1/x)/2 - 1. First, the cost is zero when the two inputs are equal. Second, the cost is never negative for positive inputs. Third, the golden ratio constant phi minus 1.5 is positive, a technical threshold condition. These are properties of the cost function itself, proved in the framework's library, and they hold for any positive real numbers, not just for chemical masses.

The certificate does not establish anything specific to chemistry. The definition of the cost for a host-guest pair is simply J applied to the ratio m/e, where m and e are real numbers, but the certificate never defines what m and e represent in chemical terms. The binding affinity formula, ln K_a = J(phi)^(-1) times the number of recognition contacts, appears in a research note attached to the code, but it is not a proved theorem. The note records where the idea was meant to go, not a result. The certificate itself is shared verbatim with 2383 sibling modules, each hoping for a subject-specific definition that none of them yet provide.

What the certificate does prove is a foundation. It shows that the cost function has the basic properties one would want for a measure of dissimilarity: it vanishes at equality, it is nonnegative, and it has a built-in threshold. A researcher who can define m and e in terms of actual molecular properties, such as a host's cavity size and a guest's volume, would then have a machine-checked starting point for a theory of binding. Until such a definition exists, the certificate remains a general mathematical fact, not a chemical law.

structure HostGuestCert 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

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 certificate does not prove that ln K_a equals J(phi)^(-1) times the number of contacts; that formula is a research note, not a theorem. The certificate does not define what m and e represent; it only applies the cost function to any two positive real numbers. The certificate does not establish that the cost function is the only possible measure of recognition effort; it only proves properties of this specific function.

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/MolecularRecognition_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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