Chemistry Molecular Recognition From Jcost

The ledger of binding

Molecular recognition is the selective binding of one molecule to another, the handshake that lets enzymes find their substrates and drugs find their targets. In chemistry, the strength of that handshake is measured by the association constant K_a, which for crown ethers and beta-cyclodextrins ranges from about 10^3 to 10^9 per molar. Recognition Science proposes to derive that binding strength from a single cost function, a measure of the price a system pays for a mismatch between two quantities.

The framework defines the cost of a host-guest pair as J(m/e), where J(x) = (x + 1/x)/2 - 1, and m and e are real numbers meant to represent the guest and host. The machine-checked library of formal theorems proves three general facts about this definition: the cost vanishes when m equals e, it is never negative for positive inputs, and the threshold value phi - 3/2 is positive. These are properties of the cost function itself, not of any particular molecule.

In Recognition Science, the framework's library shows that this cost function emerges uniquely from five plain conditions on how recognition events are recorded, and that its minimum at m = e encodes a perfect match. The intended chemistry is that the logarithm of the binding constant equals the inverse cost at the golden ratio phi times the number of recognition contacts, with 2 to 6 contacts giving binding constants near 10^7 per molar. That claim is a research note, not a proved result: the module does not define what m and e are in chemical terms.

What the module actually establishes is a template. It proves that any host-guest system described by this cost function will have a zero-cost perfect match, nonnegative costs elsewhere, and a positive threshold. The step from those facts to a binding constant requires a definition of m and e for a real molecule, which the module does not provide. The chemistry remains open, while the mathematics is settled.

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

The module proves no specific chemistry, because m and e are undefined in chemical terms. The binding constant formula is a research note, not a proved result. The framework does not claim that crown ethers or beta-cyclodextrins actually obey this cost 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)

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