Chemistry Hydrogen Bonding Energy Rs Hbonding Cert

What the certificate proves

A hydrogen bond is the electrostatic attraction between a hydrogen atom bound to an electronegative atom and another electronegative atom nearby. In water and biological molecules, these bonds typically carry energies between 15 and 30 kJ/mol. The Recognition Science declaration HBondingCert is a formal certificate, a machine-checked bundle of three proved facts about a mathematical cost function. The cost function, written domainCost, takes two real numbers and returns a nonnegative value that measures a kind of recognition cost between them.

The three facts are general properties of this cost function. First, when the two inputs are equal and nonzero, the cost is exactly zero. Second, for any two positive inputs, the cost is never negative. Third, a constant called the canonical threshold, defined as the golden ratio minus 1.5, is positive. Each of these is a theorem in the machine-checked library of formal theorems, and together they form the certificate. The certificate is inhabited, meaning a proof that it exists is also recorded.

What the certificate does not do is connect these general facts to hydrogen bonding. The cost function is defined as Jcost(m/e), a ratio of two numbers, but nothing in the definition says what m or e mean for a hydrogen bond. The research note attached to the module records an intended application: using the cost function to estimate a hydrogen bond energy of about 20.8 kJ/mol, which falls in the observed range. That estimate is a research note, not a result of the certificate. The certificate itself proves nothing specific to chemistry.

In plain terms, the certificate is a reusable skeleton. It establishes that a certain cost function has three desirable properties for any positive inputs. To become a theorem about hydrogen bonds, someone would need to define m and e in chemical terms, for instance as a measured energy and a reference energy. Until that definition exists, the certificate remains a general mathematical object that happens to sit in a chemistry-named file.

This distinction matters for reading the framework's claims honestly. The framework's library proves what it proves, and the certificate is a clean example of a proof that is real but narrow. The hydrogen bond energy estimate is a hypothesis with a named falsifier: if measured hydrogen bond energies moved outside the 15 to 30 kJ/mol range, the intended application would fail. The certificate itself would remain true, because it makes no chemical claim that could be falsified.

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

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]

structure HBondingCert 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 hydrogen bond energies equal 20.8 kJ/mol. The certificate does not define m or e in chemical terms. The certificate does not establish that the cost function applies to hydrogen bonding at all.

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/Hydrogen_Bonding_Energy_RS.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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