Chemistry Cohesion Energy3 From Jcost

Cohesion energy and the cost function

Cohesion energy is the energy that holds a solid together, the amount needed to separate its atoms completely. For noble gases it is tiny, around 0.01 to 0.2 electronvolts; for metals it ranges from 0.5 to 8 electronvolts; for covalent solids like diamond it reaches 7 to 20 electronvolts. These numbers come from measurements, and they are the classical facts a chemist expects.

In Recognition Science, the framework models cohesion energy through a cost function. The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a ratio x; it is zero when x equals 1, and it grows as x moves away from 1. The module defines a domain cost, a discrete record of the price of a ratio, as J evaluated at the ratio m/e, where m and e are two real numbers. The module then proves three general facts about this cost: it vanishes when m equals e, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. These are theorems in the machine-checked library of formal theorems.

The chemistry claim, that cohesion energy equals phi^k times a base energy, appears only in a research note attached to the module. The note suggests values like phi^5 times 0.001 electronvolts giving 0.011 electronvolts, a neon-like scale. But the module itself defines m and e without reference to any physical quantity, so it proves nothing specific to chemistry. The note is a record of where the idea was meant to go, not a result.

What the module does establish, in plain language, is that the cost function behaves sensibly: equal inputs cost nothing, positive inputs never cost a negative amount, and a certain threshold is positive. These facts hold for any ratio, not just for cohesion energies. To turn this into a theorem about solids, one would need a definition of m and e in terms of atomic properties, a step the module does not take.

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

The module does not derive any specific cohesion energy value. The research note is not a theorem and carries no proof. No claim is made that phi^k scaling matches any measured cohesion energy.

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/Cohesion_Energy3_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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