Chemistry Cohesion Energy3 From Jcost Cohesion E3 Cert

The cohesion certificate

In materials science, cohesion energy is the energy that holds a solid together, the amount needed to separate its atoms completely. Measured values span a wide range: noble gases sit around 0.01 to 0.2 electronvolts per atom, metals from 0.5 to 8, and covalent solids from 7 to 20. A theory that predicts these numbers from first principles would be a major advance.

The Recognition Science framework approaches this through a ledger, a discrete record of recognition events, and a forced cost, the price the ledger pays for each recognition. The framework's library, a machine-checked collection of formal theorems, proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The declaration CohesionE3Cert packages three general facts about this cost function applied to a ratio of two quantities, m and e.

What CohesionE3Cert actually establishes is modest and precise. First, when the two quantities are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the golden ratio minus 3/2 is a positive number. These are the three fields of the certificate structure, and the library proves each one. The certificate exists and is inhabited, meaning the three facts are consistent and provable together.

The critical limitation is that the certificate defines domainCost as Jcost (m / e) without ever defining what m and e mean for a specific solid. The docstring records an intention: cohesion energy might equal phi^k times some energy scale, with phi^5 times 0.001 eV giving 0.011 eV, a neon-like value. But that paragraph is a research note, not a theorem. The certificate proves nothing about real cohesion energies because the quantities m and e are never tied to atomic properties.

What this means for the reader is a clear boundary. The framework has a proven, general mathematical tool, the cost function J, and a certified package of its basic properties. What it does not have yet is the bridge from that tool to the physics of solids. That bridge, defining m and e in terms of measurable quantities, remains open. The certificate is a foundation stone, not a building.

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]

theorem cert_inhabited : Nonempty CohesionE3Cert := ⟨cert⟩

What this page does not claim

CohesionE3Cert does not prove any value for the cohesion energy of any real solid. The certificate does not define m and e in terms of atomic or material properties. The phi-ladder formula for cohesion energies is a research note, not a proved result.

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.

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