Chemistry Solid Electrolyte3 From Jcost

Solid electrolyte conductivity

A solid electrolyte is a material that conducts electricity through the movement of ions, not electrons, and is the working part of solid-state batteries. The module named solid electrolyte in the Recognition Science library defines a quantity called domainCost, a discrete record of recognition cost, as the function J applied to the ratio of two real numbers m and e. The Lean code proves three general facts: the cost vanishes when m equals e, it is never negative for positive inputs, and the number phi minus 3/2 is positive. These facts hold for any positive real numbers, because the definition never says what m and e mean in chemistry.

The research note attached to the module records the intended application: superionic conductors, materials with unusually high ion mobility, should have conductivity above J(phi) times 10^-2 S/cm, about 1.18e-3 S/cm. Empirical solid electrolytes range from 10^-3 to 10^-1 S/cm, which is consistent with that threshold. But this comparison is not proved. The module proves nothing about conductivity, ions, or materials, because the definition of domainCost never connects m and e to any physical quantity. The note is a plan for where the idea was meant to go, not a result about electrolytes.

What the module does establish is a reusable template. The same three facts, with the same proof, appear in 2383 sibling modules, each sharing the body verbatim. The template is stated once, universally quantified, in a central file, and each subject module instantiates it. To turn this into a theorem about solid electrolytes, someone would need to define m and e in the subject's own terms, for example as activation energy and temperature, and then prove the conductivity claim from that definition. Until then, the honest statement is that the framework provides the cost function and its basic properties, while the chemistry remains an open research target.

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)

noncomputable def cert : SolidElec3Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The module does not prove any statement about actual solid electrolytes or ionic conductivity. The conductivity threshold of 1.18e-3 S/cm is not derived from the Lean theorems; it is a research note. No claim is made that the 2383 sibling modules each have a valid subject-specific interpretation.

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/SolidElectrolyte3FromJCost.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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