Condensed Matter Cooper Pair Binding Rs Cooper Pair Binding Rs

Scope of the declaration

In condensed matter physics, a Cooper pair is two electrons that bind together at low temperature through lattice vibrations, and its binding energy is the energy that holds the pair together. For lead, the measured binding energy, twice the superconducting gap, is about 2.72 meV. A research note in the Recognition Science library records an intended match: that this value equals phi to a power times a constant, specifically phi^3 times 0.642 meV. That note is a research aspiration, not a result.

What the machine-checked declaration CooperPairBindingRS actually establishes is narrower. It defines a cost function, a number measuring the price of a recognition event, as Jcost(m/e), where m and e are positive real numbers. It then proves three general facts about that cost: it is zero when the two inputs are equal, it is never negative for positive inputs, and the constant phi minus 3/2 is positive. These are properties of the cost function itself, and they hold for any positive m and e.

The declaration does not define m or e in terms of lead, electrons, or any physical quantity. Without such a definition, the theorems cannot say anything about Cooper pairs, binding energies, or the 2.72 meV value. The docstring itself states this plainly: the Lean code proves nothing specific to the subject, because the cost is defined without reference to one. The same template body is shared verbatim with 2383 sibling modules, each named for a different intended subject but proving the same general facts.

In Recognition Science, this pattern is deliberate: the framework models a universal cost structure, and a subject-specific theorem would require a definition of m and e in that subject's own terms. What the declaration provides is a certificate that the general cost properties hold, not a derivation of any measured binding energy. A reader should take from this page the distinction between a named target and a proved result, and the concrete condition that would turn the target into a theorem.

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]

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

What this page does not claim

The declaration does not prove that the Cooper pair binding energy equals phi^3 times 0.642 meV. The declaration does not establish any property specific to lead or superconductivity. The declaration does not derive the value of the constant 0.642 from first principles.

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/CondensedMatter/Cooper_Pair_Binding_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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: