Physics Superconducting Gap3 From Jcost

The BCS gap template

The superconducting energy gap is the energy needed to break apart a Cooper pair, the bound electron pair that carries current without resistance in a superconductor. Near the critical temperature Tc, the gap follows a known temperature curve described by the BCS theory, named for Bardeen, Cooper, and Schrieffer (1957). The framework's cost function, a forced price of recognition, was intended to reproduce that curve: the research note attached to the file sketches how the gap ratio might equal a function of the golden ratio at a special temperature.

What the file actually proves is narrower. It defines a cost function on a ratio of two real numbers, domainCost, and proves three facts: the cost is zero when the two numbers are equal, the cost is never negative for positive inputs, and a certain threshold involving the golden ratio is positive. These are general properties of the cost function, true for any positive numbers, not results specific to superconductors. The file proves nothing about the BCS gap, because the quantities in the cost function are not defined in terms of superconducting physics.

The file does provide a template. Its structure bundles the three proved facts into a certificate, a packaged record that any future theory can cite. The research note records where the idea was meant to go, not a result that has arrived. A theorem about the superconducting gap would require defining the two quantities in the cost function using the gap's own variables, a step this file does not take.

In plain language: the file establishes that a certain cost function has three basic mathematical properties, and it leaves a placeholder for the physics. The BCS connection remains a research target, not a proved statement. The honest summary is that the file is a scaffold, 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]

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

What this page does not claim

This file does not derive the BCS gap equation or its temperature dependence. The golden ratio threshold is not shown to equal any physical temperature ratio. No claim is made that the cost function represents the actual energy cost of breaking a Cooper pair.

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/Physics/Superconducting_Gap3_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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