Condensed Matter Bcs Coherence Length Rs Bcscoherence Cert

What the certificate proves

The BCS coherence length is the distance over which electrons in a superconductor stay paired, typically a few nanometers to a micrometer. In the Recognition Science framework, the declaration BCSCoherenceCert is a machine-checked certificate, a packaged set of three proved facts about a mathematical cost function. The cost function, written Jcost, measures the price of recognition in the framework's ledger, a discrete record of events. The certificate proves that this cost is zero when its two inputs are equal, that it is never negative for positive inputs, and that a certain threshold involving the golden ratio is positive.

These three facts are general properties of the cost function, not results about superconductivity. The certificate's definition of cost uses the ratio of two numbers, m and e, but the declaration never defines what those numbers mean for a superconductor. The document's own notes state this plainly: the Lean code proves nothing specific to this subject. The research note mentioning phi^15 = 1364 nm for conventional superconductors and phi^7 = 29 nm for cuprates is a record of where the idea was meant to go, not a derived result.

What the certificate does establish is that the framework's cost function behaves sensibly: it vanishes at equality, stays nonnegative, and has a positive threshold. These are the building blocks for any future claim. The certificate's existence shows the framework can package such general facts into a reusable structure, but the leap from a general cost function to a physical coherence length remains open. The declaration 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]

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

What this page does not claim

The certificate does not prove that BCS coherence length follows a phi-power ladder. The certificate does not derive any specific value for a coherence length in nanometers. The certificate does not establish that the framework's cost function is the correct model for superconductivity.

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/BCS_Coherence_Length_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: