Physics Superfluidity Helium From Jcost

The helium-4 module

Superfluidity is a state of matter, reached by liquid helium-4 below about 2.17 kelvin, in which the fluid flows without measurable friction. The fraction of the liquid that behaves this way, ρ_s/ρ, rises from zero at the transition temperature T_λ toward one as the temperature drops. Near T_λ, experiments show that fraction rising as a power law, roughly 1 − (T/T_λ)ⁿ with n close to 1.618, a number that matches the golden ratio φ.

The module named for this subject, SuperfluidityHeliumFromJCost, does not prove that power law. Its ledger, a discrete record of recognition events, enters through a cost function J(x) = (x + 1/x)/2 − 1. The module defines domainCost(m, e) = J(m/e) and proves three general facts: the cost is zero when m equals e, it is never negative for positive inputs, and the threshold φ − 3/2 is positive. These statements hold for any positive real numbers m and e. Nothing in the file ties m or e to helium, to temperature, or to the superfluid fraction.

The file's own docstring says as much: it proves nothing specific to this subject. The research note records where the idea was meant to go, not a result. A theorem about superfluidity would need a definition of m and e in the physics of helium-4, such as m as a thermal energy scale and e as a condensation energy. That definition does not appear. The module is one of 2383 siblings sharing the same template body, all stating the same three generic facts about J.

What the module does establish, in plain language, is that the cost function J has the basic hygiene any useful ledger needs: matching entries cost nothing, all positive ratios cost something nonnegative, and a particular threshold derived from φ is a genuine positive number. Those are real, machine-checked results, but they are results about J, not about helium. The empirical φ-power law for the superfluid fraction remains a research aspiration recorded in a comment, not a derived consequence.

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 module does not prove the φ-power law for the superfluid fraction. The module does not establish any physical property of helium-4. The empirical exponent 1.618 is a measured value, not a derived consequence of the framework.

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/SuperfluidityHeliumFromJCost.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: