Physics Liquid Helium3 From Jcost

Helium-3 and the cost function

Helium-3 is a light, stable isotope of helium with two protons and one neutron. Below about 2.7 millikelvin, it becomes a superfluid, a state where it flows without viscosity. The transition temperature is a sharp physical fact, measured in laboratories and predicted by the theory of fermionic pairing.

The module named physics-liquid-helium3-from-jcost in the Recognition Science framework's machine-checked library of formal theorems does not establish that transition. Its own docstring says so plainly: it proves three general facts about a cost function, and nothing specific to helium-3. The cost function is cost, a measure of the forced expense of a recognition event, defined here as J(m/e), where J(x) = (x + 1/x)/2 - 1.

What the module proves, with machine-checked certainty, is three things. First, the cost vanishes when the two inputs are equal: J(r/r) = 0 for any nonzero r. Second, the cost is never negative for positive inputs. Third, the constant phi - 3/2 is positive, where phi is the golden ratio, about 1.618. These are all true of J itself, not of helium-3.

The module defines a structure that packages these three facts together and shows it is inhabited. But the definition of the cost uses only the ratio m/e, with no physical meaning attached to m or e. The docstring records an intended research direction: the author hoped to relate the helium-3 transition temperature to the helium-4 transition temperature using the golden ratio. The numbers did not work. The measured helium-4 transition is 2.17 kelvin; applying the cost ratio gives about 0.256 kelvin, off by a factor of roughly 100 from the real 2.7 millikelvin. A second attempt using phi to the power minus three gave 0.512 kelvin, still far off.

In plain language, the module is a structural placeholder. It proves general theorems about a cost function that happens to be named for helium-3, but it contains no physics of the superfluid. The honest takeaway is that the framework's library has a file with this name, and the file itself says it is not a result about its subject.

theorem cert_inhabited : Nonempty He3SuperfluidCert := ⟨cert⟩

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]

What this page does not claim

The module does not predict or explain the helium-3 superfluid transition temperature. The golden ratio does not appear in any proved theorem connecting it to helium-3 physics in this module. No claim is made that the cost function J is a model of any real physical system in this file.

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/LiquidHelium3_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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