superfluid_fraction
Superfluid fraction ρ_s(T)/ρ is defined as 1 minus (T/Tlam) raised to the RS critical exponent α ≈ 0.694. Physicists modeling temperature dependence in He-4 superfluidity cite this expression below the lambda point. It is a direct one-line definition that applies the exponent derived from the golden ratio logarithm.
claimThe superfluid fraction is given by $1 - (T/T_λ)^α$ where $α = ln φ / ln 2$ and $φ = (1 + √5)/2$.
background
Superfluidity arises in the RS framework from eight-tick coherence, with He-4 modeled as a BEC of integer-spin bosons. The upstream rs_critical_exponent is defined as Real.log ((1 + Real.sqrt 5) / 2) / Real.log 2, the logarithm of the golden ratio divided by the logarithm of 2. The T abbrev from Breath1024 supplies fundamental periods while the triangular T from Gap45.SyncMinimization remains unused here. The module sets the local context as superfluid He-4 and He-3 following RS eight-tick and four-tick coherence per the paper RS_Superfluidity.tex.
proof idea
One-line definition that applies rs_critical_exponent directly to the power term in the subtracted ratio.
why it matters in Recognition Science
This supplies the core temperature dependence for superfluid fraction, feeding the downstream theorems superfluid_fraction_at_lambda (fraction vanishes at T = Tlam), superfluid_fraction_at_zero (full superfluidity at T = 0), and superfluid_fraction_between (strictly between 0 and 1 for 0 < T < Tlam). It realizes the eight-tick coherence landmark for He-4 in the Recognition Science forcing chain and links to the phi-ladder via the critical exponent.
scope and limits
- Does not derive the critical exponent from the forcing chain.
- Does not compute the lambda point Tlam from microscopic parameters.
- Does not address quantized vortices or U(1) gauge effects.
- Does not extend to He-3 Cooper pairing.
formal statement (Lean)
117noncomputable def superfluid_fraction (T Tlam : ℝ) : ℝ :=
proof body
Definition body.
118 1 - (T / Tlam) ^ rs_critical_exponent
119
120/-- At T = 0, fully superfluid. -/