Physics Superfluidity

Superfluidity and the Eight-Tick Cycle

Physics superfluidity in Recognition Science is the study of zero-viscosity flow as a consequence of the framework's forced eight-tick cycle, the recognition cycle with eight steps that the cost function forces. The module IndisputableMonolith/Physics/Superfluidity.lean states the physical identification in its docstring: superfluid He-4 is a Bose-Einstein condensate of integer-spin (eight-tick) bosons, and superfluid He-3 is Cooper pairing of half-integer-spin (four-tick) fermions. The module then defines the standard thermodynamic objects and proves their basic properties inside Lean.

The definitions are the working tools. The Bose-Einstein occupation number is defined as 1 / (exp((ε - μ) / T) - 1), and the theorem be_occupation_positive proves it is positive when temperature is positive and the chemical potential lies below the energy. The BEC temperature is defined as (2π / m) * (n / 2.612)^(2/3), and bec_temperature_positive proves it is positive for positive mass and density. The lambda point, the transition temperature for He-4, is defined with a correction term T_BEC * (1 - 0.43 * a_s * n^(1/3)), and lambda_point_lt_bec proves it lies below the bare BEC temperature when the correction is small. The constant lambda_point_He4 is set to 2.17 kelvin, and lambda_He4_in_range proves it lies strictly between 2.0 and 2.5.

The superfluid fraction is defined as 1 - (T / T_λ)^α, where α is the RS critical exponent log(φ) / log(2) with φ the golden ratio. The theorems superfluid_fraction_at_zero and superfluid_fraction_at_lambda prove the fraction is 1 at absolute zero and 0 at the lambda point. The theorem superfluid_fraction_between proves the fraction lies strictly between 0 and 1 for temperatures strictly between zero and the lambda point. The vortex quantum is defined as 2π / m, and vortex_quantized proves that circulation is quantized in integer multiples of this value, a direct consequence of U(1) gauge invariance. The theorem he3_b_phase_global_minimum proves the existence of an order parameter equal to 1, which the docstring identifies with the global minimum of the He-3 B phase.

What the module establishes is that the standard phenomenology of superfluidity, the occupation numbers, the transition temperatures, the fraction curve, and the quantized vortices, is compatible with the RS picture of an eight-tick recognition cycle. The proofs are axiom-clean in the sense that they use only the ambient type theory's standard axioms; the physical identification of He-4 with eight-tick bosons and He-3 with four-tick fermions is a definitional choice recorded in the docstring, not a theorem. The numerical value 2.17 kelvin for the lambda point is a defined constant, and the theorem that it lies in the range (2.0, 2.5) is a check on that definition, not a derivation of the measured value.

The consequence for the subject is that superfluidity is not an exotic exception in Recognition Science but a natural expression of the cycle structure. The same eight-tick cycle that forces three spatial dimensions also provides the coherence that makes zero-viscosity flow possible. The module's contribution is to make the standard toolkit precise and to prove its basic consistency, leaving the deeper question of how the recognition-to-linking bridge produces the actual He-4 and He-3 systems as an open target.

MODEL superfluid_fraction · rs_critical_exponent · IndisputableMonolith/Physics/Superfluidity.lean

THEOREM superfluid_fraction_between · IndisputableMonolith/Physics/Superfluidity.lean

THEOREM vortex_quantum · vortex_quantized · IndisputableMonolith/Physics/Superfluidity.lean

THEOREM he3_b_phase_global_minimum · IndisputableMonolith/Physics/Superfluidity.lean

THEOREM lambda_point_He4 · lambda_He4_in_range · IndisputableMonolith/Physics/Superfluidity.lean

What this page does not claim

This answer does not claim that the physical identification of He-4 with eight-tick bosons is a theorem; it is a definitional choice recorded in the docstring. This answer does not claim that the lambda point value 2.17 kelvin is derived from first principles; it is a defined constant with a established range check. This answer does not claim that the module derives the fine-structure constant or any other coupling constant.

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/Superfluidity.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.

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