module module moderate

`IndisputableMonolith.Physics.Superfluidity`

The Superfluidity module defines the Bose-Einstein occupation number at temperature T and derives associated critical temperatures, the lambda point for helium-4, and quantized vortex circulation within the Recognition Science framework. Condensed matter physicists would cite these results to connect the J-cost formalism to macroscopic superfluid phenomena. The module consists of sequential definitions and positivity theorems built directly on the imported J-cost core.

claimThe Bose-Einstein occupation number is $n(T) = 1/(e^{J(T)} - 1)$, where $J$ is the cost function. The BEC temperature satisfies $T_{BEC} > 0$, the lambda point for $^4$He lies in an experimentally consistent range, and vortex circulation is quantized.

background

This module extends the J-cost framework imported from IndisputableMonolith.Cost.JcostCore to model superfluidity. It introduces the occupation number as the central object, followed by positivity statements for occupation and temperatures, specific results for the lambda point in helium-4, quantized vortices, and the critical exponent. The setting assumes the Recognition Composition Law and phi-ladder conventions to obtain temperature-dependent quantities from the underlying forcing chain.

proof idea

The module proceeds by defining the occupation number and critical temperatures, then establishing positivity and range constraints through direct algebraic reductions on the J function. It applies basic properties from the JcostCore import with no complex tactics, relying on sequential positivity arguments and explicit range checks for helium-4 and vortices.

why it matters in Recognition Science

This module supplies the microscopic J-cost basis for superfluidity and Bose-Einstein condensation, supporting derivations of quantized circulation and critical exponents. It aligns with the eight-tick octave and D=3 dimensions by modeling collective behavior in the Recognition framework. No downstream theorems are listed, but the declarations close the gap between the unified forcing chain and condensed-matter observables.

scope and limits

Does not derive the full superfluid density or order parameter from first principles.
Does not treat fermionic pairing or high-temperature superconductivity.
Does not compute numerical values outside the stated lambda-point range.
Does not include finite-size corrections or external trapping potentials.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Cost.JcostCore module_import

declarations in this module (19)

def be_occupation L23
theorem be_occupation_positive L27
def bec_temperature L37
theorem bec_temperature_positive L41
def lambda_point L51
theorem lambda_point_lt_bec L55
def lambda_point_He4 L73
theorem lambda_He4_in_range L76
def vortex_quantum L84
theorem vortex_quantum_positive L87
theorem vortex_quantized L92
def rs_critical_exponent L99
lemma golden_ratio_gt_one L103
theorem rs_critical_exponent_positive L110
def superfluid_fraction L117
theorem superfluid_fraction_at_zero L121
theorem superfluid_fraction_at_lambda L127
theorem superfluid_fraction_between L133
theorem he3_b_phase_global_minimum L149