IndisputableMonolith.Thermodynamics.PhaseTransitions
The PhaseTransitions module defines the J-cost landscape as a function of order parameter m and temperature T in Recognition Science thermodynamics. It establishes the critical temperature Tc separating first-order and second-order transitions with explicit mechanisms and critical ratios. Researchers modeling discrete self-similar systems cite it for symmetry-breaking analysis beyond T=0. The module structures its content through successive definitions of the landscape, transition types, and order-parameter examples.
claimThe J-cost is expressed as the function $J(m,T)$ of order parameter $m$ and temperature $T$, with a phase transition occurring at critical temperature $T_c$ that separates first-order and second-order regimes, together with associated critical ratios and spontaneous symmetry breaking.
background
Recognition Science builds thermodynamics from the absolute minimization of the universal cost functional $J(x) = ½(x + 1/x) - 1$ at T=0. This module extends that foundation to finite temperature by coupling J to an order parameter m, drawing on the cost primitives and the forcing of the golden ratio from self-similar discrete ledgers. The upstream PhiForcing module states that it proves φ is forced by self-similarity in a discrete ledger with J-cost, while Constants supplies the time quantum τ₀ = 1 tick and ExternalAnchors quarantines all empirical calibration data.
proof idea
The module first defines the J-cost landscape and proves its zero value together with positivity for T greater than 1. It then establishes the phase transition at Tc, followed by separate mechanisms for first-order and second-order transitions, the critical point with its ratios, order-parameter examples, and spontaneous symmetry breaking.
why it matters in Recognition Science
This module supplies the phase-transition machinery imported by the parent IndisputableMonolith.Thermodynamics module, which develops the statistical mechanics of RS. It extends the T=0 foundation where reality is defined by absolute minimization of J(x) = ½(x + 1/x) - 1 to finite-temperature critical phenomena. The module thereby connects the cost structure and phi-forcing to observable thermodynamic behavior.
scope and limits
- Does not derive numerical values of Tc without external anchors.
- Does not include dynamical equations for order-parameter evolution.
- Does not treat quantum corrections to the classical J-cost.
- Does not address multi-component or spatially inhomogeneous systems.
used by (1)
depends on (4)
declarations in this module (20)
-
def
jcostLandscape -
theorem
jcost_at_zero -
theorem
jcost_positive_for_T_gt_1 -
theorem
phase_transition_at_Tc -
structure
FirstOrderTransition -
theorem
first_order_mechanism -
structure
SecondOrderTransition -
theorem
second_order_mechanism -
structure
CriticalPoint -
theorem
critical_ratios -
def
orderParameterExamples -
theorem
spontaneous_symmetry_breaking -
def
ssbMechanism -
structure
MetastableState -
def
nucleationRate -
structure
QuantumPhaseTransition -
def
qptExamples -
structure
TopologicalTransition -
def
summary -
structure
PhaseTransitionFalsifier