IndisputableMonolith.Thermodynamics.PhaseTransitions
Recognition Science treatment of thermodynamic phase transitions, with the J-cost cast as a free-energy landscape in order parameter m and temperature T. Defines first- and second-order transitions, critical points, critical ratios, and spontaneous symmetry breaking in J-native language. Condensed-matter and RS auditors comparing to Landau theory would cite it. Argument is definitional scaffolding plus elementary positivity and critical-ratio facts on Cost and PhiForcing.
claimThe module introduces the J-cost landscape $J(m,T)$ in the order parameter $m$ and temperature $T$, and records RS characterizations of first-order transitions, second-order transitions, critical points (with critical ratios), order-parameter examples, and spontaneous symmetry breaking.
background
Recognition Science derives dynamics from the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. Temperature and order parameters enter as coordinates on that cost surface rather than as independent thermodynamic potentials.
This module sits in the thermodynamics layer. It imports the RS constants and external anchors (for empirical comparison only), the Cost primitives, and PhiForcing (self-similarity of the discrete ledger forcing $\varphi$). The local object is the J-cost landscape in $(m,T)$: at vanishing order parameter the cost sits at a reference value, and for $T>1$ (in RS-native units) the cost is strictly positive away from the ordered sector.
Sibling declarations sketch the standard taxonomy: first-order vs second-order mechanisms, a critical-point package with critical ratios, concrete order-parameter examples, and spontaneous symmetry breaking read off the landscape minima.
proof idea
Definition-heavy module, not a single theorem. It introduces jcostLandscape as $J$ on $(m,T)$, then records elementary facts (value at zero order parameter; positivity for $T>1$). First- and second-order transitions, critical points, critical ratios, order-parameter examples, and spontaneous symmetry breaking are packaged as structures or short lemmas that read minima, jumps, and scaling off that landscape, using Cost and the forced $\varphi$ from PhiForcing. No deep analytic machinery; the work is naming RS-native counterparts of Landau-style phase structure.
why it matters in Recognition Science
Gives the thermodynamics domain an explicit phase-transition vocabulary tied to the unique J-cost rather than to an assumed free-energy polynomial. That keeps phase structure inside the forcing chain: J from T5, $\varphi$ from self-similarity (PhiForcing), constants from the RS unit system. Downstream used-by edges are empty in the graph snapshot, so the module is presently a leaf that supplies definitions and elementary lemmas for later RS condensed-matter or cosmology arguments (order parameters, critical ratios, SSB). It also quarantines empirical comparison via ExternalAnchors, preserving the cost-first core.
scope and limits
- Does not derive a microscopic Hamiltonian or partition function from the ledger.
- Does not prove universality classes or renormalization-group flows.
- Does not fix numerical critical exponents beyond stated critical-ratio claims.
- Does not replace experimental EOS data; ExternalAnchors remain comparison-only.
- Does not assert a unique physical order parameter for every material.
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