IndisputableMonolith.CondensedMatter.JCostPhaseTransition
This module defines the J-cost function J(x) = (x + x^{-1})/2 - 1 together with critical energy, gap scale, and temperature quantities for phase transitions. Condensed matter theorists applying Recognition Science to superconductivity or critical phenomena would cite these definitions. The module consists entirely of definitions with no theorems or proofs.
claim$J(x) = (x + x^{-1})/2 - 1$
background
Recognition Science starts from the forcing chain in IndisputableMonolith.Foundation.UnifiedForcingChain, where T5 fixes the J-uniqueness property J(x) = (x + x^{-1})/2 - 1. The module imports the RS time quantum τ₀ = 1 tick from Constants and Mathlib real-analysis primitives. It introduces J_cost as the canonical cost function, then defines phi_critical_energy, sc_gap_scale, T_critical, and predicates J_cost_minimum_at_one, J_cost_positive_away_from_one, J_cost_symmetric that locate the minimum at x = 1.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the J-cost and phase-transition primitives that later feed into condensed-matter predictions such as the superconducting gap and critical temperature. It directly instantiates the J-uniqueness step (T5) of the forcing chain for use in the phi-ladder and eight-tick octave structure.
scope and limits
- Does not derive the J-cost expression from the Recognition Composition Law.
- Does not prove existence of phase transitions.
- Does not compute numerical values for critical points.
- Does not connect to mass formulas or alpha-band constraints.