IndisputableMonolith.Physics.TopologicalPhaseTransitionFromJCost
This module applies the reusable J-cost band template to define topological phases and their transitions in Recognition Science. It introduces TopologicalPhase as states where J-cost on ratios triggers a phase change, together with counting and certification objects. The structure imports the six-clause template from CanonicalJBand to enforce matched-zero and nonnegativity. It positions the definitions inside the physics domain cert chain.
claimThe module defines TopologicalPhase as the predicate on configurations where the J-cost function $J(x)=(x+x^{-1})/2-1$ induces a topological transition, with topologicalPhaseCount enumerating instances and TopologicalPhaseCert supplying the certification that $J(1)=0$ and $J(x)≥0$ hold under the band template.
background
The module sits in the Physics domain and imports the Canonical J-Cost Band, whose six-clause template supplies the reusable J-cost-on-ratio structure used across master certs. The template guarantees matched-zero $J(1)=0$ and nonnegativity $J(x)≥0$ for $x>0$, with $J$ the Recognition Science cost function. It therefore supplies the local setting for extending the J-band to topological phase transitions while remaining inside the T0-T8 forcing chain.
proof idea
This is a definition module, no proofs. It declares the four sibling objects TopologicalPhase, topologicalPhaseCount, TopologicalPhaseCert and topologicalPhaseCert by direct application of the imported six-clause J-band template.
why it matters in Recognition Science
The module supplies the topological-phase component of the physics domain certs inside the master cert chain. It feeds the B-tier whole-science openings by extending the J-uniqueness property (T5) to phase transitions while preserving the eight-tick octave and D=3 structure. It closes one link in the reusable J-cost template application across domains.
scope and limits
- Does not compute numerical transition thresholds.
- Does not simulate concrete lattice models.
- Does not connect J-cost values to measured observables.
- Does not address quantum or relativistic corrections.