IndisputableMonolith.Physics.TopologicalPhaseTransitionFromJCost
Module linking topological phase structure to the canonical J-cost on positive ratios. It introduces a phase type, a phase count, and a certificate that the count is forced by the six-clause J-band. Physicists tracking RS phase structure or cert-chain openings would cite it. Argument shape is definitional plus a thin cert wrapper over CanonicalJBand.
claimFrom the canonical cost $J$ on ratios $x>0$ (with $J(1)=0$ and $J\ge 0$), the module defines a topological phase object, a finite phase count, and a certificate that this count is the unique value compatible with the six-clause $J$-band.
background
Recognition Science builds physics from a single cost functional $J$ on positive ratios. The reusable template in CanonicalJBand packages six clauses used across domain certificates: matched zero $J(1)=0$, nonnegativity $J(x)\ge 0$ for $x>0$, and the remaining band identities that pin $J$ to the unique form $(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$).
This physics module sits downstream of that template. It treats topological phases as discrete labels whose admissible count is constrained by the same $J$-band, rather than by an independent topological invariant. The local setting is therefore cost-first: phase structure is read off the recognition cost, not postulated from a separate Chern or winding construction.
Sibling names in the module (phase type, phase count, phase certificate) mirror the usual RS cert pattern: a data type, a numeric claim, and a Prop-level certificate discharging the claim against the canonical band.
proof idea
Definition-heavy module with a thin certificate layer. Types and counts are introduced as defs; the certificate applies the six-clause CanonicalJBand template (matched zero, nonnegativity, and the remaining band clauses) to pin the admissible topological phase count. No deep tactic development is indicated; the logical work is inheritance from the upstream J-band plus packaging for the physics domain cert chain.
why it matters in Recognition Science
Gives the physics-side opening that topological phase transitions are not free parameters but consequences of the same $J$-cost that forces the forcing chain (T5 J-uniqueness onward). It feeds the master cert pattern described in CanonicalJBand: domain certs that open whole-science or Plan-style arguments by proving the six clauses in a concrete setting.
No downstream consumers are recorded yet in the graph, so the module is presently a leaf packaging step. Its place in the framework is to make phase counting cite the RCL/$J$ lineage rather than an external topological axiom, keeping the RS claim that discrete structure (ticks, dimensions, phases) descends from one functional equation.
scope and limits
- Does not derive a microscopic Hamiltonian or lattice model for the transition.
- Does not compute critical exponents, correlation lengths, or scaling functions.
- Does not identify phases with concrete Chern numbers or winding numbers from condensed-matter data.
- Does not claim experimental values for transition temperatures or gaps.
- Does not extend the six-clause band; it only applies CanonicalJBand.