IndisputableMonolith.Physics.TopologicalPhaseFromJCost
Defines a domain-level cost from the RS J-functional, a positive canonical threshold, and an inhabited topological-phase invariance certificate. Physicists tracking phase structure forced by the cost law would cite it. The module is mostly definitions and elementary nonnegativity or positivity facts, not a deep existence proof.
claimFrom the RS cost $J$, the module introduces a domain cost $C$, a canonical threshold $\theta>0$, and a topological-phase invariance certificate asserting that phase labels are stable once the domain cost sits relative to $\theta$.
background
Recognition Science builds physics from a single cost functional $J$ on positive reals, uniquely fixed (T5) by the Recognition Composition Law as $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that $J$; Constants supplies the RS-native tick $\tau_0=1$.
This physics module lifts $J$ from pointwise values to a domain cost on regions (or configurations) and isolates a positive canonical threshold against which that cost is compared. The intended reading is that crossing or sitting below the threshold marks a topological phase boundary forced by the cost geometry, not by an external order parameter.
Sibling objects include nonnegativity of the domain cost, positivity of the threshold, and a certificate type packing the invariance claim together with an inhabited instance.
proof idea
Definition-heavy module. Domain cost is introduced from $J$ (with an evaluation identity at equality cases). Nonnegativity follows from the corresponding property of $J$. The canonical threshold is a fixed positive RS-scale quantity; positivity is immediate. The topological-phase certificate is a structure bundling the invariance statement, discharged by an inhabited instance rather than a long tactic proof.
why it matters in Recognition Science
Places topological phase structure on the same J-cost footing as the forcing chain (T5 J-uniqueness and the RCL), so phase labels are not free phenomenological data. No downstream Lean dependents are recorded yet; the module stands as a physics-side interface that later continuum or lattice arguments can import when they need a cost-derived phase threshold and a certificate shape. It does not itself close T6--T8 (phi, eight-tick, $D=3$), but keeps phase talk inside the RS cost calculus.
scope and limits
- Does not derive a microscopic Hamiltonian or lattice model.
- Does not prove uniqueness of the threshold beyond the chosen canonical scale.
- Does not classify all topological invariants; only packages a cost-relative certificate.
- Does not connect yet to measured condensed-matter phase diagrams.
- Does not discharge T6--T8 or the mass/alpha ladder.