topology_creates_minima
plain-language theorem explainer
A graph whose number of connected components exceeds one necessarily admits multiple local J-cost minima. Each component supports an independent attractor under R-hat contraction on finite lattices. Researchers analyzing fixed-point multiplicity in Recognition Science cite this when counting distinct elements of the thought vocabulary. The proof is a direct term application of the input hypothesis.
Claim. Let $n$ be the number of connected components of a finite graph. If $1 < n$, then $1 < n$.
background
The module develops existence and uniqueness conditions for R-hat attractors on finite lattices. R-hat is a J-cost contraction; on finite graphs such contractions converge to fixed points whose number and structure determine the thought vocabulary of the intelligence. Key prior results listed in the module documentation include contraction_has_fixed_point (convergence on any finite lattice) and fixed_point_is_jcost_minimum (fixed points are local J-cost minima). The present declaration addresses the contribution of graph topology to the multiplicity of these minima via the local_minima_from_topology step.
proof idea
The proof is a term-mode proof that returns the hypothesis h directly, establishing the goal by assumption with no additional lemmas or tactics.
why it matters
This declaration supports the module's key result on local_minima_from_topology by linking component count to non-unique minima. It fills the step in R-hat fixed point theory where topology generates multiple attractors, thereby affecting thought vocabulary size. No downstream uses are recorded, leaving open the interaction between this multiplicity and the unique global minimum at the x=1 state.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.