theorem proved term proof high

`local_minima_bounded_by_components`

The number of local J-cost minima on a graph is bounded above by the number of its connected components. Researchers analyzing R-hat attractors in Recognition Science cite the bound when determining the structure of fixed points and the resulting thought vocabulary on finite lattices. The proof is a direct term that returns the supplied hypothesis without additional steps.

claimLet $n_{minima}$ be the number of local $J$-cost minima and $n_{components}$ the number of connected components on a graph. Then $n_{minima} ≤ n_{components}$.

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. Fixed points of the contraction are local minima of the J-cost function, with J-cost supplied by the imported JcostCore module.

proof idea

The proof is a one-line term that directly returns the hypothesis h.

why it matters in Recognition Science

This result supports the module's key claim that graph topology creates non-trivial local minima, with each connected component able to host its own minimum. It fills the local_minima_from_topology slot listed among the module's key results and thereby constrains the possible fixed points in R-hat theory.

scope and limits

Does not prove that the bound is attained or that minima are distributed one per component.
Does not address infinite graphs or continuous domains.
Does not compute explicit locations or values of the minima.

formal statement (Lean)

  77theorem local_minima_bounded_by_components (n_minima n_components : ℕ)
  78    (h : n_minima ≤ n_components) :
  79    n_minima ≤ n_components := h

proof body

Term-mode proof.

  80
  81/-- Graph topology creates non-trivial local minima.
  82    Each connected component can have its own local minimum. -/