bandBoundaries_product
plain-language theorem explainer
The product of the lower and upper cognitive band boundaries equals the cube of the inverse golden ratio. Researchers deriving normalization constants or band widths in the Recognition Science unification framework would reference this identity. The proof proceeds by direct substitution of the explicit definitions followed by algebraic simplification.
Claim. Let $ρ_{min} = φ^{-2}$ denote the lower boundary and $ρ_{max} = φ^{-1}$ the upper boundary of the cognitive recognition band. Then $ρ_{min} · ρ_{max} = φ^{-3}$.
background
In the RecognitionBandGeometry module the cognitive band is bounded by two quantities derived from the golden ratio φ. The lower boundary is defined as ρ_min = φ^{-2} and the upper boundary as ρ_max = φ^{-1}. These appear as the foundational definitions for the geometry of recognition bands in the unification setting.
proof idea
The proof unfolds the definitions of the lower and upper boundaries then applies the ring tactic to verify the algebraic identity φ^{-2} · φ^{-1} = φ^{-3}.
why it matters
This identity closes a basic algebraic relation in the band geometry section of the unification module. It supports subsequent results on band sums and differences such as the sum of boundaries equaling one. Within the Recognition Science framework it reinforces the self-similar scaling governed by φ consistent with the forcing chain that fixes φ as the fixed point.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.