rhoBandUpper
rhoBandUpper sets the upper boundary of the recognition band to the reciprocal of the golden ratio. Researchers deriving band ratios or products in the unification layer cite it to close the complementary pair with the lower boundary. The declaration is a direct abbreviation with no lemmas or computation.
claimThe upper boundary of the recognition band satisfies $ρ_{upper} = φ^{-1}$.
background
The RecognitionBandGeometry module defines a recognition band on the real line whose endpoints are built from the golden ratio phi, the self-similar fixed point forced by the Recognition Composition Law. rhoBandUpper supplies the upper endpoint directly as the multiplicative inverse of phi, pairing with the lower endpoint to satisfy the sum-to-one identity shown in bandBoundaries_sum_to_one. This construction sits inside the broader unification setting that imports constants and applies the phi-ladder geometry.
proof idea
The declaration is a one-line abbreviation that directly assigns the value phi inverse.
why it matters in Recognition Science
It supplies the upper edge required by parent results bandBoundaries_sum_to_one, bandBoundaries_product, bandUpper_ratio, and one_sub_rhoBandUpper_eq_rhoBandLower. These theorems establish the geometric constraints on recognition bands and feed the unification layer. The definition anchors the phi-derived boundaries that appear in the eight-tick octave and D=3 spatial structure of the Recognition Science framework.
scope and limits
- Does not derive the boundary value from the forcing chain T5-T8.
- Does not assign a physical observable to the band.
- Does not prove positivity or location inside the unit interval.
- Does not address invariance under the Recognition Composition Law.
formal statement (Lean)
16def rhoBandUpper : ℝ := phi⁻¹
proof body
Definition body.
17