rhoBandUpper_eq
The upper recognition band boundary equals the reciprocal of the golden ratio phi by direct definition. Researchers deriving band widths or phi-ladder scalings in the unification setting cite this equality when fixing the upper limit at phi inverse. The proof is a single reflexivity step that follows immediately from the definition of the boundary quantity.
claimThe upper boundary of the recognition band satisfies $ρ_{upper} = φ^{-1}$.
background
In the RecognitionBandGeometry module the upper cognitive band boundary is introduced as the real number equal to the reciprocal of phi, where phi is the self-similar fixed point obtained from the forcing chain. This definition supplies the starting value for all subsequent band relations, including the lower boundary, positivity statements, and the sum-to-one identity for complementary boundaries. The module imports the core constants and works inside the geometric layer of the unification framework that follows the eight-tick octave and D=3 results.
proof idea
The proof is a one-line reflexivity wrapper. Because the upper band boundary is defined to be exactly phi inverse, the stated equality holds by construction with no additional lemmas or reductions required.
why it matters in Recognition Science
This equality fixes the upper edge of the recognition band and thereby anchors the complementary relations that appear in the same module, such as the lower-boundary equality and the strict inequality between the two boundaries. It directly instantiates the phi value required by the self-similar fixed-point step of the forcing chain and supplies the concrete numerical anchor used in mass-ladder and Berry-threshold calculations. No open scaffolding remains at this point.
scope and limits
- Does not prove any property of phi beyond the reciprocal equality.
- Does not derive the lower band boundary or their difference.
- Does not connect the band to physical constants or particle spectra.
- Does not address higher-dimensional or non-octave extensions.
formal statement (Lean)
21theorem rhoBandUpper_eq : rhoBandUpper = phi⁻¹ := by
proof body
Decided by rfl or decide.
22 rfl
23