phi_golden_recursion
plain-language theorem explainer
The golden ratio φ, serving as the preferred aspect ratio for minimal J-cost rectangles, satisfies φ(φ − 1) = 1. Aesthetic theorists and architects modeling proportions in the Recognition Science framework would cite this result. The proof proceeds by unfolding the definition and reducing via the quadratic identity φ² = φ + 1 using linear arithmetic.
Claim. Let $φ$ be the golden ratio serving as the aesthetically preferred aspect ratio. Then $φ(φ - 1) = 1$.
background
The module on Golden Section in Architectural Proportion derives the golden section from the J-cost functional in Recognition Science. It predicts that the minimum J-cost rectangle under area constraint has aspect ratio φ:1, where J(r) = (r + 1/r)/2 - 1. This yields the recursion φ = 1 + 1/φ as the self-similar fixed point on the lattice.
proof idea
The term proof unfolds the definition of the aesthetically preferred aspect ratio to obtain phi. It introduces the hypothesis from phi_sq_eq asserting phi ^ 2 = phi + 1. A further hypothesis rewrites the square as a product, substitutes, and closes with linarith on the resulting linear equation.
why it matters
This theorem supplies the golden recursion component in the certificate record for the golden section properties. It realizes the structural identity φ = 1 + 1/φ highlighted in the module documentation for deriving beautiful proportions from J-cost minimization. Within the framework it supports the self-similar fixed point at T6 and the eight-tick octave structure for aesthetic predictions.
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