goldenDivision
The golden section of unit length is defined as the reciprocal of the golden ratio phi. Art historians and Recognition Science researchers cite this value when deriving sub-rectangle area ratios that lie on the phi-ladder for classical compositions. The definition is introduced by a direct one-line assignment to phi inverse with no lemmas or tactics applied at that step.
claimThe golden section of a unit-length interval is the real number $1/phi$, where $phi$ is the golden ratio satisfying the fixed-point equation $phi = 1 + 1/phi$.
background
The module ArtHistory.FibonacciInComposition treats the historical recurrence of the golden section in artistic composition from antiquity onward. Recognition Science predicts that the optimal division point for a length-L composition lies at L/phi, yielding sub-segments whose ratio is phi:1 and whose areas become integer powers of phi. The supplied definition supplies the concrete real number 1/phi that serves as the base unit for these ratios.
proof idea
The declaration is a direct definition that equates the symbol to the multiplicative inverse of phi. No lemmas are invoked; the three immediate dependent results unfold the definition and apply standard facts on positive reals such as inv_pos and mul_inv_cancel.
why it matters in Recognition Science
This definition anchors the structure FibonacciCompositionCert that packages the positivity, upper bound, and ratio properties required for certifying golden-section divisions. It realizes the RS prediction for artistic composition stated in the module documentation and connects to the phi-ladder landmark by providing the base whose powers generate the allowed area ratios. The module falsifier is any large-N eye-tracking study showing equal fixation density on golden-section versus non-golden-section layouts.
scope and limits
- Does not establish empirical preference for golden-section layouts in human vision.
- Does not derive phi from the forcing chain inside this module.
- Does not address compositions whose aspect ratio differs from phi.
- Does not quantify artistic optimality beyond the stated geometric ratios.
formal statement (Lean)
38def goldenDivision : ℝ := phi⁻¹
proof body
Definition body.
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