phi
plain-language theorem explainer
Golden ratio φ is defined as the positive real (1 + sqrt(5))/2. Workers on self-similar fixed points in J-cost geometry cite it when minimizing total bond cost or deriving the eight-tick octave. The declaration is a direct noncomputable assignment with no lemmas or proof steps.
Claim. Let $φ = (1 + √5)/2$.
background
The module develops log-domain J-cost geometry for foundation paper F1. It collects identities for the canonical cost J(x) = ½(x + x⁻¹) − 1, proves J(e^ε) = cosh(ε) − 1, and shows that the geometric mean minimizes total bond cost while simultaneous and sequential descent differ.
proof idea
The declaration is a direct noncomputable assignment of the closed-form algebraic expression for the golden ratio in the reals.
why it matters
This supplies the self-similar fixed point required at forcing-chain step T6. It underpins the phi-ladder mass formula, the eight-tick octave, and geometric-mean optimality in the F1 paper. The value also enters bounds on the fine-structure constant inside the alpha band.
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papers checked against this theorem (showing 1 of 1)
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"A complexity map for spotting forbidden patterns in ordered graphs"
"There exists a reduction from the P-Detection problem ... to the detection of a k-clique in a graph G' with kn vertices ... complexity nω(⌊k/3⌋,⌈k/3⌉,⌈(k−1)/3⌉)"