cos_pi_5_is_phi_2
The identity states that cosine of pi over five equals half the golden ratio. Researchers deriving constants from discrete eight-fold symmetries or linking circular measures to self-similar ratios would cite it. The proof rewrites via the standard cosine evaluation at that angle together with the definition of phi, then reduces the equality algebraically.
claim$cos(π/5) = φ/2$ where $φ = (1 + √5)/2$ is the golden ratio.
background
The module develops π from 8-tick geometry, where the circle arises as the continuous limit of eight discrete phases. Phi enters as the self-similar fixed point forced by the Recognition Composition Law and the J-uniqueness condition. The supplied doc-comment lists the exact pentagonal ties: golden angle 2π/φ², interior angle 3π/5, and the two cosine and sine identities that connect the circle directly to φ.
proof idea
One-line wrapper that applies the Mathlib identity for cosine at π/5, substitutes the definition of phi, and finishes with the ring tactic for algebraic simplification.
why it matters in Recognition Science
The result supplies the exact geometric bridge between π and φ inside the MATH-002 development. It instantiates the eight-tick octave constraint (T7) and the forced appearance of φ (T6) by exhibiting a concrete pentagonal relation that any 8-tick model must satisfy. The declaration therefore anchors the claim that π emerges from the same discrete symmetry that produces the golden ratio.
scope and limits
- Does not derive the cosine identity from Recognition Science axioms.
- Does not produce numerical bounds or series expansions for π.
- Does not treat other angles or higher multiples of π/5.
- Does not address convergence of the 8-tick discretization to the continuous circle.
formal statement (Lean)
130theorem cos_pi_5_is_phi_2 :
131 Real.cos (π / 5) = phi / 2 := by
proof body
Term-mode proof.
132 -- cos(π/5) = (1 + √5)/4 (Mathlib)
133 -- φ = (1 + √5)/2, so φ/2 = (1 + √5)/4
134 rw [Real.cos_pi_div_five, phi]
135 ring
136