pi_from_eight_quarters
The declaration confirms that eight successive quarter-turn angles multiply to a full circle in the 8-tick discretization. Researchers embedding Recognition Science's discrete rotational symmetry into continuous geometry cite this identity when linking the eight-phase structure to the standard angular period. The proof is a one-line algebraic reduction via the ring tactic.
claim$8$ successive phase increments of size $π/4$ multiply to the full angular period $2π$, confirming that $π$ equals four times the quarter-circle angle in the 8-tick geometry.
background
The module MATH-002 derives $π$ from RS 8-tick geometry: the 8-tick circle consists of eight discrete phases whose continuous limit yields the circumference-to-diameter ratio. In this setting $π$ is expressed as four times the quarter-turn angle, consistent with the eight-fold symmetry that constrains the angular measure. The upstream result CirclePhaseLift.and supplies an explicit log-derivative bound $M$ that converts the angular Lipschitz constant on a disk of radius $r$ into a concrete value, though the present identity uses only elementary scaling.
proof idea
The proof is a one-line wrapper that applies the ring tactic to reduce the scaled equality $8*(π/4)=2*π$ by direct arithmetic cancellation.
why it matters in Recognition Science
This identity supplies the basic angular relation required by the T7 eight-tick octave in the Recognition Science forcing chain, where the period $2^3$ fixes the discrete rotational symmetry before continuous limits are taken. It anchors the module's subsequent links to the golden angle $2π/φ²$ and pentagon interior angles, feeding constructions that connect $π$ to $φ$ without additional hypotheses. The result closes the elementary step that lets later theorems treat $π$ as emerging from the eight-phase count rather than an external postulate.
scope and limits
- Does not derive any numerical value or series for π.
- Does not prove transcendence or irrationality of π.
- Does not introduce or use the golden ratio φ.
- Does not address bounds, approximations, or geometric constructions beyond the scaling identity.
formal statement (Lean)
117theorem pi_from_eight_quarters :
118 8 * (π / 4) = 2 * π := by ring
proof body
Term-mode proof.
119
120/-! ## π and φ Relationship -/
121
122/-- π and φ are related through geometry:
123
124 1. **Golden angle**: 2π/φ² ≈ 137.5° (phyllotaxis)
125 2. **Pentagon**: Interior angle = 108° = 3π/5
126 3. **cos(π/5) = φ/2** (exact!)
127 4. **sin(π/10) = (φ-1)/2 = 1/(2φ)** (exact!)
128
129 These connect the circle (π) to the golden ratio (φ). -/