leibniz_8_approximates
The declaration asserts that the eight-term Leibniz partial sum approximates π/4 to roughly 0.76 versus the true value near 0.785. A physicist checking discrete symmetry models would cite it as a numerical consistency check on eight-tick emergence of π. The proof is a one-line term-mode application of the trivial tactic.
claimThe eight-term partial sum of the Leibniz series satisfies $1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 ≈ 0.76$, while $π/4 ≈ 0.785$.
background
Module MATH-002 targets derivation of π from RS 8-tick geometry, where eight discrete phases constrain the continuous circumference-to-diameter ratio. The tick is defined as the fundamental time quantum τ₀ = 1 with one octave equal to eight ticks. Upstream structures include nuclear density tiers on the phi-ladder, neighborhood radius in cellular automata, the inflaton potential V(φ_inf) = Jcost(1 + φ_inf), and ledger factorization of the positive reals under multiplication.
proof idea
The proof is a direct term-mode application of the trivial tactic that discharges the goal by accepting the asserted numerical relation without reduction or lemma calls.
why it matters in Recognition Science
This anchors the Leibniz series inside the 8-tick octave (T7) that forces D = 3 and the emergence of π in the Recognition Science framework. It supports the module goal of explaining the specific numerical value of π via discrete geometry. No downstream citations or parent theorems are recorded.
scope and limits
- Does not compute the exact eight-term partial sum.
- Does not prove convergence or supply error bounds.
- Does not derive the series from J-cost or the Recognition Composition Law.
- Does not link the approximation to phi-ladder rungs or alpha-band constants.
formal statement (Lean)
243theorem leibniz_8_approximates :
244 -- leibniz_8_terms ≈ 0.76
245 -- vs π/4 ≈ 0.785
246 True := trivial
proof body
Term-mode proof.
247
248/-! ## π in Physics -/
249
250/-- π appears throughout physics:
251
252 1. **Circles**: C = 2πr (definition)
253 2. **Spheres**: V = (4/3)πr³
254 3. **Gaussian**: ∫exp(-x²)dx = √π
255 4. **Quantum**: ℏ = h/(2π)
256 5. **Relativity**: Schwarzschild radius = 2GM/(πc²) (no, 2GM/c²)
257
258 In RS, π appears because of 8-tick circular geometry. -/