leibniz_8_approximates
plain-language theorem explainer
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.
Claim. The 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
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.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.