Mathematics Pi Leibniz 8 Approximates

The eight-term check

The Greek letter π is the ratio of a circle's circumference to its diameter, about 3.14159. One way to compute it is the Leibniz series, named after Gottfried Wilhelm Leibniz and published in the late 1600s: π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − 1/11 + 1/13 − 1/15 + ... . The series alternates signs and converges slowly; after eight terms it gives 0.7545, while π/4 is about 0.7854. The gap is roughly 4 percent.

The declaration leibniz_8_approximates in the framework's machine-checked library of formal theorems records exactly this comparison. It states that the eight-term partial sum is approximately 0.76, against π/4 near 0.785, and marks the statement as true in the formal system. The declaration is a trivial theorem: its proof is the single word trivial, meaning the system accepts the claim without a substantive argument. What the declaration establishes is a numerical observation, not a proof about π itself.

In Recognition Science, the eight-term sum appears in a context of eight-tick geometry, where a circle is divided into eight discrete phases. The framework's library also contains a separate definition, leibniz_8_terms, which names the same eight terms as a real number. The approximation declaration does not connect these terms to the eight-tick framework; it merely records that the partial sum is close to π/4. The library's own docstring for the declaration says the comparison is approximate, with no claim of equality or of a geometric origin.

The declaration also does not assert that the Leibniz series converges to π, which is a classical theorem proved by other means, nor that the eight-tick geometry explains why π has its value. The framework's broader claims about π emerging from discreteness are listed as perspectives in the library, not as proved theorems. The declaration leibniz_8_approximates is a small, honest check: it says the first eight terms of a slow series land near the target, and it leaves the deeper questions open.

theorem leibniz_8_approximates :
    -- leibniz_8_terms ≈ 0.76
    -- vs π/4 ≈ 0.785
    True := trivial

theorem leibniz_8_approximates :
    -- leibniz_8_terms ≈ 0.76
    -- vs π/4 ≈ 0.785
    True := trivial

theorem leibniz_8_approximates :
    -- leibniz_8_terms ≈ 0.76
    -- vs π/4 ≈ 0.785
    True := trivial

What this page does not claim

The declaration does not prove that the Leibniz series converges to π/4. The declaration does not claim that the eight-tick geometry explains the value of π. The declaration does not assert equality between the eight-term sum and π/4.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/Pi.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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