Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Gaussian integral ∫−∞∞e−x2 dx=π

Statement

∫−∞∞e−x2 dx=π.

Facts & Assumptions

Given: Write I:=∫−∞∞e−x2 dx.

[L1]

The integral I exists as a finite positive real number (The improper integral of e−x2 over R is finite and positive).

[L2]

One has I2=∫R2e−(x2+y2) d(x,y) (The square of the one-dimensional Gaussian integral is the plane Gaussian integral).

[L3]

The plane integral equals π (The plane Gaussian integral equals π by polar coordinates).

[L4]

Every nonnegative real has a unique nonnegative square root (Square roots exist: a unique a≥0 with (a)2=a; the positives are {x2:x≠0}).

Proof

technique · direct
1.1L1L2L3

By [L1], I>0, and [L2] with [L3] gives I2=π.

2.1step 1.1L4∎

Since I is nonnegative, uniqueness in [L4] identifies it with π.

Depends on

Dependency tree · 2 levels

Sources