Catalan's Constant: Reported Irrationality Proof

About this video

Catalan's constant is irrational
Can this number ever be an exact fraction? OpenAI's manuscript reports that Catalan's constant G = sum from j=0 to infinity of (-1)^j/(2j+1)^2 is irrational. The paper uses a determinant argument with arithmetic lower bounds and analytic upper bounds. A long decimal expansion or a converging finite sum alone does not prove irrationality.

Paper: Catalan's constant is irrational
github.com/openai/math/blob/main/preprints/Catalans-constant-…

About the animation: Gold and cyan ribbons distinguish the alternating positive and negative terms of the defining series. Rounded surfaces, luminous edges and moving reflection highlights make the sculptural metaphor readable in depth. Two bright tips draw the ribbons and meet at an artistic target on the soundtrack accent at 7.6333 seconds. Their dimensions and endpoint are not a numerical embedding of the terms, the partial sums or G. The displayed partial sums are genuinely computed from 6 through 96 terms, then reversed to close the loop. This illustrates the defining series, not the determinant argument or a visual proof of irrationality.

GitHub repository: github.com/openai/math
Related formalization scope: github.com/openai/math/blob/main/lean/docs/005.md
The linked scope documents irrationality of Catalan's constant. The formalization was not executed or audited in this production.

Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit.

'Inflection' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
scottbuckley.com.au/library/inflection/
Licence: creativecommons.org/licenses/by/4.0/
Music edits: excerpted, crossfaded for looping, and gain adjusted.