Noam Zeilberger's comments on the Jackson-Richmond Attempted Proof of the Four Color Theorem

These are Noam Zeilberger's comments on the Jackson-Richmond attempted human-readable proof of the Four Color Theorem . I thank him for permission to post them here.

Just wanted to let you know that I read the paper by Jackson & Richmond, and I think their proof attempt has major bugs. I explained why in a discussion with John Baez and Simon Pepin Lehalleur over here, that Baez also linked to over here.

A major issue with the Jackson-Richmod proof is that they claim to have constructed a subclass of 4-colorable maps having positive density in the class of *all* rooted planar maps (as originally counted Tutte, which include maps with loops and bridges)...but that's impossible! By 4CT, the density of 4-colorable planar maps is equal to the density of loopless planar maps (or equivalently bridgeless planar maps), and by Tutte's enumeration results, that density is equal to

A000260(n) / A000168(n)

which grows exponentially small as a function of n. (Wolfram Alpha tells me that asymptotically it tends to c*(64/81)n, where c =4*sqrt(2/3)/9.) So the claim at the end of 2212.09835 to have constructed a family 4-colorable maps having density at least 1/1458 in the class of all planar maps must be false.

I sent an email to David Jackson and Bruce Richmond asking about this issue, but didn't get a response yet.

Otherwise, there are other major issues with the proof, discussed in the thread I linked to above.

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