Though it's hard once again to understand. When are two nodes coincident? By definition you can only have one character in one place on the screen. Is he talking about indentation, that the same level of indentation can mean different things? That seems true of ETN as well, from what I can tell.

The pencil scratchings on the screenshots don't help either.

Sorry, the lines in the geometric mapping of the source code are coincident. In drawing B) in the visual proof, the edges which connect the child nodes to their parents intersect and/or are coincident. So when you put Lisp source code onto graph paper, and draw boxes around the nodes, and line segments for edges, it shows why Lisp source is not a geometric language (I define a geometric language as one where there are no intersecting or coincident line segments).

Now, figure A) shows the same Lisp code, formatted differently, in a way that is a geometric language. But as you can see, that code is standard TN/ETN. Or perhaps another way to put it is ETNs are just Lisps with a whitespace syntax and no parentheses. Another reader on HN pointed me to I expressions (https://srfi.schemers.org/srfi-49/srfi-49.html), which I hadn't seen before and is 90% of the way there to TN and ETNs. The creator of I-Expressions have communicated briefly over email now and are going to be talking soon.

Anyway, perhaps another term for Tree Notation/ETNs is "Geometric Lisp", or "2-Dimensional Lisp". I'm not wedded to the terms TN/ETNs, although I do think it's better to have new terms, because I think these will come to dominate the usage of Lisp.

Still working on more updates and evidence on why I think these will be so big.