New Definition

The 9 pseudoline counterexample means my prior definition of linear space is inadequate for constructibility. A sufficient condition for constructibility is that the space has a cospace. Then the proof of constructibility is perhaps possible as follows.

If pseudolinear space S with m boundaries has cospace C then S is constructible as hyperplanes.

Assume the conclusion for spaces with only m-1 boundaries. Construct the hyperplanes for the subspace S' of S with one boundary b removed. The subspace C' of C with vertex b removed is a cospace of S'. C' is constructible, so choose a point in the superregion. That is a consruction of b, as required.

 The cospace condition is also necessary because a constructible space has a constrctible cospace.

This is still not complete, because the cospace is not unique. My technique for constructing a cospace produces a particular cospace which may be different from the subspace of the cospace obtained by removing a vertex. I guess I could patch this by requiring the space to have one of the cospaces that my cospace construction would produce, but I have to express that without reference to construction. Another option that might exist is to prove that every cospace of a constructible space is constructible.

Counterexample

identify

 Among the most satisfying for me interactions on social media such as Bluesky could be described as codependent. We argue incessantly without convincing each other. One such climate denier was fooled by anti-doomists into believing the keeling curve is exponential when it is obviously parabolic. The climate denier prefers for it to be exponential because anti-doomists falsely believe worse is less motivating than less than worse. Anyway, after trouncing the climate denier with the rolling average of the derivative of the keeling curve, I suggested taking me down with a counter example to my proof of the constructibility of linear spaces, and the climate denier took me up on the challenge. By then, I was arguing with copy pasted images of professionally formatted text from AI. After some undefined insistence from the AI, I got a reference out of it, and it turns out there is a counter example buried in the literature that I never could have found on my own. One thing that nagged me about my proof was that it did not depend on my definition of linear. Now I believe my proof is still good, and contains a better definition of linear. I just need to state precisely what that is.