Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

A non-equivalent collapsed quadrilateral

One certified four-face slicing exchange › A non-equivalent collapsed quadrilateral · lines 2641–2666

Consider

A = (- 1, 0), B = (1, 0), C = (0, 1 - t), D = (0, t - 1) .

For $t \neq = 1$ , the written boundary order $A B C D$ has crossing segments $A B$ and $C D$ . The signed triangle areas are

Area_{sgn} (A, C, D) = t - 1, Area_{sgn} (B, C, D) = 1 - t .

The convex hull has area $2 ∣ 1 - t ∣$ , which vanishes at $t = 1$ ; the entire two-dimensional patch collapses to a segment. The proposed replacement $A B$ has constant length $2$ , rather than being born from zero. Relabelling orientations does not force an exchange of geometric diagonals.

With spacetime coordinate $T = t$ , the $A, B$ trajectories are timelike and the $C, D$ trajectories are null. These causal classifications are not by themselves a prohibition on polygonal coordinates. The failures relevant here are the crossing boundary, the collapse of the whole patch, and the absence of a replacement edge emerging from zero. This quadrilateral is not the certified exchange.

Displayed formulas in this passage

unlabeled · line 2644
$A = (- 1, 0), B = (1, 0), C = (0, 1 - t), D = (0, t - 1) .$
unlabeled · line 2650
$Area_{sgn} (A, C, D) = t - 1, Area_{sgn} (B, C, D) = 1 - t .$