Here is a list of statements that can serve as axioms for the catastrophetheoretic study of syntax. Some of these are in fact theorems. Compare with Thom's Résumé des thèses on p. 321 of Stabilité ....
the simple minimum  x^{2} 
the fold  x^{3} + ax 
the cusp  x^{4} + ax^{2} + bx 
the swallowtail  x^{5} + ax^{3} + bx^{2} + cx 
the butterfly  x^{6} + ax^{4} + bx^{3} + cx^{2} + dx 
the parabolic umbilic  x^{2}y + y^{4} + ax^{2} + by^{2}  cx + dy 
the elliptic umbilic  x^{3}y  3xy^{2} + a(x^{2} + y^{2})  bx  cy 
the hyperbolic umbilic  x^{3} + y^{3} + axy  bx  cy 

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