The Schloss Toolkit: Exact Criteria for Linear Flow Embedding, Defect Detection, Degeneracy and Collision

We state four exact criteria extracted from the Jonah family of pedal maps. The first is a linear flow-embedding criterion: a real 2 times 2 matrix with two distinct real eigenvalues equals e to the L for some real L if and only if both eigenvalues are positive. The determinant obstruction det (e to the L)=e to the tr L >0 follows directly. The second is a defect detector for an algebraic correspondence, based on the factorisation F(s,s)=s squared (1 minus s squared ) Delta t(s). The third is a degeneracy test for singular 2 times 2 matrices, which is a restatement of the Cayley–Hamilton theorem. The fourth is a transcritical collision certificate: an analytic fixed-point function factorises as x(x minus x J( mu ))k( mu,x) with k is not equal to 0 on a neighbourhood of the collision. Applied to the Jonah branch, the linear criterion gives a positive certificate exactly for t in (t 0,1), with t 0=(3+ the square root of 3)/6. The first and fourth criteria are substantive; the second is an exact factorisation for this family; the third carries no independent content. The multiplier formulas are imported from the companion papers and are not re-derived here. The nonlinear embedding question for t in (t 0,1) is open.