Example. A rational motor checks every projective coordinate layer [fcap-002T]
Example. A rational motor checks every projective coordinate layer [fcap-002T]
Take \[c=\frac 35,\qquad s=\frac 45,\qquad t=(4,0).\] Then \[U=\frac {3-4e_{12}}5,\qquad R_U=\frac 1{25}\begin {pmatrix}-7&-24\\24&-7\end {pmatrix},\] and the ordered motor is \[M=T(4,0)U =\frac 35-\frac 45e_{12}-\frac 85e_{20}-\frac 65e_{01}.\] Its full named tuple and the ascending core tuple supplied by Proposition [fcap-002R] are \[\begin {aligned} \operatorname {named}(M) &=\left (\frac 35,0,0,0,-\frac 45,-\frac 85,-\frac 65,0\right ),\\ \operatorname {core}(M) &=\left (\frac 35,0,0,-\frac 65,0,\frac 85,-\frac 45,0\right ). \end {aligned}\] Thus the calculation enters the algebra through all eight-coordinate conversion data, including the \(e_{20}=-e_{02}\) sign.
The point and line \[P_0=P(1,3,-1),\qquad L_0=L(3,-1,0)\] are incident. The motor action gives \[\begin {aligned} MP_0M^{-1}&=P\left (1,\frac {103}{25},\frac {79}{25}\right ),\\ ML_0M^{-1}&=L\left (\frac {47}{25},\frac 7{25},-\frac {24}{25}\right ). \end {aligned}\] The output incidence check is \[\frac {47}{25} +\frac 7{25}\frac {103}{25} -\frac {24}{25}\frac {79}{25}=0.\] The output point has tuples \[\begin {aligned} \operatorname {named}(MP_0M^{-1}) &=\left (0,0,0,0,1,\frac {103}{25},\frac {79}{25},0\right ),\\ \operatorname {core}(MP_0M^{-1}) &=\left (0,0,0,\frac {79}{25},0,-\frac {103}{25},1,0\right ). \end {aligned}\] This checks the geometric action and the signed coordinate adapter on the same exact example.
Reversal gives the algebra inverse, and sandwiching the two outputs by \(\widetilde M\) and \(M\) returns both original representatives. In the same ordered factorization, \[M^{-1}=M\left (\left (\frac {28}{25},\frac {96}{25}\right ),U^{-1}\right ).\] Composition order is visible: reversing the factors gives \[UT(4,0):P(1,3,-1)\longmapsto P(1,-1,7),\] which differs from the output above. These are exact rational identities, not floating-point or private-software evidence. FCAP does not yet expose the named executable product and conversion theorems needed to certify this calculation as a public implementation refinement.