Proposition. The signed projective adapter preserves products and incidence [fcap-002R]
Proposition. The signed projective adapter preserves products and incidence [fcap-002R]
For the two orders in the preceding Convention, define \[\begin {aligned} \operatorname {toCore} &(s,c_0,c_1,c_2,c_{12},c_{20},c_{01},c_{012})\\ &=(s,c_0,c_1,c_{01},c_2,-c_{20},c_{12},c_{012}),\\ \operatorname {toNamed} &(r,r_0,r_1,r_{01},r_2,r_{02},r_{12},r_{012})\\ &=(r,r_0,r_1,r_2,r_{12},-r_{02},r_{01},r_{012}). \end {aligned}\] They are inverse linear maps on every eight-tuple. The minus sign is exactly \(e_{20}=-e_{02}\); substituting either displayed tuple into the other proves both round trips coordinate by coordinate.
This signed permutation also preserves the Clifford product, rather than merely being declared to do so. Encode the ascending blades by masks \(0,\ldots ,7\), with bit \(i\) representing \(e_i\), and put \[\sigma =(0,1,2,4,6,5,3,7),\qquad \epsilon =(1,1,1,1,1,-1,1,1).\] Then the \(i\)-th named blade is \(S_i=\epsilon _iC_{\sigma _i}\). For masks identified with subsets of \(\{0,1,2\}\), set \[g(m,n)=(-1)^{\#\{(i,j)\in m\times n:i>j\}} \prod _{i\in m\cap n}q_i, \qquad (q_0,q_1,q_2)=(0,1,1).\] The strict-inversion formula of Theorem [fcap-002O] gives \(C_mC_n=g(m,n)C_{m\mathbin {\triangle }n}\). If \(\sigma _k=\sigma _i\mathbin {\triangle }\sigma _j\), it follows that \[S_iS_j=\epsilon _i\epsilon _j\epsilon _k g(\sigma _i,\sigma _j)S_k.\] This checks every one of the \(64\) basis pairs, including the zero products caused by a repeated \(e_0\); bilinearity proves multiplication compatibility for arbitrary tuples. Addition, scalar multiplication, and the unit are preserved immediately.
The permutation \(\sigma \) preserves grade. Reversal multiplies a grade-\(k\) blade by \((-1)^{k(k-1)/2}\), so it commutes with the adapter. For exterior multiplication, the same crossing sign applies to disjoint masks and both sides vanish when masks overlap. Hence the adapter also preserves reversal and exterior multiplication. It therefore preserves inverses wherever they exist and, in particular, \[\operatorname {toCore}(MX\widetilde M) =\operatorname {toCore}(M)\operatorname {toCore}(X) \widetilde {\operatorname {toCore}(M)}.\] Under the ordered basis of Proposition [fcap-002P], these equations give the corresponding abstract-coordinate commuting square. They do not supply a named executable coordinate type or software-storage theorem.
For the projective representatives of the preceding Convention, direct exterior multiplication gives \[L(d,a,b)\wedge P(w,x,y)=(dw+ax+by)I.\] Since \(I\) is a nonzero basis blade, the point lies on the line precisely when \(dw+ax+by=0\); no inverse pseudoscalar is used. This agrees with Sokolov's incidence equation [sokolov2013foundations, (37), p. 25]. The four coordinate rows are \[\begin {aligned} [L(d,a,b)]_S&=(0,d,a,b,0,0,0,0),\\ [L(d,a,b)]_C&=(0,d,a,0,b,0,0,0),\\ [P(w,x,y)]_S&=(0,0,0,0,w,x,y,0),\\ [P(w,x,y)]_C&=(0,0,0,y,0,-x,w,0). \end {aligned}\] Thus the abstract incidence calculation and both coordinate presentations agree through the full adapter.
The ideal-direction sign is fixed by the same convention. Namely, \[e_0\wedge (ue_1+ve_2) =ue_{01}+ve_{02}=ue_{01}-ve_{20}=P(0,-v,u).\] The input pair \((u,v)\) is a line normal, while the resulting homogeneous point has direction \((-v,u)\). Direction \((u,v)\) itself is represented by \(P(0,u,v)\). This distinction keeps the algebraic encoding separate from a drawn arrow-orientation convention.
The result synthesizes Sokolov's projective encoding with the explicit blade kernel of Theorem [fcap-002O] and the finite coordinate refinement of Proposition [fcap-002P]. FCAP's current exterior-basis transport is noncomputable and does not yet expose these named conversion functions, an executable multiplication kernel, or compatibility theorems. Those implementation and external storage-order claims remain separate work.