Antiunitary AZ symmetry classes give Clifford extension problems

Apply [F1]. On the underlying real inner-product space, $I^2=-1$, $I^{\ast }=-I$ and $IH=HI$. Each antiunitary is orthogonal and anticommutes with $I$. An orthogonal generator of square $ϵ\in \{1,-1\}$ has adjoint $ϵ$ times itself. Thus positive generators are symmetric and negative generators skew-symmetric in this same metric. In the simultaneous case $TC$ is unitary and anticommutes with $H$; its square is $tc$, so the chiral involution is $TC$ for $tc=1$ and $iTC$ for $tc=-1$.

Put $r=\mathrm{diag}(1,-1)$, $s=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and $J=rs$. Then $r^2=s^2=1$, $J^2=-1$, and $J$ anticommutes with $r,s$. Send sign-reversed old generators to $x\otimes J$ and two new positive generators to $1\otimes r,1\otimes s$. Their squares and anticommutators prove a homomorphism ${\mathrm{C}\mathrm{l}}_{b+2,a}\to {\mathrm{C}\mathrm{l}}_{a,b}\otimes M_2(\mathbb{R})$. Its image contains $1\otimes J$, hence all $x\otimes 1$ and both tensor factors. Equal PBW dimensions [F2] make it an isomorphism. An old negative added mass becomes a positive added mass under this map. Together with [F3], this reduces a positive-mass problem with fixed signature $(a,b)$ to index $a-b$, and a negative-mass problem to index $b+2-a$. The matrix-unit functors extracting $E_{11}W$ and reconstructing $W$ preserve metrics, fixed representation identifications, continuous homotopies and direct-sum reference stabilization. Therefore this is equivalence of mass problems, not only an equality of algebra dimensions.

If only $C$ is present use positive mass $M=H$ and fixed generators $e_1=C,e_2=IC$. Their squares are both $c$, they anticommute, and both anticommute with $M$. Conversely $I=c{e}_2{e}_1$ is an orthogonal complex structure, $C=e_1$ is antiunitary for it, and a positive symmetric mass anticommuting with both fixed generators commutes with $I$ and anticommutes with $C$. Thus this is an exact invertible dictionary: class D has fixed signature $(2,0)$ and class C has fixed signature $(0,2)$, both with positive mass.

If only $T$ is present use negative mass $M=IH$ and fixed generators $e_1=T,e_2=IT$. Their squares are both $t$, they anticommute with one another and with $M$. Conversely $I=t{e}_2{e}_1$ and $H=-IM$ recover a complex-linear self-adjoint involution commuting with the antiunitary $T=e_1$. The matrix $M$ is skew-symmetric and has square $-1$, so these formulas recover all adjoint conditions too. Thus AI has fixed signature $(2,0)$ and AII fixed signature $(0,2)$, with a negative mass.

When both antiunitaries occur use positive mass $M=H$ and $e_1=C,e_2=IC,e_3=ITC$. The fixed squares are $(c,c,-tc)$. They pairwise anticommute: the first pair was checked in step 2.1, $e_{3}C=cIT$ while $C{e}_3=-cIT$, and $e_3$ commutes with $I$, so it also anticommutes with $IC$. Each anticommutes with $H$. Conversely set $C=e_1$, $I=c{e}_2{e}_1$ and $T=-cI{e}_{3}C$. Multiplying these formulas using the three Clifford relations gives $I^2=-1$, $CI=-IC$, $TI=-IT$, $C^2=c$, $T^2=t$, $TC=CT$; the mass commutes with $I,T$ and anticommutes with $C$. This recovers the exact normalized input algebra. The fixed signatures are BDI $(2,1)$, DIII $(3,0)$, CII $(0,3)$ and CI $(1,2)$.

Substituting the signatures of step 2.1, step 2.2 and step 3.1 gives exactly the eight indices in the Statement, with the eightfold identification of [F2]. Each additional fixed negative Dirac generator increases $b$ by one in the positive-mass model and therefore decreases its index by one. These are fixed anticommuting matrices, not suspension coordinates of an arbitrary parameter family. For the complex classes, A has no fixed generator and a positive Hermitian mass, while AIII has the fixed chiral involution and a positive anticommuting mass. The same Pauli construction over $\mathbb{C}$ gives twofold matrix periodicity; multiplication of a generator by $i$ changes its square, so the complex index is modulo two. This proves all class labels and the dimension shift with their declared signs.