Positive Clifford mass differences give the actual KO classes

Transport $H_0$ by $w$ so that both masses act on the same fixed representation. Define first an auxiliary comparison ${\mathrm{\Phi }}_q$ with the positive-rank convention, and finally set ${\mathrm{\Psi }}_q=-{\mathrm{\Phi }}_q$ in EVERY degree. For $q=0$ put ${\mathrm{\Phi }}_0=\dim W_H^{+}-\dim W_{H_0}^{+}$. Equal total dimensions give $-{\mathrm{\Phi }}_0=\dim W_H^{-}-\dim W_{H_0}^{-}$. The uniform negation is part of the comparison convention; it will also negate integer-valued Hopf classes in degrees divisible by four.

For $q=1$ split $W=W^{+}\oplus W^{-}$ by $e_1$. Write the masses as $H=\left(\begin{array}{cc}0& g^{\ast }\\ g& 0\end{array}\right)$ and $H_0=\left(\begin{array}{cc}0& g_0^{\ast }\\ g_0& 0\end{array}\right)$, where $g,g_0:W^{+}\to W^{-}$ are isometries. Clutch on $S^1$ using the two values $I,g_0^{-1}g$ on $S^0$ and subtract the common rank. This defines ${\mathrm{\Phi }}_1$. The original reference identification remains present in $g_0^{-1}g$; replacing it by an abstract isomorphism class of one extended module would discard this datum.

For $q\ge 2$, use $e_2$ to identify $W^{-}$ with $W^{+}$, making $e_2=\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right)$. The remaining fixed generators and masses have blocks $e_{i+2}=\left(\begin{array}{cc}0& -J_i\\ J_i& 0\end{array}\right)$ and $H=\left(\begin{array}{cc}0& -A\\ A& 0\end{array}\right)$, with an analogous $A_0$. Direct multiplication gives $J_i^{\ast }=-J_i$, $J_i^2=-I$, pairwise anticommutation, $A^{\ast }=-A$, $A^2=-I$, and $A{J}_i+J_{i}A=0$, with the same relations for $A_0$. For $x=(x_0,x_1,\dots ,x_{q-2},t)\in S^{q-1}$ define $F(x)=x_{0}I+{\sum }_i{x}_i{J}_i+tA$ on $t\ge 0$ and $F(x)=x_{0}I+{\sum }_i{x}_i{J}_i-t{A}_0$ on $t\le 0$. The formulas coincide at $t=0$ and satisfy $F^{\ast }F=I$ on each hemisphere by the anticommutation identities. They therefore define a based orthogonal clutching map. Let ${\mathrm{\Phi }}_q$ be its reduced clutching class on $S^q$.

Choose positive orthogonal generators $a_1,\dots ,a_8$ on the sixteen-dimensional irreducible ${\mathrm{C}\mathrm{l}}_{8,0}$ module $S$, and put ${\mathrm{\Gamma }}_8=a_1\cdots a_8$. Then ${\mathrm{\Gamma }}_8^2=I$, ${\mathrm{\Gamma }}_8^{\ast }={\mathrm{\Gamma }}_8$ and ${\mathrm{\Gamma }}_8{a}_j=-a_j{\mathrm{\Gamma }}_8$. The graded negative $C_8$ module $B$ with grading ${\mathrm{\Gamma }}_8$ and generators $b_j={\mathrm{\Gamma }}_8{a}_j$ has ordered volume $b_1\cdots b_8={\mathrm{\Gamma }}_8$. It is therefore the volume-graded irreducible symbol module of [F4], with $s_8(B)=\beta$. Define $T$ by old fixed generators $E_i=e_i\otimes {\mathrm{\Gamma }}_8$, eight new fixed generators $F_j=1\otimes a_j$, masses $H^{\mathrm{\prime }}=H\otimes {\mathrm{\Gamma }}_8$, $H_0^{\mathrm{\prime }}=H_0\otimes {\mathrm{\Gamma }}_8$, and reference $w^{\mathrm{\prime }}=w\otimes 1$. All Clifford and mass identities hold by direct multiplication.

If $A=A_0$, the clutching map factors through the closed upper hemisphere by $t\mapsto \mathrm{\mid }t\mathrm{\mid }$. That disk contracts to the chosen equatorial basepoint while fixing the point, so its clutching map is based nullhomotopic. Common reference blocks contribute zero. Mass homotopies give homotopies of clutching maps; simultaneous coordinate changes conjugate the clutching map and give isomorphic bundles; direct sums give block sums and hence sums of reduced classes. These conclusions hold also in step 1.1 and step 1.2. Reversing the two masses reflects the clutching sphere and negates its class by the reflection/block-rotation proof in [F2]. Thus ${\mathrm{\Phi }}_q$, and consequently ${\mathrm{\Psi }}_q$, respect stable difference relations and inverses.

We compute the actual component generators rather than infer them from the KO table. At $q=0$, real involutions are classified up to orthogonal conjugacy by their eigenspace dimensions; their fixed-rank orbits are real Grassmannians. At $q=1$ the mass parameter is $g\in O(n)$ relative to $g_0$. At $q=2$ it is an orthogonal complex structure $A$, with orbit $O(2n)\mathrm{/}U(n)$: an adapted basis consists of pairs $(v,Av)$, and its stabilizer is precisely the complex-linear isometry group. The latter is connected by complex plane rotations, whereas $O(2n)$ has its two determinant components. At $q=4$, the fixed $J_1,J_2$ give a quaternionic multiplicity space. Put $K=J_1{J}_2$; then $A=KS$ with $S=-KA$ a quaternionic-linear symmetric involution. Its eigenspace orbit is $Sp(k+m)\mathrm{/}(Sp(k)\times Sp(m))$. Hence the nonzero stable difference groups in these rows are the real rank difference, the determinant parity, the complex-orientation parity, and the quaternionic rank difference. Their elementary differences are respectively one opposite real line, one reflected real isometry, one reversed complex line, and one opposite quaternionic line.

To identify the exact symbol, first take an OPPOSITE mass pair $(H,-H)$ on a summand. For $q\ge 2$, $A_0=-A$, so its transition is the globally linear map $h(x)=x_{0}I+{\sum }_i{x}_i{J}_i+tA$. On $W$ put grading $e_1$ and negative-square Clifford generators $c_1=e_1{e}_2,\dots ,c_{q-1}=e_1{e}_q,c_q=e_{1}H$. Each squares to $-I$, is skew-adjoint and odd for this grading, and the generators anticommute. Identify the odd space with the even space by $-e_2$. The even-to-odd blocks then become $I,J_1,\dots ,J_{q-2},A$, respectively. Thus the actual ABS symbol of this graded $C_q$ module $M$ is EXACTLY $h$, and ${\mathrm{\Phi }}_q(H,-H)=s_q(M)$ whereas ${\mathrm{\Psi }}_q(H,-H)=-s_q(M)$. At $q=1$ use grading $e_1$ and generator $e_{1}H$; the two opposite boundary isometries normalize to $I,-I$ and give the same statement for an elementary reflected line. At $q=0$ the auxiliary positive-rank convention corresponds to the elementary graded real line. This use of a module is confined to an actual opposite DIFFERENCE pair whose reference is retained.

This stabilization is an equivalence of fixed-representation mass problems. The $F_j$ generate $1\otimes M_{16}(\mathbb{R})$. An operator anticommuting with every $F_j$, multiplied by $1\otimes {\mathrm{\Gamma }}_8$, commutes with that matrix algebra and hence is $K\otimes 1$; consequently the original operator is $K\otimes {\mathrm{\Gamma }}_8$. Matrix units extract its multiplicity space, and multiplication by those units reconstructs it. This recovers the old fixed generators and masses, with their homotopies, sums and reference identifications. The classification of [F1] supplies the same decomposition for every admissible representation of the stabilized fixed algebra. Hence $T:D_q\to D_{q+8}$ is an isomorphism. In particular it transports the elementary component generators from step 2.2 and common folded-null blocks from step 2.1 to all nonnegative degrees.

For completeness, the other four multiplicity models have connected mass spaces. At $q=3$, a further complex structure anticommuting the fixed one is a quaternionic structure on a complex space, with orbit $U(2n)\mathrm{/}Sp(n)$. At $q=5$, the two quaternionic isotypic halves must have equal dimension and a mass is an isometry between them, giving $Sp(n)$. At $q=6$, an orthogonal quaternionic-linear complex structure has orbit $Sp(n)\mathrm{/}U(n)$: its complex $+i$ eigenspace and its right-$j$ image provide an adapted quaternionic frame. At $q=7$, a further anticommuting structure determines an antiunitary involution on that eigenspace, hence a real form, with orbit $U(n)\mathrm{/}O(n)$. These descriptions follow by diagonalizing the fixed Clifford algebra of [F1] and taking adapted frames; their stabilizers are exactly the transformations commuting with the indicated structures. Each quotient is connected because $U$ and $Sp$ are connected, the latter by quaternionic-frame induction over the connected sphere fibrations. Thus $D_3,D_5,D_6,D_7$ vanish. The parity and equal-rank conditions are imposed on admissible representations before stabilization.

A reflected real line in step 1.2 gives transition values $+1,-1$ on $S^0$, so transport around the clutched line returns a vector to its negative. This is the Möbius line and is nonzero stably because its determinant is negative. At $q=2$ take $A=J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $A_0=-J$ on one complex line. The two hemispheres in step 1.3 combine to $F(x_0,t)=x_{0}I+tJ$, the one-turn rotation on $S^1$ and the underlying real transition of the complex Hopf line. Its lift to $S^3\to SO(3)$ is $\cos (\theta \mathrm{/}2)+k\sin (\theta \mathrm{/}2)$, which runs from $1$ to $-1$, so it is not nullhomotopic. The stable inclusion preserves its nonzero class by [F3]. Replacing the direction of the difference negates this class, which does not change a group of order two. Thus ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ send the actual parity generators to the stated actual bundle generators.

For $q\ge 1$, apply step 2.3 to the stabilized opposite pair, using grading $E_1=e_1\otimes {\mathrm{\Gamma }}_8$. Its negative generators, in order, are the old $c_1\otimes 1,\dots ,c_{q-1}\otimes 1$, then $e_1\otimes b_1,\dots ,e_1\otimes b_8$, then $c_q\otimes 1$. Move the last coordinate across the eight new coordinates. This is an even permutation, so its orthogonal action is homotopic to identity and introduces no suspension sign. The resulting ordered module has generators $c_i\otimes 1$ and ${\mathrm{\Gamma }}_M\otimes b_j$, and grading ${\mathrm{\Gamma }}_M\otimes {\mathrm{\Gamma }}_8$, where ${\mathrm{\Gamma }}_M=e_1$. It is literally $M\widehat{\otimes }B$. The actual symbol product in [F4] gives ${\mathrm{\Phi }}_{q+8}(T\delta )=s_{q+8}(M\widehat{\otimes }B)={\mathrm{\Phi }}_q(\delta )⊠\beta$ for every elementary opposite pair. Uniform negation gives the same identity for $\mathrm{\Psi }$, since $-(u⊠\beta )=(-u)⊠\beta$.

Check the integer base case and its sign separately. Stabilizing the POSITIVE-rank elementary difference $(+1,-1)$ produces opposite masses ${\mathrm{\Gamma }}_8,-{\mathrm{\Gamma }}_8$ on $S$. Its module from step 2.3 has grading $a_1$ and negative generators $a_1{a}_2,\dots ,a_1{a}_8,a_1{\mathrm{\Gamma }}_8$. Their ordered product is $a_1$: multiplying eight pairs contributes $(-1{)}^{28}=1$, and $(a_2\cdots a_8{)}^2=-1$ cancels the sign from moving $a_1$ past seven factors. Thus it is the volume-graded irreducible $C_8$ module. Classification identifies it with $B$, and the grading intertwines because it is the same volume element; polar normalization makes an algebraic intertwiner orthogonal without changing its Clifford action. Consequently its RAW ${\mathrm{\Phi }}_8$ class is $+\beta$. Reversing the difference to the NEGATIVE-rank generator $(-1,+1)$ gives raw ${\mathrm{\Phi }}_8=-\beta$, hence final ${\mathrm{\Psi }}_8=+\beta$. Since ${\mathrm{\Psi }}_0(-1,+1)=+1$, this proves ${\mathrm{\Psi }}_8(T\delta )={\mathrm{\Psi }}_0(\delta )⊠\beta$ with the stated native negative-rank convention. There is no volume-sign obstruction and no silent reversal of $\beta$.

At $q=4$ use $J_1=L_i,J_2=L_j,K=L_k$ on one quaternionic line. For $S=+1,S_0=-1$, the transition is $F(x_0,x_1,x_2,t)=L_{x_0+x_{1}i+x_{2}j+tk}$, exactly the quaternionic Hopf transition. Thus ${\mathrm{\Phi }}_4$ gives the positive Hopf class for a POSITIVE-rank difference. The negative-rank generator is instead $S=-1,S_0=+1$: its ${\mathrm{\Phi }}_4$ value is the negative Hopf class, and its ${\mathrm{\Psi }}_4=-{\mathrm{\Phi }}_4$ value is the positive Hopf class. The source-minus-target/volume convention of [F4] identifies the latter with the chosen generator. Arbitrary involutions are connected to diagonal ones with the same real or quaternionic ranks; common sign blocks give the folded-null transition in step 2.1. Therefore these elementary computations prove ${\mathrm{\Psi }}_q$ is an isomorphism in rows $0,1,2,4$. Steps 3.2 and [F5] prove it for the four zero rows.

In each of the first eight rows, every stable component difference is a sum of the elementary opposite differences and common null blocks identified in step 2.1, step 2.2, step 3.2 and step 4.1. Step 3.1 transports this generating description through every eighth row. Additivity, homotopy invariance and step 3.4 and step 3.5 therefore prove ${\mathrm{\Psi }}_{q+8}T=(-{\mathrm{\Phi }}_{q+8})T=\beta ⊠(-{\mathrm{\Phi }}_q)=\beta ⊠{\mathrm{\Psi }}_q$ on every class. Since $T$ and product by $\beta$ are isomorphisms, induction from the low rows in step 4.1 proves that every ${\mathrm{\Psi }}_q$ is an isomorphism. All constructions retained the fixed reference identification $w$; abstract isomorphism of the modules attached to two separate masses was never used as a substitute for the difference. This proves the stated finite-model comparison with the exact negative-eigenspace and volume conventions.