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Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the comments are not editable after a couple of minutes).

The approach suggested by vap in the comment of the linked question should work. This is more-or-less depicted in Loday's book Cyclic Homology, §8.4.13. Let me briefly explain it here.

Let $k$ be the base commutative ring (e.g. $k=\mathbb C$), and $A$ a unital $k$-algebra. The Chern characters $\DeclareMathOperator\ch{ch}\DeclareMathOperator\HC{HC}\DeclareMathOperator\HP{HP}\ch_{n,i}\colon K_n(A)\to\HC_{n+2i}(A)$ are induced by the maps $\HP_n(A)\to\HC_{n+2i}(A)$ and the composite map $K(A)\to\HC^-(A)\to\HP(A)$, where the first map is the (enhanced) Dennis trace map valued in the negative cyclic homology, and the second map is the canonical map from the negative cyclic homology to the periodic cyclic homology.

Combining with the canonical pairing $\HC_{n+2i}(A)\otimes_k\HC^{n+2i}(A)\to k$, one gets the Chern–Connes pairing. To identify with Connes' original description (maybe up to a constant multiplier), we invoke loc. cit., Prop 8.4.9, which describes the Dennis trace map $K_1(A)\to\HC_1^-(A)$. Let me mention that the $-1$ appearing in Connes' original description does not matter in cyclic homology, cf. the proof of Thm 3.9 in Mesland's master thesis.

Now I believe that the $S$-invariance comes from the fact that the Chern characters factors through the first (i.e. odd) periodic cyclic homology $\HP_1(A)$.

Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the comments are not editable after a couple of minutes).

The approach suggested by vap in the comment of the linked question should work. This is more-or-less depicted in Loday's book Cyclic Homology, §8.4.13. Let me briefly explain it here.

Let $k$ be the base commutative ring (e.g. $k=\mathbb C$), and $A$ a unital $k$-algebra. The Chern characters $\DeclareMathOperator\ch{ch}\DeclareMathOperator\HC{HC}\DeclareMathOperator\HP{HP}\ch_{n,i}\colon K_n(A)\to\HC_{n+2i}(A)$ are induced by the maps $\HP_n(A)\to\HC_{n+2i}(A)$ and the composite map $K(A)\to\HC^-(A)\to\HP(A)$, where the first map is the Dennis trace map, enhanced by Goodwillie–Jones, valued in the negative cyclic homology, and the second map is the canonical map from the negative cyclic homology to the periodic cyclic homology.

Combining with the canonical pairing $\HC_{n+2i}(A)\otimes_k\HC^{n+2i}(A)\to k$, one gets the Chern–Connes pairing. To identify with Connes' original description (maybe up to a constant multiplier), we invoke loc. cit., Prop 8.4.9, which describes the Goodwillie–Jones trace map $K_1(A)\to\HC_1^-(A)$. Let me mention that the $-1$ appearing in Connes' original description does not matter in (periodic) cyclic homology, cf. the proof of Thm 3.9 in Mesland's master thesis.

Now I believe that the $S$-invariance comes from the fact that the Chern characters factors through the first (i.e. odd) periodic cyclic homology $\HP_1(A)$.

Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the comments are not editable after a couple of minutes).

The approach suggested by vap in the comment of the linked question should work. This is more-or-less depicted in Loday's book Cyclic Homology, §8.4.13. Let me briefly explain it here.

Let $k$ be the base commutative ring (e.g. $k=\mathbb C$), and $A$ a unital $k$-algebra. The Chern characters $\DeclareMathOperator\ch{ch}\DeclareMathOperator\HC{HC}\DeclareMathOperator\HP{HP}\ch_{n,i}\colon K_n(A)\to\HC_{n+2i}(A)$ are induced by the maps $\HP_n(A)\to\HC_{n+2i}(A)$ and the composite map $K(A)\to\HC^-(A)\to\HP(A)$, where the first map is the (enhanced) Dennis trace map valued in the negative cyclic homology, and the second map is the canonical map from the negative cyclic homology to the periodic cyclic homology.

Combined with the canonical pairing $\HC_{n+2i}(A)\otimes_k\HC^{n+2i}(A)\to k$, one gets the Chern–Connes pairing. To identify with Connes' original description (maybe up to a constant multiplier), we invoke loc. cit., Prop 8.4.9, which describes the Dennis trace map $K_1(A)\to\HC_1^-(A)$. Let me mention that the $-1$ appearing in Connes' original description does not matter in cyclic homology, cf. the proof of Thm 3.9 in Mesland's master thesis.

Now I believe that the $S$-invariance comes from the fact that the Chern characters factors through the first (i.e. odd) periodic cyclic homology $\HP_1(A)$.

Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the comments are not editable after a couple of minutes).

The approach suggested by vap in the comment of the linked question should work. This is more-or-less depicted in Loday's book Cyclic Homology, §8.4.13. Let me briefly explain it here.

Let $k$ be the base commutative ring (e.g. $k=\mathbb C$), and $A$ a unital $k$-algebra. The Chern characters $\DeclareMathOperator\ch{ch}\DeclareMathOperator\HC{HC}\DeclareMathOperator\HP{HP}\ch_{n,i}\colon K_n(A)\to\HC_{n+2i}(A)$ are induced by the maps $\HP_n(A)\to\HC_{n+2i}(A)$ and the composite map $K(A)\to\HC^-(A)\to\HP(A)$, where the first map is the (enhanced) Dennis trace map valued in the negative cyclic homology, and the second map is the canonical map from the negative cyclic homology to the periodic cyclic homology.

Combining with the canonical pairing $\HC_{n+2i}(A)\otimes_k\HC^{n+2i}(A)\to k$, one gets the Chern–Connes pairing. To identify with Connes' original description (maybe up to a constant multiplier), we invoke loc. cit., Prop 8.4.9, which describes the Dennis trace map $K_1(A)\to\HC_1^-(A)$. Let me mention that the $-1$ appearing in Connes' original description does not matter in cyclic homology, cf. the proof of Thm 3.9 in Mesland's master thesis.

Now I believe that the $S$-invariance comes from the fact that the Chern characters factors through the first (i.e. odd) periodic cyclic homology $\HP_1(A)$.

Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the comments are not editable after a couple of minutes).

The approach suggested by vap in the linked question should work. This is more-or-less depicted in Loday's book Cyclic Homology, §8.4.13. Let me briefly explain it here.

Let $k$ be the base commutative ring (e.g. $k=\mathbb C$), and $A$ a unital $k$-algebra. The Chern characters $\DeclareMathOperator\ch{ch}\DeclareMathOperator\HC{HC}\DeclareMathOperator\HP{HP}\ch_{n,i}\colon K_n(A)\to\HC_{n+2i}(A)$ are induced by the maps $\HP_n(A)\to\HC_{n+2i}(A)$ and the composite map $K(A)\to\HC^-(A)\to\HP(A)$, where the first map is the (enhanced) Dennis trace map valued in the negative cyclic homology, and the second map is the canonical map from the negative cyclic homology to the periodic cyclic homology.

Combined with the canonical pairing $\HC_{n+2i}(A)\otimes_k\HC^{n+2i}(A)\to k$, one gets the Chern–Connes pairing. To identify with Connes' original description (maybe up to a constant multiplier), we invoke loc. cit., Prop 8.4.9, which describes the Dennis trace map $K_1(A)\to\HC_1^-(A)$. Let me mention that the $-1$ appearing in Connes' original description does not matter in cyclic homology, cf. the proof of Thm 3.9 in Mesland's master thesis.

Now I believe that the $S$-invariance comes from the fact that the Chern characters factors through the first (i.e. odd) periodic cyclic homology $\HP_1(A)$.

Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the comments are not editable after a couple of minutes).

The approach suggested by vap in the comment of the linked question should work. This is more-or-less depicted in Loday's book Cyclic Homology, §8.4.13. Let me briefly explain it here.

Let $k$ be the base commutative ring (e.g. $k=\mathbb C$), and $A$ a unital $k$-algebra. The Chern characters $\DeclareMathOperator\ch{ch}\DeclareMathOperator\HC{HC}\DeclareMathOperator\HP{HP}\ch_{n,i}\colon K_n(A)\to\HC_{n+2i}(A)$ are induced by the maps $\HP_n(A)\to\HC_{n+2i}(A)$ and the composite map $K(A)\to\HC^-(A)\to\HP(A)$, where the first map is the (enhanced) Dennis trace map valued in the negative cyclic homology, and the second map is the canonical map from the negative cyclic homology to the periodic cyclic homology.

Combined with the canonical pairing $\HC_{n+2i}(A)\otimes_k\HC^{n+2i}(A)\to k$, one gets the Chern–Connes pairing. To identify with Connes' original description (maybe up to a constant multiplier), we invoke loc. cit., Prop 8.4.9, which describes the Dennis trace map $K_1(A)\to\HC_1^-(A)$. Let me mention that the $-1$ appearing in Connes' original description does not matter in cyclic homology, cf. the proof of Thm 3.9 in Mesland's master thesis.

Now I believe that the $S$-invariance comes from the fact that the Chern characters factors through the first (i.e. odd) periodic cyclic homology $\HP_1(A)$.