The Koszul-Hochschild comparison respects crossing differentials and trigradings

The local quotient modules. At $a=0$, the two-arc Koszul resolution has quotient the identity bimodule $R^{\mathrm{\prime }}$, while the wide-edge resolution has quotient $B_s^{\mathrm{\prime }}=\mathbb{Q}[x_1,x_2,x_3,x_4]\mathrm{/}(x_1+x_2-x_3-x_4,x_1{x}_2-x_3{x}_4)\mathrm{.}$ Choose the left strand coordinates $(x_4,x_3)$ and right strand coordinates $(x_1,x_2)$; this order agrees with the arcs $x_1=x_4$, $x_2=x_3$. Put $y_L=x_4-x_3$ and $y_R=x_1-x_2$. The sum relation implies $x_4-x_2=(y_L+y_R)\mathrm{/}2$ in $B_s^{\mathrm{\prime }}$. The common invariant coordinate and the other strand coordinates pass through this local calculation unchanged. The regular-layer quotient and diagonal identification in [L4] carry these coefficient modules to the corresponding termwise Hochschild contributions; removing the universal zero row recovers the original $a$-theory with its fixed $\{-1,1\}$ correction.

Read the induced maps from the matrices. The degree-zero Koszul homology is the quotient of the even coefficient summand. The even matrices of [L1] have first entries $x_4-x_2$ for ${\chi }_0$ and $1$ for ${\chi }_1$. Consequently ${\chi }_0$ induces $1\mapsto (y_L+y_R)\mathrm{/}2$ from the identity module to $B_s^{\mathrm{\prime }}$, while ${\chi }_1$ induces the quotient map $B_s^{\mathrm{\prime }}\to R^{\mathrm{\prime }}$ setting $y_L=y_R$, namely multiplication. The first map is a bimodule map because $(y_L-y_R)(y_L+y_R)=0$ in the wide-edge quotient and the invariant generators already balance. After removing the common invariant polynomial factor, these are exactly $r{b}_s\mathrm{/}2$ and $b{r}_s$ on the reduced modules. In particular $r{b}_{s}b{r}_s$ is multiplication by $y_L+y_R$ on $B_s$, rather than multiplication by the single scalar $y_s$; $b{r}_{s}r{b}_s=2y_s$ on $R$. Rescaling the source of the positive two-term complex by a nonzero rational scalar turns $r{b}_s\mathrm{/}2$ into $r{b}_s$, and the negative map already equals $b{r}_s$. All shifts are those in [L2] and [L3].

Assembly over the layers. The local quotient maps of step 2.1 are tensored with the identity on every other layer. The regular-layer augmentations and the diagonal Koszul comparison are natural for these coefficient maps by [L4], so the resolution-wise identifications intertwine each crossing edge. Multiplying a resolution's coefficient module by the product of the source-rescaling constants for its positive arc terms makes all positive maps exactly $r{b}_s$ simultaneously: changing one resolution choice changes precisely the factor belonging to that crossing. The scalars commute, so the two routes around each cube face agree, and the ordinary cube signs are preserved. These isomorphisms therefore assemble into a chain isomorphism between the corrected reduced resolution-homology complex and the termwise Hochschild complex of $F(\sigma )$. For the universal-row reduction, the normal-form differential is $a(e_0\wedge -)+d_K$, and the crossing matrices and homogeneous row changes contain no $a$. Comparing the coefficient of $a$ in the chain-map equation forces each crossing map to commute with $e_0\wedge -$. It therefore preserves the image of this wedge operator, the copy retained by the original $(a,0)$ row after polynomial cancellation. Its induced coefficient map there is the one computed above. The extra zero-row copy of the full $a=0$ theory is not included.

The first grading. On the Khovanov-Rozansky side the first bigrading of a class of the folded complex is the negative of the number of Koszul symbols of the closure block that produced it, because each relation of the closure block contributes the shift $(-1,1)$ by [L5], while the layer block contributes nothing on homology: it is a regular sequence whose Koszul complex is a resolution of $B^{\mathrm{\prime }}(D)$. Under the identification of step 1.1 the closure block is exactly the Hochschild block of the termwise complex, whose homology degree is $h$; hence $k=-h$ after the global correction, which does not involve the first bigrading of the moving classes beyond the fixed shift $(1,-1,0)$.

The second and third gradings. By [L5] a relation of internal degree $d$ contributes the shift $(-1,d-1)$ to the folded complex, and the surviving class has internal degree $p$ equal to the sum of the degrees of the relations and of the coefficient bimodules of the resolution; therefore its second bigrading is $l=p-h$, because each of the $h$ closure factors contributes $d$ to the internal degree $p$ and $d-1$ to the second bigrading, a difference of exactly $h$ down from $p$. The cohomological degree $j$ of the assembled complex is the position in the cube of resolutions, which is the same index as the Rouquier degree $c$ of the termwise complex; hence $j=c$. The three displays combine to $k=-h$, $l=p-h$, $j=c$, that is $a=-h$, $q=p-h$, $t=c$ in the variables of the statement.

The global correction and the constants. The source records that the Khovanov-Rozansky trigrading carries a built-in shift by $(-1,1,0)$ coming from the variable $a$, that both the one-strand classes lie in $(0,0,0)$ after undoing it, and that after the correction the third gradings match, the Hochschild grading equals the Koszul grading with the minus sign, and the second grading equals the $x$-degree grading minus the Hochschild grading; these are exactly the three displays of steps 4.1 and 4.2, with the correction $(k,l)\mapsto (k+1,l-1)$ applied to the first two. The one-strand class is $(-1,1,0)$ on the Khovanov-Rozansky side and $(0,0,0)$ on the Hochschild side, so the correction is fixed, and the $(2,n)$ computation with its alternating differentials of degrees $2$ confirms that no further constant is needed.