Gts framing comparison bracket rank one check

Modulo $s^2$, $e^{s\delta }=1+s\delta$, so the coefficient of $s$ after conjugation is $e^A\delta e^{-A}$. Differentiating $e^{uA}\delta e^{-uA}$ gives $[A,e^{uA}\delta e^{-uA}]$ with initial value $\delta$, hence the coefficient is $e^{\mathrm{ad}A}\delta$.

The actual cross-deformation complex used above can be seen directly in a dg resolution: deforming a differential $d$ by $d+ϵa$ gives $da+ad=0$, while a first-order change of basis changes $a$ by a commutator with $d$. Thus cocycles and coboundaries are the endomorphism complex shifted by one. Off-diagonal blocks for ${\mathcal{P}}_i\oplus {\mathcal{Q}}_i$ give the stated ordered $R\mathrm{Hom}({\mathcal{P}}_i,{\mathcal{Q}}_i)[1]$; taking the shifted dual gives ${\mathrm{\Theta }}_i$. Applying $R\mathrm{Hom}$ successively to the two actual extension triangles gives additivity of its Grothendieck class in each argument:

This is an equality of classes of the actual complexes. Their differentials include the actual extension classes; no direct-sum quasi-isomorphism or split extension is asserted. The same construction works for perfect families because derived Hom and these fibre/cofibre triangles commute with the stipulated base changes.

If $\alpha$ is a primitive class with no positive proper decomposition, its coefficient is exactly $[z_{\alpha },\delta ]$, with diagonal coefficient one. If $\alpha =\beta +\gamma$ with distinct positive factors $\beta \mathrm{\ne }\gamma$, the two ordered contributions are $\frac{1}{2}[z_{\gamma },[z_{\beta },\delta ]]$ and $\frac{1}{2}[z_{\beta },[z_{\gamma },\delta ]]$. If $\beta =\gamma$, the contribution is $\frac{1}{2}[z_{\beta },[z_{\beta },\delta ]]$. The terms retain their exponential factorials and cannot be replaced by an unordered term.

The given concentration says $R\mathrm{Hom}(S_i,F)\simeq V_i(F)[-1]$. CY4 duality gives

Sphericity gives $[R\mathrm{Hom}(S_i,S_i)]=[\mathbb{C}]+[\mathbb{C}[-4]]=2$. Both odd shifts in the cross terms have negative Grothendieck sign. Negating the dual of the equality in 1.2 therefore proves, rather than stipulates,

Fixed trace/source corrections, if present in the chosen original convention, are retained on both sides and do not change the displayed framing contribution.

It follows that the actual ordered framing contribution changes by

Thus after identifying the old cross-complexes by a supplied old obstruction compatibility, the change is $H_2-H_1$; without that identification the term $K_2-K_1$ must remain. In particular removing an additional framing summand $U$ removes $P\otimes U(F{)}^{\vee }$ in the left-source/right-old position and $U(E)\otimes Q^{\vee }$ in the reversed position. For a formal direct-sum framing these two terms can equally be recorded as $W_U^{\vee }+{\iota }^{\ast }{W}_U$, where $W_U=P^{\vee }\otimes U(F)$. This last bookkeeping does NOT assert that a new spherical source realizing that direct-sum framing has been constructed. On the old-old sector $P=Q=0$ and on the pure-source/pure-source sector $E=F=0$, the framing contribution vanishes.

For left factor $\delta$ and right old factor $z_{\beta }$, set $P=\mathbb{C}$, $Q=0$, $E=0$: the additional framing contribution is $U(F{)}^{\vee }$. In the reversed order set $P=0$, $Q=\mathbb{C}$, $F=0$: it is $U(E)$. Thus dualization cannot be dropped when reversing the factors. For the rank-one calculation let $\mathrm{\ell }=c_1(P)$ in the first order and $x=c_1(U(F))$. The added cross line has first Chern class $\mathrm{\ell }-x$; in the exchanged order it has class $x-\mathrm{\ell }$. Consequently its formal total-Chern multipliers are

Deleting the component gives the formal inverse. Multiplying these factors for formal Chern roots gives the higher-rank calculation. This proves the ordered framing terms and their Chern calculation, not a geometric transfer or an injectivity statement.

At old charge zero the conjugation coefficient is exactly $\delta$, with scalar coefficient one. It agrees with the positive unit normalization of a source point: after normalizing a one-dimensional source and taking old object zero, its classical model is $\mathrm{Spec}\mathbb{C}$, whose unit class is one. This fixes the formal normalization; comparison with the virtual class of any particular secondary CY4 master still requires that master's construction and compatible orientation. The source-degree-one terms in 2.1 retain precisely one such $\delta$, and their two-letter coefficients remain $1\mathrm{/}2$ in the stated order. The actual extension-complex calculations in steps 1.2, 2.2, 3.1, and 4.1 change neither this unit coefficient nor those exponential factorials.

These calculations establish the pure-source normalization, ordered cross-kernel framing changes from the actual extension complexes, and first triangular coefficients requested in this example. They do not prove that a geometric comparison maps to a nonzero multiple of the expression: the actual nondegenerate comparison remains pending. Thus this completed formal check is not an index-independence proof and does not supply the held geometric seed.