The Laurent determinant cocycle has Virasoro central weight two

Since f has a finite negative Laurent cutoff, $D_f$ maps $H_{+}$ into $H_{+}$ plus a fixed finite-dimensional negative-power space. Consequently its block $A_{-+}$ has finite rank. Its other off-diagonal block may have infinite rank, but each product in beta still has finite rank because its right factor does. Algebraic finite-rank trace is well defined: a factorization $F=uv$ through a finite-dimensional space gives $\mathrm{Tr}F=\mathrm{Tr}(vu)$, independent of the factorization by enlarging the finite-dimensional image; it satisfies $\mathrm{Tr}(CF)=\mathrm{Tr}(FC)$ whenever F has finite rank. No topology on $\mathbb{C}((z))$ is used.

To verify the cocycle without manipulating traces of infinite operators, consider pairs $(A,Q)$ with $Q-A_{++}$ finite rank. Under componentwise commutator they form a Lie algebra, since $[A,B{]}_{++}=[A_{++},B_{++}]+A_{+-}{B}_{-+}-B_{+-}{A}_{-+}\mathrm{.}$ Its kernel over A is the Lie algebra of finite-rank operators. Its trace-zero subspace is an ideal: its bracket with Q is a finite-rank commutator of trace zero, by step 1.1. Quotienting by this ideal gives a central extension by the trace scalar. The section $A\mapsto (A,A_{++})$ has central bracket error exactly beta. Jacobi for the commutator Lie algebra and its quotient therefore proves the two-cocycle identity. This also specifies the sign of the extension, rather than invoking an unspecified determinant convention.

A monomial operator satisfies $D_m{e}_k=-k{e}_{k+m}$, where $e_k=z^k$, and $[D_m,D_n]=(m-n)D_{m+n}$. If $m+n\mathrm{\ne }0$, either off-diagonal product has total shift m+n, so its finite-rank matrix has zero diagonal and zero trace. Suppose $m>0$ and $n=-m$. Then $(D_m{)}_{-+}=0$ and $(D_{-m}{)}_{+-}=0$. The product $(D_m{)}_{+-}(D_{-m}{)}_{-+}$ acts diagonally on $e_k$, $0\le k\le m-1$, by $k(k-m)$ and is zero on the other basis vectors. Thus $\beta (D_m,D_{-m})=-{\sum }_{k=0}^{m-1}k(k-m)=\frac{m(m-1)(m+1)}{6}\mathrm{.}$ Antisymmetry gives the negative-m cases and $m=0$ gives zero. This proves the formula on the Laurent modes. The cocycle on all $D_f$ is established algebraically in step 2.1; no continuity or passage to a topological completion is asserted.

Comparing step 3.1 with F1 gives beta equal to twice the normalized Virasoro cocycle. In particular it vanishes on the three-dimensional span of $D_{-1},D_0,D_1$; the linear term has not been discarded as an unnamed coboundary. Any global use must still identify the determinant line and its isotropy splitting. The computation here proves precisely the asserted local central weight and no global descent claim.