Actual principal W extension splitting gives semisimplicity in every specified admissible grading

Let M be generated by finitely many homogeneous vectors $w_i$ in its specified admissible grades $r_i$. Its grade n is the sum of the fixed-degree spans $H_{n-r_i}(w_i)$, hence is finite-dimensional by F1. Let G act by n on M(n). The conformal commutator and F3 give $[L0,a_m]=[G,a_m]=(wt(a)-m-1)a_m$, so $D=L0-G$ commutes with every mode. It preserves each finite-dimensional $M(r_i)$. A finite product of the characteristic polynomials of D on these finitely many spaces annihilates all $w_i$, hence all M since D commutes with its generating modes. Thus D has finitely many generalized eigenvalues ${\delta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\delta }_s$ with a uniform finite annihilating polynomial. On M(n), $L0=D+n$, so L0 is locally finite, its spectrum lies in the finite union ${\delta }_i+Z_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$ and is lower bounded in real part. A fixed generalized L0 space receives contributions from at most these finitely many indices $n=h-{\delta }_i$, each finite-dimensional. Therefore M is an actual lower-bounded generalized module with finite-dimensional generalized conformal-weight spaces; neither diagonalizability nor finite length has been assumed.

Let Lambda be the finite set of scalar values of the central conformal class [omega] on the simple modules of the finite-dimensional semisimple algebra Z0. Every nonzero weak subquotient Q of M has a nonzero generalized conformal space of weight in Lambda. Indeed submodules and quotients decompose into generalized L0 spaces: polynomial projection on the finite spectral support of each vector gives exactness of this decomposition. Split Q into its finitely many weight cosets modulo Z and choose a nonempty coset. Its support is bounded below, so it has a least occupied weight h. This coset component has the admissible grading Q(n)=Q_[h+n] even if L0 is not yet semisimple, so the top-action hypothesis of F2 applies. All energy-lowering modes kill Q_[h], giving the actual Z0 action by F2. That finite nonzero space is semisimple over Z0, so [omega]=L0 is diagonalizable there and h lies in Lambda. In particular every nonzero weak subquotient has a nonzero component in the finite window Lambda. The integer measure sum_(h in Lambda) dim M_[h] is finite. It strictly decreases on a nonzero quotient of a submodule, because generalized-weight spaces are exact. Consequently it bounds every strict ascending or descending chain of weak submodules. Choosing a submodule of least positive measure gives a simple submodule; then induction on that measure gives a finite composition series. This proves finite length without a module-Noetherian assumption.

Each simple composition factor is ordinary. Its lowest generalized conformal space carries a nonzero simple Z0 submodule U, on which [omega] acts as a scalar h. The negative-energy modes kill U. Actual top induction in F2 maps the corresponding highest module into that simple factor; its image is the whole factor. The conformal commutator makes L0 scalar h+n on every generating homogeneous mode word of degree n. Thus L0 is semisimple on the factor. Its conformal spaces are finite-dimensional and lower bounded because it is a subquotient of M, so it is ordinary. The assumed pairwise extension splitting therefore applies to all its simple factors. Induct on length to make M semisimple as a weak module: after choosing a simple kernel, the quotient is a finite sum of simple factors. Pull back each summand to obtain an extension by that kernel, choose its splitting, and sum the lifted sections. The sum is injective because its composite to the direct-sum quotient is the identity. It supplies a complement to the kernel. This is an actual splitting argument, not an appeal to semisimple Zhu alone.

Recover the specified grading. Weak semisimplicity in step 3.1 makes L0 semisimple on M, as all its simple summands are ordinary. G is semisimple and commutes with L0, so D=L0-G is semisimple as well, as can be checked in each finite M(n). Its eigenspaces are mode-stable and are also specified-graded. In the D=delta eigenspace the specified grade n is exactly the L0 eigenspace of eigenvalue delta+n. Every weak simple submodule there is therefore specified-graded: polynomial interpolation in L0 on the finite support of a vector isolates each of its specified components. The D eigenspace, a submodule of a finite semisimple weak module, is semisimple; choose its simple summands. Doing this for the finitely many D eigenvalues gives an internal simple decomposition of M which preserves its original specified grading. No new grading was chosen on the summands.

Now let M be an arbitrary admissible module, with no finiteness of its given grades or multiplicities. Each homogeneous vector generates a specified-graded cyclic submodule, to which steps 1.1–4.1 apply; any vector has finite homogeneous support, and those cyclic submodules span M. Thus M is a sum of specified-graded weakly simple submodules. By A1 choose a maximal family of such submodules whose sum is internal and direct. If their sum S were not all M, one of the graded simple submodules spanning M would not be contained in S. Its intersection with S is a weak submodule of that simple module, hence is zero. Adjoining it contradicts maximality. Therefore the sum equals M. Every selected submodule inherits its nonnegative grading, and it is weakly simple; arbitrary cardinal multiplicities are allowed. The zero module is the empty sum. This is precisely rationality in F3.

Finally apply step 5.1 to the regular module A with its actual state grading. It is cyclic under the vacuum: $a=a_{-1}1$. In its graded simple decomposition every nonzero summand has nonzero vacuum projection, since otherwise the corresponding module projection would vanish on every $a_{-1}1$. Each summand therefore meets $A_0$. Since $A_0=C1$, at most one nonzero summand exists. Thus A is simple as its own weak module, hence as a vertex algebra. The nonzero graded quotient $W^k$ to A must then be its specified unique graded simple quotient: in a CFT-type vertex algebra every proper graded ideal has zero grade-zero part, their sum remains proper, and the quotient by their sum is the unique graded simple quotient. Hence A identifies with $W_k$, and $W_k$ has the rationality just proved in the original specified-grading convention. No separate plus-DS simplicity input has been used.