The loop filtration on the type-A finite W-algebra has associated graded U(g^e)

Actual ambient leading term. Give a matrix unit $e_{a,b}\in \mathfrak{p}$ its good degree $\mathrm{col}(b)-\mathrm{col}(a)$. In the explicit signed path formula a term of canonical degree $r+1$ with $k$ matrix-unit factors has good degree at most $r+1-k$; a diagonal scalar replacement only lowers that degree. Therefore only single-factor terms contribute in good degree $r$. Those terms sum to $(-1{)}^r\sum e_{a,b}$ over $\mathrm{row}(a)=i$, $\mathrm{row}(b)=j$, $\mathrm{col}(b)-\mathrm{col}(a)=r$, namely $(-1{)}^r{e}_{i,j;r}$. Products introduced by Gauss factorization or its inverse have at least two factors and have smaller good degree. Thus the leading homogeneous component of $T_{i,j}^{(r+1)}$ in the actual graded algebra $U(\mathfrak{p})$ is exactly $(-1{)}^r{e}_{i,j;r}$. This proves the leading-symbol claim using the path degrees rather than a claim that Gauss inversion alone supplies its sign.

Independence in the ambient enveloping algebra. The elements $e_{i,j;r}$ form a homogeneous basis of the actual Lie subalgebra ${\mathfrak{g}}^e\subseteq \mathfrak{p}$. Extend that finite basis to a homogeneous basis of $\mathfrak{p}$, with an order extending the prescribed order of the centralizer basis. Enveloping PBW implies that the ordered monomials in the $e_{i,j;r}$ are linearly independent in $U(\mathfrak{p})$, and that the natural homomorphism $U({\mathfrak{g}}^e)\to U(\mathfrak{p})$ is injective. Their products and brackets are the actual ambient products and brackets, including all boundary terms of the centralizer strings. No projection of a current bracket with an arbitrary out-of-range convention is needed.

Strictness of the loop basis. The finite $T$ monomials form a vector-space basis of $W(\lambda )$ by [F1]. For a monomial with total loop degree $d$, step 1.1 says that its degree-$d$ homogeneous component in $U(\mathfrak{p})$ is, up to the product of its displayed signs, the corresponding PBW monomial in $U({\mathfrak{g}}^e)$. In any nonzero finite linear combination of $T$ monomials, take the largest total loop degree $d$. The degree-$d$ homogeneous component is nonzero by step 1.2, even when degree-zero generators occur. Thus that combination belongs to ${⨁}_{s\le d}U(\mathfrak{p}{)}_s$ and not to ${⨁}_{s<d}U(\mathfrak{p}{)}_s$. Consequently the intersection filtration in the Statement is exactly the span of $T$ monomials of total loop degree at most $d$, and their degree-$d$ symbols form a basis of its associated graded. This verifies symbol independence directly, rather than inferring it from independence of filtered elements.

The graded algebra isomorphism. For any subalgebra of a nonnegatively graded algebra with its intersection filtration, taking highest homogeneous components embeds its associated graded into that graded algebra: if a representative's component of degree $d$ is zero it already lies in filtration $d-1$, and multiplication agrees with multiplication of homogeneous components. Apply this to $W(\lambda )\subseteq U(\mathfrak{p})$. Step 2.1 identifies its image with the span of the ordered monomials in ${\mathfrak{g}}^e$, which is precisely the embedded algebra $U({\mathfrak{g}}^e)$ from step 1.2. Hence ${\mathrm{gr}}^{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}W(\lambda )\cong U({\mathfrak{g}}^e)$, and the identification sends the symbol of $T_{i,j}^{(r+1)}$ to $(-1{)}^r{e}_{i,j;r}$ as asserted.

Filtration and the second graded algebra. Each generator has finite nonnegative good degree, and each element is a finite sum of $T$ monomials, so the loop filtration is exhaustive and bounded below with $F_{-1}=0$. The actual homogeneous decomposition in $U(\mathfrak{p})$ and step 2.1 give separatedness. The finite basis is already established in [F1], and step 3.1 establishes its filtered deformation property. Independently, the Kazhdan filtration uses positive weights $\mathrm{col}(b)-\mathrm{col}(a)+1$ on $\mathfrak{p}$ and has associated graded $\mathbb{C}[S]$ by [F4]; in the doubled convention its weights are twice these. That graded algebra is commutative, whereas $U({\mathfrak{g}}^e)$ is noncommutative whenever ${\mathfrak{g}}^e$ is nonabelian. These are the two claimed filtrations and their respective graded algebras.