The shifted character observables form a basis of $A$, with the Kerov weight filtration

Membership and basis: let $A_Y$ be the source's algebra of functions on Young diagrams in [F3]. Its profile-moment generators are algebraically independent by [F3]. Restriction from the algebra $A$ of profile polynomials on $D_0$ onto $A_Y$ is surjective, and is injective: a polynomial that vanishes on $D_0$ vanishes on all Young profiles, so independence in $A_Y$ forces its coefficients to be zero; likewise a function in $A$ vanishing on all Young profiles is the zero polynomial. Hence restriction is an isomorphism. Each $p_{\rho }^{\mathrm{\#}}$ has a unique polynomial extension to $D_0$, and [F3] gives the linear basis of these extensions. The empty partition labels $p_{\mathrm{\varnothing }}^{\mathrm{\#}}=1$. This proves (i), including that $A$ is a polynomial algebra.

Structure constants and the two filtrations: by [F2] the structure constants of $A$ in the basis $\{p_{\rho }^{\mathrm{\#}}\}$ are $f_{\sigma \tau }^{\rho }=\frac{z_{\sigma }{z}_{\tau }}{z_{\rho }}{g}_{\sigma \tau }^{\rho }$, and by [F4] the degrees ${\mathrm{deg}}_J$ are compatible with multiplication; taking $J=\{1\}$ gives ${\mathrm{deg}}_1(p_{\rho }^{\mathrm{\#}})\le {\mathrm{deg}}_1(p_{\sigma }^{\mathrm{\#}})+{\mathrm{deg}}_1(p_{\tau }^{\mathrm{\#}})$ whenever $f_{\sigma \tau }^{\rho }\mathrm{\ne }0$. Taking $J=\mathbb{N}$ gives weight compatibility, and [F4] identifies the ${\mathrm{deg}}_{\mathbb{N}}$ filtration with the weight filtration $\mathrm{wt}({\tilde{p}}_k)=k$, so $\mathrm{wt}(p_{\rho }^{\mathrm{\#}})=\mathrm{\mid }\rho \mathrm{\mid }+\mathrm{\ell }(\rho )$ and $\mathrm{wt}$ is an algebra filtration. Since $m_1(\rho )\le \mathrm{\ell }(\rho )$, ${\mathrm{deg}}_1\le \mathrm{wt}$ on each basis element, hence on $A$. This is (ii).

Unique top term: by [F1] the only partition with $\mathrm{\mid }\rho \mathrm{\mid }=\mathrm{\mid }\sigma \mathrm{\mid }+\mathrm{\mid }\tau \mathrm{\mid }$ and $g_{\sigma \tau }^{\rho }\mathrm{\ne }0$ is $\rho =\sigma \cup \tau$, and by [F2] its coefficient is $f_{\sigma \tau }^{\sigma \cup \tau }=1$. Any other contributing $\rho$ satisfies ${\mathrm{deg}}_{\mathbb{N}}(p_{\rho }^{\mathrm{\#}})<{\mathrm{deg}}_{\mathbb{N}}(p_{\sigma \cup \tau }^{\mathrm{\#}})$: otherwise, if the ${\mathrm{deg}}_{\mathbb{N}}$-degrees were equal, the equality case recorded in [F4] (with $J=\mathbb{N}$) would force $\rho =\sigma \cup \tau$ by the same imported argument. Since ${\mathrm{deg}}_{\mathbb{N}}$ is the weight, all other terms have strictly smaller weight. Hence $p_{\sigma }^{\mathrm{\#}}{p}_{\tau }^{\mathrm{\#}}=p_{\sigma \cup \tau }^{\mathrm{\#}}+(\text{lower weight})$, which is (iii).

Triangularity: [F3] gives ${\tilde{p}}_k=k{p}_{k-1}$ plus lower canonical degree and $p_{\rho }^{\mathrm{\#}}=p_{\rho }$ plus lower canonical degree, where $p_{\rho }={\prod }_i{p}_{{\rho }_i}$. Thus the monomial ${\prod }_i{\tilde{p}}_{{\rho }_i+1}$ has leading canonical component ${\prod }_i({\rho }_i+1)p_{\rho }$, so its expansion in the shifted-character basis is triangular with nonzero diagonal. These monomials, including the empty product, form a basis because the generators are algebraically independent. The equality of weight filtrations in [F4] makes this change compatible with weight levels. The generators themselves are algebra generators, rather than a linear basis. This proves the final assertion. No choice principle is used in the finite algebraic operations or cited algebraic results.