Green positivity specializes the integral half into the composition algebra

Let $\mathcal{T}$ be the free $A$-algebra on $X_{i,n}$, $i\in I,n\ge 1$, with degree $n{\alpha }_i$. The map $X_{i,n}\mapsto F_i^{(n)}$ has image the actual integral half $U_A^{-}$. Define $\mathrm{\Phi }:\mathcal{T}\to C_K$ by coefficient evaluation $t\mapsto c={\nu }^{-1}$ and $X_{i,n}\mapsto u_i^n\mathrm{/}[n{]}_c!$. The denominator evaluations are nonzero by [F3]. Its image includes every $u_i$, so its $K$-linear span is $C_K$.

For ordinary words $w,z$ of degree $\alpha$ and height $h$, the primitive Hall coproduct and Green adjunction evaluate their Hall pairing by deleting successive letters. Deleting a letter at position $r$ crosses the preceding letters that remain in the second tensor factor and contributes ${\nu }^{{\sum }_{s<r}{a}_{i,j_s}}$. Each generator pairing contributes $1\mathrm{/}(Q-1)$. Thus repeated deletion gives exactly $(w(u),z(u){)}_H=(Q-1{)}^{-h}{B}_t(w(F),z(F)){\mathrm{\mid }}_{t=c}$, since $c^{-a}={\nu }^a$. Equivalently both sides have the same first-letter recursion, vacuum value and generator norms after the height rescaling. By dividing by the evaluated factorials and taking finite linear combinations, for homogeneous $P,R\in \mathcal{T}$ one obtains $(\mathrm{\Phi }(P),\mathrm{\Phi }(R){)}_H=(Q-1{)}^{-\mathrm{ht}\alpha }{B}_t(P(F),R(F)){\mathrm{\mid }}_{t=c}\mathrm{.}$ Every evaluation is legitimate by the actual form's localization bound in [F2]. Unequal degrees give zero on both sides.

If homogeneous $P$ is in the actual kernel of $\mathcal{T}\to U_A^{-}$, then $P(F)=0$ as an element of the actual generic half. Since [F2] has constructed a well-defined bilinear form on that actual half, its norm $B_t(P(F),P(F))$ is the zero rational function. Step 2.1 and positive definiteness of the real Green form imply $\mathrm{\Phi }(P)=0$. The kernel is graded, so applying this to its finitely many homogeneous components kills every kernel element. The map $\mathrm{\Phi }$ consequently factors through an $A$-algebra map $U_A^{-}\to C_K$, and extension of scalars gives an algebra map whose image contains $K$ and every $u_i$, hence the surjection $\mathrm{\Psi }:K{\otimes }_{A,t\mapsto c}{U}_A^{-}\to C_K$ asserted in the Statement. The evaluated form on $U_A^{-}$ is $A$-balanced and induces a $K$-bilinear form on its scalar extension. Step 2.1 passes through the quotient and scalar extension, proving the stated form-preservation assertion, with the factor $(Q-1{)}^{-\mathrm{ht}\alpha }$.

Fix $\alpha$. The finite set of ordinary words $W_{\alpha }$ spans $C_K[\alpha ]$, since $C_K$ is generated by the homogeneous simple classes. Let $E:K^{W_{\alpha }}\to C_K[\alpha ]$ be word evaluation. Its pullback Green Gram matrix is $(Q-1{)}^{-\mathrm{ht}\alpha }{G}_{\alpha }(c)$ by step 2.1. The kernel of this matrix is exactly $\ker E$: one inclusion is immediate; if $v$ is in the matrix kernel then $(E(v),E(v){)}_H=0$, and positivity in the real Hall space gives $E(v)=0$. All coefficients lie in the specified real field $K=\mathbb{Q}(c)=\mathbb{Q}(\nu )$. Finite rank-nullity therefore gives ${\dim }_K{C}_K[\alpha ]={\mathrm{rank}}_K{G}_{\alpha }(c)\le p_{\alpha }\mathrm{.}$ The final inequality is [F2]. This proves the dimension consequence without asserting freeness of $U_A^{-}[\alpha ]$, nondegeneracy of the generic or specialized form, or injectivity of any numerical Serre-presentation map.