Finite source normal form over the Dubrovnik coefficient extension

Define $f:R\to T$ by $q\mapsto q$, $r\mapsto a$, $\mathrm{\Delta }\mapsto d$. Its defining relation maps to zero by [F3], and its unit generators map to units. Conversely define $h:T\to R$ by $a\mapsto r$, $d\mapsto \mathrm{\Delta }$, $q\mapsto q$, and $z\mapsto q-q^{-1}$. The relation of $S$ maps to the relation of $R$, and $q-q^{-1}-z$ maps to zero. On each displayed ring generator, both composites are the identity. Thus $f,h$ are inverse ring isomorphisms, without inverting $z$ or requiring a flatness argument.

Transport the source presentation along $f$. Send each $g_i^{\pm 1}$ and $e_i$ to the identically named generator over $T$. The braid, commutation and cup-slide relations are literally identical, while the remaining coefficients become $q-q^{-1}$, $a^{-1}$, $a$, and $d$, exactly as in [F2] and [F3]. The inverse coefficient map $h$ gives an inverse map of presented algebras, because it fixes the generator names and reverses each coefficient substitution. This verifies the source comparison without assuming that either tangle realization is injective.

Transport the basis in [F1] through this algebra isomorphism. If $b_m$ denotes the transported basis element with matching index $m$, every source element has one expression ${\sum }_m{t}_m{b}_m$ with $t_m\in T$, and these expressions are unique. This is the finite source normal form supplied by the already proved local universal theorem, whose ideal/shuffle proof concerns the abstract algebra rather than a dimension inference from a tangle quotient. A perfect matching pairs the first endpoint in $2n-1$ ways and then matches the remaining endpoints; hence there are $N_n=(2n-1)!!$ indices, and one empty index at $n=0$. The scalar definitions give the same rank-one conclusion at $n=1$.