The generic Hecke algebra comparison does not extend to every parameter

In the degenerate quotient $x_1=m$ and $s{x}_2=ms+1$ give $x_2=m+s$. Hence the quotient is generated by $s$ with $s^2=1$. Conversely the assignments $x_1\mapsto m$, $x_2\mapsto m+s$, $s\mapsto s$ in $\mathbb{C}[s]\mathrm{/}(s^2-1)$ satisfy both cross relations and commutativity, so they define an inverse to the map sending $s$ into the quotient. Evaluation at $s=1,-1$ identifies this algebra with $\mathbb{C}\oplus \mathbb{C}$, with inverse $(a,b)\mapsto (a+b)\mathrm{/}2+(a-b)s\mathrm{/}2$. In particular it has no nonzero nilpotents and has precisely the two one-dimensional simple modules $s=1,-1$.

In the nondegenerate level-one quotient [F3], $X_1=Q$ and $X_2=Q{\xi }^{-1}{T}^2$, so it is generated by $T$. Conversely in the finite algebra $H_{\xi }(2)$ of [F1], $T^{-1}={\xi }^{-1}(T-(\xi -1))$. The assignments $X_1\mapsto Q$, $X_2\mapsto Q{\xi }^{-1}{T}^2$, $T\mapsto T$ therefore give commuting units satisfying $T{X}_{1}T=\xi X_2$ and the cyclotomic relation. They define the inverse of the map from $H_{\xi }(2)$ into the quotient: both composites fix $T,X_1,X_2$ and their inverses. Hence the whole nondegenerate level-one rank-two quotient is exactly $H_{\xi }(2)$ for every $\xi \mathrm{\ne }0$ and $Q={\xi }^m$. This uses only the arbitrary-base affine and monic definitions, not a finite-$e$ theorem in the generic branch.

At $\xi =-1$, step 1.2 and [F1] give $H_{-1}(2)=\mathbb{C}[T]\mathrm{/}((T+1{)}^2)$. Setting $\epsilon =T+1$ identifies it with $B=\mathbb{C}[\epsilon ]\mathrm{/}({\epsilon }^2)$, whose basis is $1,\epsilon$ by division by the monic polynomial ${\epsilon }^2$. Thus $\epsilon$ is nonzero and nilpotent. Every $a+b\epsilon$ with $a\mathrm{\ne }0$ has inverse $a^{-1}-b{a}^{-2}\epsilon$; those with $a=0$ are nonunits. Consequently $(\epsilon )$ is the unique maximal ideal and is the nonzero radical. Every simple $B$-module is killed by $\epsilon$: its image under this central operator is a submodule, and if the image were the whole nonzero simple module, its square could not vanish. The unique simple is therefore $B\mathrm{/}(\epsilon )\cong \mathbb{C}$.

At rank two the limitation can be seen in an actual comparison formula. For $\xi \mathrm{\ne }-1$ put $a=(\xi -1)\mathrm{/}2$ and $b=(\xi +1)\mathrm{/}2$. The map $T\mapsto a+bs$ respects [F1] because its values at $s=1,-1$ are $\xi ,-1$; its inverse is $s\mapsto (T-a)\mathrm{/}b$, since the quadratic relation implies $(T-a{)}^2=b^2$. At $\xi =-1$ this inverse is undefined and the specialized map sends $T$ to $-1$, killing the nonzero element $T+1$. This explicit generic comparison therefore does not specialize to an isomorphism. Other rank-two parameters can yield isomorphisms, including some roots of unity; that does not repair the universal assertion.

The two algebras cannot be isomorphic: an isomorphism would send the nonzero square-zero element of step 2.1 to a nonzero nilpotent in the algebra of step 1.1. Their decomposition behavior also differs directly. The regular module of $\mathbb{C}\oplus \mathbb{C}$ is the direct sum of its two simples. The regular module $B$ has the filtration $0\subset (\epsilon )\subset B$ with both subquotients equal to its unique simple; it is indecomposable since an idempotent $a+b\epsilon$ satisfies $a^2=a$ and $2ab=b$, forcing it to be $0$ or $1$. Thus even projective and simple classes cannot be identified across these two specializations by algebra transport.

Steps 1.1, 2.1 and 3.1 disprove the claimed universal algebra transport and step 2.2 exhibits its failure in a concrete comparison formula. The witness uses the entire rank-two algebras, with the degenerate full integer residue set; it is not a comparison with a supported cut retaining only the charge $m$. For a finite degenerate cut containing this entire example the root interval must contain $m-1,m,m+1$, since $x_2=m\pm 1$ on its two simples. The calculation is choice-free and entirely over $\mathbb{C}$. It neither extends characteristic-zero canonical-basis identifications to positive characteristic nor replaces the separate root-of-unity comparison and grading theorems.