Generic cyclotomic local dictionaries glue to a tower

Bind the actual proved suppliers. The arbitrary-base affine presentations in the Statement specialize the common defining generators and relations over k, with first-strand monic polynomials ${\prod }_j(t-m_j)$ and ${\prod }_j(t-{\xi }^{m_j})$; the latter has unit constant term. Thus the finite-dimensional spanning and spectral-support proof [F1] applies at every level and rank. Its line datum satisfies the native quadratic symmetry and homogeneous extremal conditions with $d_i=1$, and its standard exponents are exactly ${\lambda }_i=\mathrm{\#}\{j:m_j=i\}$. On every finite interval its support proof gives actual whole-algebra equality, global nilpotent dots and polynomial projectors over k, not just a supported corner. The two local suppliers in [F2] use the SAME native $\tau$ and set $\psi =-\tau$: both have $\psi u-v\psi =-1$, $\psi v-u\psi =1$, quadratic $Q_{a,a+1}=v-u$, $Q_{a+1,a}=u-v$, and braid difference ${\psi }_1{\psi }_2{\psi }_1-{\psi }_2{\psi }_1{\psi }_2=[Q(u,v)-Q(w,v)]\mathrm{/}(w-u)$ on $(a,b,a)$. Their pair series are fixed for all integer pairs and all ranks; the nondegenerate symbol q is our $\xi$. Their proofs give faithful ACTUAL affine polynomial presentations before completion and before any quotient, then identify the true central completions with one formal dot ring per word. They prove both map directions on all five equality patterns of three colors, including all equal, and inverse composites with the displayed right multiplication order. Therefore they apply to exactly the affine blocks under consideration; no finite cyclic quantum characteristic, generic-rank faithfulness or cyclotomic polynomial faithfulness is assumed.

Evaluate a formal series in commuting nilpotent dots by its finite Taylor sum. This evaluation is independent of the nilpotence exponent chosen: every larger truncation differs only by zero monomials. A series with nonzero constant term is a unit by a terminating geometric expansion. Thus all replacements in the Statement are defined. By [F1] only finitely many words survive in any rank. The local relation verifications in [F2], applied on those words, give homomorphisms between the affine presentations before checking the first-strand quotient. No cyclic wraparound or reduction of integer colors is used.

On a first color $i$, write the degenerate cyclotomic polynomial as $f(i+y)=y^{{\lambda }_i}{U}_i(y)$, where $U_i(y)={\prod }_{m_j\mathrm{\ne }i}(i-m_j+y)$. Its constant term is nonzero in characteristic zero. In the other branch $g({\xi }^i(1-y))=(-{\xi }^i{)}^{{\lambda }_i}{y}^{{\lambda }_i}{V}_i(y)$, where $V_i(y)={\prod }_{m_j\mathrm{\ne }i}({\xi }^i-{\xi }^{m_j}-{\xi }^{i}y)$. Infinite multiplicative order makes its constant term nonzero. Both auxiliary factors are units after nilpotent evaluation. Thus the Hecke cyclotomic relation on each projector is equivalent to $y_1^{{\lambda }_i}e(i,\dots )=0$. If ${\lambda }_i=0$, the unit polynomial kills the entire projector. This proves both directions of cyclotomic IDEAL transport: in each completed word corner the transformed Hecke generator is an invertible scalar and unit power series times $y_1^{{\lambda }_i}e(i,\dots )$. Multiplication by that factor gives containment in the KLR-generated two-sided ideal; multiplication by its inverse gives the reverse containment. Summing the finitely many orthogonal first-color corners proves equality of the full two-sided ideals. On either nilpotent quotient these units evaluate by finite Taylor sums, so the actual affine maps descend to homomorphisms of the entire cyclotomic algebras in both directions. This does not assume that a restricted supported subalgebra is the full quotient.

Denote the descended reverse map by ${\sigma }_d:H_d\to R_d^{\lambda }$. On $e_R(\nu )R_d^{\lambda }$, each ${\sigma }_d(x_r)$ has sole generalized eigenvalue ${\nu }_r$, or ${\sigma }_d(X_r)$ has sole generalized eigenvalue ${\xi }^{{\nu }_r}$. Distinct colors have distinct values, and the orthogonal word idempotents sum to one. The simultaneous primary projector is therefore precisely left multiplication by $e_R(\nu )$. The Hecke polynomial projector maps to this idempotent: its Chinese-remainder polynomial is one modulo the corresponding primary factors and zero modulo all other factors. Hence ${\sigma }_d(e_H(\nu ))=e_R(\nu )$. The forward map already sends $e_R(\nu )$ to $e_H(\nu )$.

Both composites fix the word projectors by step 4.1 and the dots by direct substitution $x_r-{\nu }_r=y_r$, or $1-{\xi }^{-{\nu }_r}{X}_r=y_r$. On a word, the crossing composite is $({\psi }_r{q}_{ab}-p_{ab}+p_{ab})q_{ab}^{-1}={\psi }_r$, or $({\psi }_r{Q}_{ab}-P_{ab}+P_{ab})Q_{ab}^{-1}={\psi }_r$. Conversely the Hecke composite sends $s_r{e}_H(\nu )$ to $((s_r+p_{ab})q_{ab}^{-1}{q}_{ab}-p_{ab})e_H(\nu )=s_r{e}_H(\nu )$; the formula with $T,P,Q$ is identical. The polynomial generators are fixed, and the units $X_r^{-1}$ are fixed because inverses are unique. Thus the homomorphisms are inverse on every generator, without commuting a crossing through its series.

For ranks through $D>0$ at positive level use the common interval $W_D=[\min S-D,\max S+D]\cap \mathbb{Z}$. It contains the support through rank $D$ and one unused ghost color beyond each endpoint of $B_D$. A crossing permutes existing colors and never creates one; nevertheless these endpoint colors keep every boundary edge of a surviving color represented by the integer-line convention. For operations reaching rank $D+t$, use $W_{D+t}$, and enlarge further to contain any specified color and its neighbors. All extra word idempotents vanish by [F1]. Since [F2] fixes the same series for each ordered pair in every interval, evaluation on surviving words is unchanged by enlargement. The resulting maps are independent of windows and glue across all finite rank bounds.

Let ${\iota }_H:H_d\to H_{d+1}$ send each existing Hecke generator to the generator with the same index. It is a homomorphism because all rank-$d$ relations, including the first-strand relation, hold in rank $d+1$. Its spectral projector satisfies ${\iota }_H(e_H(\nu ))={\sum }_a{e}_H(\nu ,a)$: the first $d$ polynomial generators give a representation of $H_d$ on the rank-$d+1$ regular module, so their primary projector is the sum of exactly these joint projectors. The minimal polynomial of each rank-d algebra generator on its regular module annihilates that algebra element itself, hence every representation of $H_d$; the same is true of the polynomial primary projectors. Thus the first-d primary decomposition on the rank-d+1 regular module contains exactly the rank-d allowed eigenvalues, and every prefix violating rank-d support is zero. This is the algebra-identity consequence of [F1], rather than an unsupported extension of a regular-module assertion. This argument also holds for $d=0$, where the empty projector is one.

Define ${\iota }_R={\sigma }_{d+1}{\iota }_H{\varphi }_d$. The established inverse maps make this a homomorphism. Steps 4.1 and 7.1 give ${\iota }_R(e_R(\nu ))={\sum }_a{e}_R(\nu ,a)$. The unchanged polynomial generators give the usual prefix dots. For $r<d$, the rank-independent pair series in [F2] and cancellation in step 5.1 give ${\iota }_R({\psi }_r{e}_R(\nu ))={\sum }_a{\psi }_r{e}_R(\nu ,a)$. These are exactly the standard right-appending KLR maps, so the individually constructed isomorphisms commute with the actual tower maps. Injectivity of the tower maps is neither needed nor inferred.

Put ${ϵ}_i^R={\sum }_{\nu }{e}_R(\nu ,i)$ in rank $d+1$, and define ${ϵ}_i^H$ by the analogous spectral sum. Step 4.1 gives ${\varphi }_{d+1}({ϵ}_i^R)={ϵ}_i^H$. Each last-color projector commutes with every prefix generator, since prefix crossings permute only the first $d$ colors and prefix dots preserve all colors. On left modules color restriction is $N\mapsto {ϵ}_{i}N$, with the rank-$d$ action through its prefix map. Color induction is $M\mapsto A_{d+1}{ϵ}_i{\otimes }_{A_d}M$. Step 8.1 identifies both prefix actions, and the algebra isomorphism identifies these induction bimodules on both sides. Explicitly $a{ϵ}_i\otimes m\mapsto {\varphi }_{d+1}(a){ϵ}_i\otimes m$ is a balanced bijection, with inverse obtained from ${\sigma }_{d+1}$. Restriction agrees by the same projector equality. These formulas are natural in module maps.

Composing the two comparisons through the same $R_d^{\lambda }$ proves the claimed degenerate/nondegenerate tower equivalence. Naturality from step 9.1 intertwines repeated induction and restriction, while step 6.1 covers every specified finite rank cut. At level zero use the identity on rank zero and the unique maps between zero algebras in positive rank; the sole map from rank zero to rank one is the map to zero, and all positive-rank functors vanish. Thus all ranks and levels are included. The actual presentation and convention binding in step 1.1 discharges [F1]–[F2], so this proves the entire all-rank, all-level tower comparison. No based or standard-class theorem is inferred merely from these algebra isomorphisms.