Cyclotomic extension of scalars respects the based quotient

The raw extension map on divided words. Right append sends a word ${\nu }_1\cdots {\nu }_n$ to $F_{{\nu }_n}\cdots F_{{\nu }_1}{v}_{\mathrm{\Lambda }}$, whereas the generic multiplication convention assigns $F_{{\nu }_1}\cdots F_{{\nu }_n}$; on actual divided word classes the map is therefore ${\pi }_{\mathrm{\Lambda }}\circ \ast$. Their $A$-span is all generic $K_0$ by [F1], so this determines the map integrally, giving $\varphi [q_{\mathrm{\Lambda }}(P)]={\pi }_{\mathrm{\Lambda }}(\ast ({\gamma }^{-1}[P]))$; composing with the generic mirror twist of [F2], which supplies exactly $\ast$ and commutes with projective Hom duality, gives ${\pi }_{\mathrm{\Lambda }}$ without star.

Positive star on the global basis. The generic mirror equivalence of [F2] preserves indecomposability and commutes with projective Hom duality, hence permutes the normalized generic covers. Once the generic global theorem in [F1] is instantiated, their classes are exactly the positive lower global vectors. Its induced quantum star therefore permutes these vectors with positive sign: $\ast G^{-}(b)=G^{-}(b^{\ast })$. This is a consequence of the generic global theorem used here, not an input to its geometric or sign proof.

Survival and non-merging. For a surviving normalized generic simple $S$, quotient Hom base change in the cover-descent lemma [F3] sends its normalized projective cover to the cyclotomic normalized cover, commutes with $\mathrm{\#}$, and preserves its unscaled head pairing; if its simple is killed, the quotient is zero, because any nonzero finite quotient of the cover has a simple head and adjunction would produce a surviving generic head of the original cover. Thus raw extension of a generic normalized indecomposable projective is zero or a single normalized indecomposable cyclotomic projective, and distinct surviving heads do not merge; the image is ${\pi }_{\mathrm{\Lambda }}{G}^{-}(b^{\ast })$ respectively $G_{\mathrm{\Lambda }}^{-}({\pi }_{\mathrm{\Lambda }}{b}^{\ast })$ under the projections of [F1].

All-color label comparison in exact string quotients. In the scope where the generic global theorem [F1] has been established, define $\eta (S)$ by ${\gamma }^{-1}[P(S)]=G^{-}(\eta (S))$; equality of positive basis sets makes $\eta$ a bijection, not yet a crystal comparison. For every color $i$ the LEFT-string ideal equality of [F4], together with the adapted global basis, gives ${\epsilon }_i(\eta (S))={ϵ}_i^L(S)$. If this common number is $r>0$, put $T=(e_i^L{)}^{r}S$, so ${ϵ}_i^L(T)=0$ and $T$ has smaller height. The coefficient-$1$ projective expansion of [F4] gives $F_i^{(r)}{G}^{-}(\eta (T))-G^{-}(\eta (S))\in I_{(i,\ge r+1)}$. The exact adapted quantum transition in [F4] gives $F_i^{(r)}{G}^{-}(\eta (T))-G^{-}({\tilde{f}}_i^r\eta (T))\in I_{(i,\ge r+1)}$. Subtract these congruences. Both global vectors have exact $i$-position $r$, and their classes belong to the free adapted basis of $I_{(i,\ge r)}\mathrm{/}{I}_{(i,\ge r+1)}$; equality of these classes therefore forces $\eta (S)={\tilde{f}}_i^r\eta (T)$, rather than merely equality of tail lengths. Replacing $S$ by $e_i^{L}S$ in the same calculation with $r-1$ gives $\eta (e_i^{L}S)={\tilde{e}}_i\eta (S)$, and the partial-inverse identities give the corresponding $f_i^L$ equality. This holds for EVERY color with positive first tail. The unit maps to $1$, and every nonvacuum simple has a positive first tail since some first-word corner is nonzero; finite height induction now proves compatibility along every first-end lowering word. No path choice enters: $\eta$ was already defined by its actual global vector, and the exact quotient equality proves that all paths compute this same label. The mirror equivalence of [F2] exchanges FIRST and RIGHT operators on simple shift orbits, fixes the vacuum and induces star on normalized cover classes. Hence, once the general canonical RIGHT-label identification of [F3] is supplied, ${\gamma }^{-1}[P(S_c)]=G^{-}(c^{\ast })$ and step 1.1 cancels the two stars to give $G_{\mathrm{\Lambda }}^{-}({\pi }_{\mathrm{\Lambda }}c)$. Every normalized cyclotomic cover is obtained by inflating its simple and descending its generic cover. The Axiom of Choice is inherited through the generic global-basis, adapted quantum-string and quantum-projection contracts in [F1], [F4] and the canonical-extraction supplier of [F3]; no additional infinite choice is used. This proves the exact all-color adapted comparison conditional on the actual generic global theorem and canonical graph identification, and does not establish either unresolved supplier by an equality of basis sets.