Tensor bars on finite products of based integrable modules

Actual support bounds and finite correction closure. The $O_{\mathrm{i}\mathrm{n}\mathrm{t}}$ hypothesis includes finite upper-cone support: for each factor $M_j$, $\mathrm{supp}{M}_j\subseteq {\bigcup }_a({\lambda }_{j,a}-Q_{+})$. Fix a total label $\mu$ and a choice of these finitely many upper labels. A tensor label tuple satisfies ${\mu }_j={\lambda }_{j,a_j}-{\beta }_j$, ${\beta }_j\in Q_{+}$ and ${\sum }_j{\beta }_j={\sum }_j{\lambda }_{j,a_j}-\mu$. Independent simple-root coordinates bound every coordinate of every ${\beta }_j$, so there are finitely many tuples, regardless of multiplicities at their labels. Every strict root transfer increases a prefix label by a nonzero element of $Q_{+}$; hence a chain of transfers visits no tuple twice and has bounded length. Each evaluated bar column is a finite sum of raw basis vectors: the tuple bounds permit only finitely many root degrees, each degree has a finite algebra expansion, and vectors in a field basis have finite support. Starting from a finite set of raw seeds, recursively adjoin the raw support of their bar columns. This is a finitely branching tree with the bounded strict-transfer depth just proved, so its closure is finite. The correction and inverse unitriangular transition therefore take place in a finite raw-basis set, even if some label spaces have infinite multiplicity. In finite Cartan type there is also a global label bound for every upper-cone integrable factor. Indeed a nonzero vector at $\eta$ has a nonzero admissible $i$-string position $F_i^{(n)}v$ with top $\eta +n{\alpha }_i$ and top coroot $a=⟨\eta ,h_i⟩+2n$; its mirrored position $a-n$ is nonzero by [F5] and has label $s_i\eta$. Thus support is stable under every $s_i$, including on nonseparating chosen label gradings. Applying the longest element $w_0$ shows that the same support lies in ${\bigcup }_b(w_0{\lambda }_b+Q_{+})$. The intersection of an upper cone ${\lambda }_a-Q_{+}$ and a lower cone $w_0{\lambda }_b+Q_{+}$ consists of labels $w_0{\lambda }_b+\gamma$ with $0\le \gamma \le {\lambda }_a-w_0{\lambda }_b$ coefficientwise in the simple-root basis, so is finite or empty. The finite union of these intersections bounds the entire support. No finite-dimensional-weight-space inference for arbitrary based integrable modules is used. These arguments apply to the stated bounded category; an unrestricted two-sided-unbounded based pair still needs a separate finite-action and supported-correction proof.

The field tensor bar. For the empty sequence take the tensor unit $\mathbf{1}=\mathbb{Q}(q)1$ of weight zero, with $u\cdot 1=\epsilon (u)1$: the counit of the quantum-group structure underlying [F1] has $\epsilon (E_i)=\epsilon (F_i)=0$ and $\epsilon (K_h)=1$. Its basis is $\{1\}$, its crystal operators vanish, and its bar is ${\mathrm{\Psi }}_{\mathrm{\varnothing }}(f(q)1)=f(q^{-1})1$. These generator actions show integrability and bar compatibility; the divided-power lattice is $A1$, stable because positive divided powers act by zero and each $K_h$ acts by one and each $T_{i,c,t}$ acts by the Gaussian scalar ${\left(\genfrac{}{}{0px}{}{c}{t}\right)}_i\in A$. Its zero and infinity charts are $A_{0}1$ and $A_{\mathrm{\infty }}1={\mathrm{\Psi }}_{\mathrm{\varnothing }}(A_{0}1)$. A Laurent polynomial regular at zero has no negative powers and one regular at infinity has no positive powers, so $A\cap A_0\cap A_{\mathrm{\infty }}=\mathbb{Z}$ and $A\cap A_0=\mathbb{Z}[q]$. Thus the balanced map of [F3] is $\mathbb{Z}1\to \mathbb{Z}[q]1\mathrm{/}q\mathbb{Z}[q]1$, an isomorphism, and its unique global lift is $1$, with no correction. Every positive-root component of $\mathrm{\Theta }$ annihilates a tensor pair containing this unit, since its positive or negative quantum half acts trivially there, while ${\mathrm{\Theta }}_0=1$ acts as the identity; hence adjoining a unit leaves the tensor bar unchanged under the canonical unit identifications. The empty case is therefore vectorwise finite, involutive, integral, balanced and independent of parenthesization. For arbitrary integrable finite-Cartan factors, the supplier's finite-action clause proves tensor integrability and successive vectorwise finite quasi-$R$ evaluation without support or multiplicity bounds; the same field-bar calculations below therefore apply to those factors as well. For a single factor use its given bar. Define the binary bar ${\mathrm{\Psi }}_{M,N}=\mathrm{\Theta }({\psi }_M\otimes {\psi }_N)$ and, for a finite sequence, the iterated composition; on each $O_{\mathrm{i}\mathrm{n}\mathrm{t}}$ factor pair the action of $\mathrm{\Theta }$ is vectorwise finite by the explicit label-tuple bound in step 1.1; for the separately admitted ordered lowest inputs, the supplied lower-cone bounds hold before evaluation. $\mathrm{\Psi }$ is semilinear because $\psi$ sends $q$ to $q^{-1}$. Since ${\mathrm{\Theta }}^{-1}=\bar{\mathrm{\Theta }}$ by [F1], applying the binary bar twice gives $\mathrm{\Theta }\bar{\mathrm{\Theta }}=\mathrm{i}\mathrm{d}$, so ${\mathrm{\Psi }}^2=\mathrm{i}\mathrm{d}$; and $\mathrm{\Delta }(u)\mathrm{\Theta }=\mathrm{\Theta }\bar{\mathrm{\Delta }}(u)$ gives $\mathrm{\Psi }(ux)=\psi (u)\mathrm{\Psi }(x)$ for every $u$ and every tensor $x$ on which the action is defined.

Integral preservation and triangularity. In all the integral action cases of the supplier, including arbitrary based finite-type pairs, $\mathrm{\Psi }$ preserves the divided-power integral tensor lattice and is an $A$-bijection because ${\mathrm{\Psi }}^2=\mathrm{i}\mathrm{d}$; every positive-root component of $\mathrm{\Theta }$ raises the first label and lowers the second, and only the degree-$0$ component fixes a label pair, so the action on raw tensor lower basis vectors is triangular with diagonal $1$ in the weight-transfer order. Finite raw-basis correction closure additionally holds in the bounded category by step 1.1 and in the separately supported ordered cases of [F2]; it is not inferred from unrestricted finite-type action integrality.

Independence of parenthesizations. For three factors the two bracketings compute $((\mathrm{\Delta }\otimes \mathrm{i}\mathrm{d})\mathrm{\Theta }){\mathrm{\Theta }}_{12}$ and $((\mathrm{i}\mathrm{d}\otimes \mathrm{\Delta })\mathrm{\Theta }){\mathrm{\Theta }}_{23}$ composed with the factor bars, and the field cocycle of [F1] makes them equal coefficientwise; since every rebracketing of a finite sequence is generated by binary reassociations, all parenthesizations give the same $\mathrm{\Psi }$, with all sums finite in each total root degree. This is the field-level assertion and requires no integral action.

The corrected tensor global vectors. Within the finite raw-basis closure proved in step 1.1, or the separately supplied bounded ordered interval, apply the semilinear correction algorithm of [F2]: in each such finite transfer interval solve $g_t=e_t+{\sum }_{s>t}{a}_{ts}{e}_s$ with $a_{ts}\in q\mathbb{Z}[q]$ for the bar-fixed lift of the raw vector $e_t$. Involutivity forces the remainder to satisfy $f+\bar{f}=0$, whose unique positive $q$-part supplies the next coefficient; finite induction yields existence and uniqueness of the corrected family, which is an $A$-basis of the tensor lattice congruent to the raw basis modulo $q$, and an $A_0$-basis whose residues are precisely the raw tensor crystal basis supplied by [F4]. The raw vectors themselves are generally not bar fixed, which is why the correction is needed.

The integral balanced tensor triple. Since the corrected family is $\mathrm{\Psi }$-fixed, its $A_{\mathrm{\infty }}$-span is the actual infinity chart $\mathrm{\Psi }(L_0^{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}})$ and its $A_0$-span is the original tensor zero chart; intersecting the Laurent coefficient rings in this common basis gives the $\mathbb{Z}$-span as the balanced intersection $L_0^{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}}\cap \mathrm{\Psi }(L_0^{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}})\cap A$-lattice, so the integral globalization isomorphisms of [F3] identify the corrected family as the genuine tensor global basis and prove integral balancedness. This uses the crystal lattice only as the residue datum and does not assert that $\mathrm{\Psi }$ preserves $A_0$ or that the bar chart equals $L_0^{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}}$.

Based subquotients and scope. Exact based subobjects are saturated basis subsets and based quotients have basis-subset kernels; both commute with bars and crystal operators, and the image of a corrected global vector is integral and bar fixed with one raw residue or $0$, so balanced uniqueness [F3] proves the basis-or-$0$ compatibility and the corrected bases descend to exact tensor subquotients and direct sums within the stated bounds. Outside the listed cases no integral action is claimed, and the affine counterexample of the supplier shows it can fail; no universal braiding or semi-infinite evaluation is asserted.

The unrestricted finite-type globalization boundary. The actual integral rank-one bar in [F6] is algebraic and involutive but has no finite normalized lift of its seed: projecting a putative finite lift to a cut $J$ beyond its support by two identifies it, by finite-cut uniqueness, with a cut lift having a nonzero coordinate of index at least $J-1$. For the natural integral balanced map, an element $g$ of the zero/bar intersection has both $g$ and $\mathrm{\Psi }g$ with polynomial raw coordinates. The anti-invariant difference $\mathrm{\Psi }g-g$ must vanish: at its least first weight its polynomial coefficient satisfies $\bar{s}=-s$, hence is zero. The prescribed residue and the least-first-weight argument eliminate every polynomial correction at or below the seed weight. Thus a balanced lift of the seed would have the finite triangular normalization just contradicted. The resulting failure concerns the unrestricted based extension; steps 3.2–5.1 retain the entire supported full-tensor construction and its inheritance.

Conclusion. Step 1.1 proves the bounded-category finite-support bridge; steps 2.1 and 3.1 prove the well-defined semilinear involutive field bar and its parenthesization independence; steps 2.2, 3.2 and 4.1 prove integral preservation, the triangular correction, the corrected global basis and the balanced tensor triple in the supported construction cases; step 5.1 proves subquotient inheritance, and step 6.1 refutes unrestricted finite-type globalization while retaining the finite integral bar action.