The tensor-product bar on the Fock module and its truncation compatibility

The bar on a finite cut. On a finite cut take the reduced finite product of the tensor factors $V({\mathrm{\Lambda }}_{m_j})$ and equip each factor with its own normalized bar of [F3]. The finite-product theorem [F1] produces an involutive semilinear bar on the tensor product, compatible with the factor bars; by [F2] the basis of this module is the set of column monomials $M_A$ of the cut. This defines $\psi$ on each finite cut.

Triangularity. In the actual positive-first-factor quasi-$R$ convention, every nonconstant term raises the weight in an earlier tensor prefix and lowers the complementary suffix by the same positive root. The prefix-dominance definition therefore puts every resulting label strictly below $A$. The constant term is one, and each factor bar fixes its ordered minuscule wedge vector. The finite-product divided-power integrality makes every coefficient a Laurent integer, proving the displayed triangular formula.

Involution and parenthesization independence. Both properties are inherited from the finite-product theorem [F1]: applying the tensor bar twice returns the original element because each factor bar is involutive and the quasi-$R$ twist is inverted by the inverse factorization; reassociating the tensor factors replaces the product of quasi-$R$ twists by its twisted product, and the intertwining relations [F1] identify the corresponding bars.

Integral monomial lattice. Since $\psi M_A$ differs from $M_A$ by integral combinations of lower monomials (step 2.1), the $\mathbb{Z}[s^{\pm 1}]$-span of the $M_A$ is stable under $\psi$; the same holds for ${\psi }^{-1}=\psi$.

Upper cuts. Consider a larger upper coordinate interval and a tensor supported on the old alphabet. A positive operator in an earlier prefix can only replace a letter $a+1$ by $a$, never create a letter above the old upper bound. Every quasi-$R$ term involving a new upper-boundary root therefore kills that prefix. Terms supported on old roots agree with their old-cut terms, and the fundamental bars fix the same ordered vectors. Reassociate successive prefixes using step 2.2. Thus the larger bar restricts to the old bar, rather than relying on an incorrect letter-replacement description of the factor bars.

Common lower tails. After appending $k-1$ to every factor, each earlier tensor prefix has every wedge occupied at $k-1$. Any positive-root word involving the new boundary must apply $E_{k-1}$ at some stage. Earlier positive replacements on old roots preserve that occupancy; $E_{k-1}$ either encounters no $k$ or attempts to create a duplicate $k-1$, and kills each factor, hence the prefix. The entire quasi-$R$ correction involving the new boundary is zero. The old-root toral weights of the appended tail are zero, so all surviving old-root terms and factor bars agree with the old calculation. This proves $\psi {\iota }_k={\iota }_k\psi$ on finite products. Common-root string operators and integral lattices are preserved; uniqueness of balanced lifts therefore preserves the actual charts and bases. No assertion says all new generators vanish on the image.

The direct limit. Every vector of the full Fock module lies in the image of some finite double cut, and every calculation in the expansion of $\psi$ is finite and stabilizes in a cut containing the residue and the input and output supports by steps 3.1, 3.2. Hence the compatible family of bars defines a bar on the direct limit, still involutive and triangular, and it fixes each ordered fundamental wedge. No infinite series is taken.

Conclusion. On every finite cut the Fock module has the ordinary compatible bar $\psi$ of (1)-(2); the upper-cut and common-tail embeddings commute with it by (3), so it descends to the full bounded or semi-infinite module. All uses are confined to actual finite-Cartan based fundamental factors; neither a universal two-sided-unbounded tensor-bar theorem nor an arbitrary integral quasi-$R$ action is asserted.