The semi-infinite q-wedge Fock space and its q-adic topology

Finite-stage spanning and ordered-tail stabilization. Start with a finite word of length $r$ and choose $S\ge r$ so that the last appended ground index $L=M-S+1$ is below all its original indices. Its image at stage $S$ has $S$ entries, all at least $L$. By [F1], its finite straightening is a finite sum of normal tuples $n_1>\cdots >n_S\ge L$. For each resulting tuple put ${\lambda }_j=n_j-(M-j+1)$. Then ${\lambda }_S\ge 0$ and ${\lambda }_j\ge {\lambda }_{j+1}$, so extending by zeros gives a partition. Every subsequent connecting map appends ground indices $M-S,M-S-1,\dots$, strictly below the normal tuple's last entry; that image is already normal and is the longer prefix of the same partition word. Thus every direct-limit element is a finite combination of partition words with a stable representative under these ordered-tail extensions. Maps with different finite codomains are not equated.

Direct verification of the vacuum weights. Let $M=te+r$ with $t\in \mathbb{Z}$ and $0\le r<e$, and put ${\mathrm{\Omega }}_M={\mathrm{\Lambda }}_r-[et(t-1)\mathrm{/}2+tr]\delta$. It gives ${\mathrm{\Omega }}_0={\mathrm{\Lambda }}_0$. If $r>0$, subtraction of the formula for $M-1=te+(r-1)$ gives ${\mathrm{\Omega }}_M-{\mathrm{\Omega }}_{M-1}={\mathrm{\Lambda }}_r-{\mathrm{\Lambda }}_{r-1}-t\delta ={\lambda }_r-t\delta$. Here $u_M=z^{-t}{v}_r$, so this is exactly $\mathrm{w}\mathrm{t}(u_M)$ by [F2]. If $r=0$, use $M-1=(t-1)e+(e-1)$: subtraction gives ${\mathrm{\Omega }}_M-{\mathrm{\Omega }}_{M-1}={\mathrm{\Lambda }}_0-{\mathrm{\Lambda }}_{e-1}+(1-t)\delta ={\lambda }_e+(1-t)\delta$, and $u_M=z^{1-t}{v}_e$ has precisely this weight. These integer identities also hold for negative $t$. The base value and this recurrence uniquely determine the weights for all integer charges. The partition label is consequently the stipulated ${\mathrm{\Omega }}_M-{\sum }_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}}{\alpha }_{c-r+M}$, whose $d$-evaluation is ${\mathrm{\Omega }}_M(d)-m_0$ by the explicit datum; size remains the separate grading. No fractional change of an integer loop exponent or infinite unregularized weight sum is used.

Partition indexing and direct-limit independence. An eventually-ground normal word has $i_k=M+{\lambda }_k-k+1$ with ${\lambda }_k$ weakly decreasing and eventually zero; hence the ${\lambda }_k$ are nonnegative and form a partition, and every partition gives such a word. For finitely many distinct partitions choose a common prefix length $N$ containing all their nonzero rows and every position where two words differ. Each partition prefix is normal and has last index $i_N\ge M-N+1$. Its next appended index is $M-N$, strictly smaller, so every later image remains a normal basis word, and the distinct partition images remain distinct. If a finite linear combination vanished in the algebraic direct limit, it would vanish at some common later finite stage: a colimit relation uses finitely many connecting-map relations, which can all be moved to that common stage. Independence of the finite normal basis in [F1] then forces every coefficient to vanish. Together with step 1.1, this proves the partition basis without assuming that arbitrary finite words survive tail attachment.

The charts and the topology. Each displayed chart is free on the partition basis over its own ring. Both $F_A$ and $F_0$ contain $F_{\mathbb{Z}0}$, and $F_A\cap F_0=F_{\mathbb{Z}0}$ coefficientwise: an integral Laurent polynomial is regular at zero exactly when it has no negative powers. Neither chart is identified with the other. A nonzero vector of the local $A_0$-chart has finitely many nonzero coefficients, each of finite $q$-valuation, so ${\bigcap }_{n\ge 0}{q}^n{F}_0=0$. On each finite label component the completion of $A_0=\mathbb{Q}[q{]}_{(q)}$ is $\mathbb{Q}[\text{\hspace{0.17em}⁣}[q]\text{\hspace{0.17em}⁣}]$, and inverting $q$ gives the field completion $\mathbb{Q}((q))$; the corresponding full-field component is $\mathbb{Q}((q)){\otimes }_{\mathbb{Q}(q)}{F}_{\mu }^{(M)}$. Separately the completion of the integral $\mathbb{Z}[q]$-chart is $\mathbb{Z}[\text{\hspace{0.17em}⁣}[q]\text{\hspace{0.17em}⁣}]$. The Laurent lattice has $q{F}_A=F_A$ and therefore supplies no separated $q$-adic filtration. These statements concern the componentwise completions stipulated in the Definition; no full infinite direct sum is identified with a finite-rank completed lattice.

Conclusion. The charge-$M$ Fock space is the algebraic direct limit of the finite wedge spaces along the ground tails, with a basis indexed by all partitions, the three charts $F_A,F_0,F_{\mathbb{Z}0}$ are distinct free lattices, the separated topology is induced by $q^n{F}_0$ (never by $q^n{F}_A$), and the vacuum weights are ${\mathrm{\Omega }}_M={\mathrm{\Lambda }}_r-(\frac{et(t-1)}{2}+tr)\delta$ for $M=te+r$.