Spanning by analytic nested universal tensor components with real weights

The mode weight rule and the objectwise nilpotence bounds express every scalar tree series, in the logarithms of its sewing coordinates, as a real-exponential series with one finite bound on each logarithmic degree. For several sewing coordinates, the normality proof of F3 applies on a product rectangle: interpolate on a finite tensor product of imaginary-translate grids to separate the log-polynomial coefficients, then dominate each exponential on a smaller rectangle by its value at one of the $2^r$ real corners of the larger rectangle. The coefficient sums at those corners are finite by the given absolute convergence. Factors $\mathrm{\mid }{n}_1{\mathrm{\mid }}^{k_1}\cdots \mathrm{\mid }{n}_r{\mathrm{\mid }}^{k_r}$ from derivatives are bounded by an exponential margin, since ${\prod }_j\mathrm{\mid }{n}_j{\mathrm{\mid }}^{k_j}{e}^{-ϵ{\sum }_j\mathrm{\mid }{n}_j\mathrm{\mid }}$ is bounded. Thus the full scalar projection sums and all sewing derivatives are locally normally convergent. This proves the derivative manipulations used below from the actual all-index absolute convergence, rather than from an iterated grouped sum.

If the indicated span were proper, choose a nonzero restricted functional $f$ on one finite-dimensional doubly homogeneous root component annihilating that span. For every collection of leaves its scalar coefficient therefore vanishes at the specified configuration. Absolute leaf-position derivatives are linear combinations of coefficients with $L(-1)$ inserted on leaves: at each vertex the first-input derivative differentiates its relative sewing coordinate, and the translation bracket expresses the derivative at its root as the sum of derivatives at its two children. This triangular system has unit diagonal, so induction down the tree expresses every absolute derivative as a finite linear combination of such leaf insertions. The same holds for every finite repeated derivative. They vanish at the specified configuration, since the annihilation hypothesis holds for all leaf vectors, including those inserted vectors. By step1.1 these are the actual Taylor derivatives. Taylor uniqueness makes each coefficient zero in a neighborhood. Analytic continuation back along the specified path makes it zero throughout its initial connected sewing domain, by finitely many overlapping neighborhoods and the identity theorem.

Fix all sewing variables except one deepest variable, and fix all leaf generalized-weight components. Its formal scalar expansion is ${\sum }_h{q}^{h-c}{\sum }_{j=0}^N{a}_{h,j}(\log q{)}^j$, with real $h,c$ and finite $N$; its coefficients are precisely the finite intermediate-weight projections prescribed by the tree. It converges absolutely on a complex logarithm neighborhood by step1.1 and is zero there by step2.1. Apply F3 after $s=\log q$. Every coefficient $a_{h,j}$ is zero. Thus $f$ annihilates every tree coefficient in which that deepest universal map has first been projected to one finite generalized-weight component. There is no need to separate exponents by monodromy or assume finitely many exponent classes: F3 performs exactly this coefficient extraction for arbitrary real weights.

By F1 the deepest map's homogeneous components span its binary product algebraically. The zero assertion of step3.1 therefore holds with that subtree replaced by an arbitrary vector of its root object. Repeat step3.1 on a deepest remaining vertex. After finitely many repetitions only the top binary product remains; F1 then says $f$ annihilates all of the original root module. This contradicts the nonzero choice of $f$. Hence the nested universal components span, as asserted.