Entire Logarithmic Signatures of Bounded-Variation Paths in Finite Dimensions

We classify signatures of bounded-variation paths in finite-dimensional real normed spaces whose logarithms are entire, in the sense of superexponential homogeneous decay. The zero-first-level fibre is trivial; if the first level is $v\mathrm{\ne }0$, the signature is $a{\text{e}}^v{a}^{-1}$ with $a$ a bounded-variation signature, and every such conjugate is entire. For a given path, $a$ may be chosen from a prefix. For a tree-reduced representative, a gate-selected prefix gives weak path conjugacy to a line. The proof uses a finite-dimensional spectral-growth statement: if $q\ge 1$, $A:\text{C}\to M_q(\text{C})$ is entire, and $\text{norm}{\text{e}}^{A(z)}\le C{\text{e}}^{\tau \text{abs}z}$, then the $k$th characteristic-polynomial coefficient of $A(z)$ has degree at most $k$ for $1\le k\le q$. Universal matrix isospectrality, resonant developments, metric-tree fixed points and compactness, and Stieltjes–Fourier reconstruction establish the bounded-variation modified Lyons–Sidorova conjecture under a hypothesis imposed only on the whole path.