Entire Logarithmic Signatures of Bounded-Variation Paths in Finite Dimensions

Abstract

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 $v\ne 0$, the signature is $a\e^v a^{-1}$ 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 $q\ge 1$, $A:\C\to M_q(\C)$ is entire, and $\norm{\e^{A(z)}}\leq C\e^{τ\abs{z}}$, 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.