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 $v\neq0$, 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 $q\geq1$, $A:\C\to M_q(\C)$ 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 $1\leq k\leq q$. 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.

Disclosure

“emma A.1 to x and Lx gives Equation (56); the first and last pieces have the common length d(x, A). Summing their lengths proves Equation (57), and equality with τ holds exactly on the closed line A, proving Equation (58). Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the development and preparation of this work, the author used OpenAI’s ChatGPT 5.6 Sol as a conversational aid for exploring possible proof routes, critically testin”

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Classification

Proof ideas or individual proof-step assistance

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8

Verified

2026-08-23T19:11:22.402687Z

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