Resonant Fourier--Tree Factorisation for the Modified Lyons--Sidorova Conjecture

Let $\gamma$ be a continuous bounded-variation path in a finite-dimensional real normed vector space, with signature $g=S(\gamma)$, logarithmic signature $l=\log g$, and increment $v=\gamma_T-\gamma_0$. We prove the modified Lyons--Sidorova conjecture in this setting. If $R(l)=\infty$, then $g=1$ when $v=0$; when $v\neq0$, an actual prefix $\alpha$ of the centred path satisfies $S(\gamma) = S(\alpha)\e^v S(\alpha)^{-1}$. Conversely, every bounded-variation signature of this form has an entire logarithmic signature. For tree-reduced paths, a possibly different canonical prefix gives the equivalent weak path conjugacy to a line segment. The proof combines exact matrix isospectrality and loop rigidity with resonant Fourier developments. Fixed-point geometry in the Le Donne--Z\"ust signature tree produces a common rational-frequency prefix, Fourier uniqueness reconstructs the ordinary signature, and invariant-axis geometry gives the path-level reduction.