Quadratic and $p$-th variation of random signed Takagi--Landsberg bridges

We study random signed Takagi--Landsberg bridges with independent Rademacher Faber--Schauder coefficients. At index $H=1/2$, we prove that every fixed deterministic refining sequence of partitions with vanishing mesh yields quadratic variation $t$, almost surely and uniformly in $t\in[0,1]$. For partitions that need not be refining, the mesh condition $o(1/\log n)$ is sufficient, and its order is sharp. Our proofs use an operator representation of the quadratic sums and moment bounds for Rademacher chaos. We also show that asymmetric signs can destroy this invariance. At every index $H\ne1/2$, we construct a deterministic refining sequence along which the critical $p$-power sums, with $p=1/H$, have two distinct finite accumulation points almost surely. At $H=1/4$, uniform thirds grids provide an explicit alternative to the dyadic limit. Thus $H=1/2$ is the unique index in the symmetric family at which critical variation is invariant along every fixed deterministic refining sequence. Nevertheless the variation index equals $1/H$ almost surely across a large sequence of partitions.