Collapse of Metric Rigidity in Framed Holonomy Spaces: A Word-Derivative and Gram-Operator Approach

In lattice gauge theory and loop quantum gravity, it is commonly (and silently) assumed that the induced metric on the framed holonomy configuration space is flat (rigid) and independent of the holonomy variables. This paper rigorously disproves this assumption. We introduce the Word-Derivative Lemma, which expresses the logarithmic derivative of any group word in terms of its suffix tails. Combined with the Tail-Holonomy Lemma, we derive the exact block form of the Gram operator A(U) = LL*, which governs the pullback metric g(U) = A(U)^{-1}. Unlike the Abelian case, for non-Abelian groups (e.g., SU(2)), the off-diagonal blocks of A(U) contain Ad terms, explicitly proving that the induced metric depends genuinely on the holonomies and exhibits curvature. We perform the first explicit calculation on the "Glasses" graph, proving its determinant is strictly positive. Furthermore, using Jacobi's formula, we show that the drift term is exact, revealing that the measure and drift are governed entirely by a single scalar potential.