Foundations of Limbertwig 3: Supramanifold Torsion, Logic Vectors, and the Limbertwig Kernel

We recast the expression$$[\Sfun(\LS) ;=; \OmL!\left( \tan\psi \diamond \theta ;+; \Psi \star\sum_{[\LS]\star[\nLS]\to\infty}\frac{1}{|\LS|^2-|\nLS|^2}\right)\cdot \prod_{i\in \LS}\Fcal(i)]$$as a functorial analytic torsion attached to a logic vector $\LS$ living in a derived sheaf-theoretic avatar of logic. Negation becomes Verdier duality, $|\cdot|^2$ is an Euler-type quadratic form, $\tan\psi\diamond\theta$ is a curvature scalar on a differentiable thickening of logic space, the summation is a zeta-regularized duality-asymmetry series, and $\OmL$ is a scale coend implemented by a Limbertwig heat kernel. The resulting invariant $S_M$ bridges logic, topology, and analysis; we formulate existence, invariance, and stationary-point principles, and propose a Limbertwig torsion hypothesis relating stationary points to critical-line phenomena for $L$-functions.