02 — The Mathematics Intrinsic Curvature as Scheduling Signal The constraint graph induces a discrete Riemannian manifold in the sense of Regge calculus — not a metaphor, but a genuine geometric structure with a metric tensor, connection, and holonomy group. The local curvature field measures three independent scalar invariants of the constraint fiber bundle.
Local Curvature $$K_{\text{loc}}(v) = w_s \cdot \sigma(v) + w_r \cdot \rho(v) + w_c \cdot \kappa(v)$$ $\sigma$ = saturation (boundary curvature) · $\rho$ = scarcity (fiber dimension) · $\kappa$ = coupling norm (connection rigidity)
Information Value $$V(v) = \frac{K_{\text{loc}}(v) + \displaystyle\sum_{u \in N(v)} K_{\text{loc}}(u)}{\left|D(v)\right|}$$ Curvature-weighted variable ordering — selects the vertex whose resolution propagates the most information globally.
Davis Energy Functional $$E[\gamma] = \lambda_1 \int_\gamma ds + \lambda_2 \int_\gamma K_{\text{loc}}(s)\, ds + \lambda_3 \int_\gamma \left\| \mathrm{Hol}_\gamma(s) - I \right\| ds$$ Path length + curvature-weighted complexity + holonomy deficit. A geodesic of this functional resolves constraints with minimal total effort.
The trichotomy parameter $\Gamma = \dfrac{m \cdot \tau}{\hat{K}_{\max} \cdot \log |S|}$ classifies instances by the ratio of assigned structure to geometric complexity, automatically gating them into the optimal phase combination.