Continuous but Non-Differentiable Support

This paper introduces a geometric refinement to the Theory of Axiomatic Necessity (TNA) by defining a formal regularity distinction between the domains $N_0$ and $N_1$. The author argues that realizable systems require an external support structure ($N_1$) that is continuous in existence ($C^0$) but non-smooth in its directional evolution (non-$C^1$). While the observable local dynamics ($N_0$) evolve through smooth and locally differentiable processes, the discontinuity in the first derivative of $N_1$ prevents the system from being entirely derived or anticipated from within $N_0$. This geometric condition mathematically secures the failure of local closure, preserving the structural openness necessary for multiplicity, historical discontinuity, human consciousness, and freedom. The framework is subsequently applied to analyze the limits of artificial intelligence, institutional collapse, suffering, and the metrics of creatio continua at the Planck scale. The paper proposes that the non-differentiable structure of N₁ finds its most fundamental realization at the Planck scale, which anchors the ontological continuity of the support of realizability. Nevertheless, this does not mean that all relevant discontinuities are confined to that scale. Rather, the fundamental non-smoothness of N₁ generates — via nonlinear dynamics and criticality — effective discontinuities across multiple emergent levels: ranging from quantum collapses to biographical, geological, and historical breaks.