Gradient Indeterminacy. From Classical Gradients to Quantum Uncertainty

Heisenberg's indeterminacy is here proposed not as a foundation but as a limiting case of a deeper, geometric constraint: gradient indeterminacy (I-G). Physical reality is taken to operate with finite regions and gradients rather than with mathematical points and sharp values. For conjugate fields $A,B$ over a region $\Omega$ of volume $V(\Omega)$, the relation $\|\nabla A\|_\Omega\,\|\nabla B\|_\Omega\,V(\Omega)\ge(\hbar/2)\,\mathcal{F}(\Omega)$ recovers $\Delta x\,\Delta p\ge\hbar/2$ in the point limit $V(\Omega)\to 0$, while the contextual factor $\mathcal{F}(\Omega)$ ties the bound to the shape, topology, and curvature of the region. Because the relation is anchored in real regions rather than in an idealization, the bound adapts to the system and extends naturally beyond the canonical pairs Heisenberg treated. We give the point-limit derivation and a consistency check, a classical toy model that reaches an indeterminacy floor without assuming the uncertainty principle, a typology of conjugate relations, the differential-% geometric formalization, and a set of falsifiable predictions (shape- and curvature-dependent vacuum fluctuations, small-region corrections, and an environment-dependent Hubble rate). The divergences of point-based theories are proposed to be artefacts of the point idealization rather than features of the physics. This is offered as a proposal of structure, not a completed derivation.