This work lays the foundation for a framework of cortical learning based on the idea of a competitive column, which is inspired by the functional organization of neurons in the cortex. A column describes a prototypical organization for neurons that gives rise to an ability to learn scale, rotation, and translation-invariant features. This is empowered by a recently developed learning rule, conflict learning, which enables the network to learn over both driving and modulatory feedforward, feedback, and lateral inputs. The framework is further supported by introducing both a notion of neural ambiguity and an adaptive threshold scheme. Ambiguity, which captures the idea that too many decisions lead to indecision, gives the network a dynamic way to resolve locally ambiguous decisions. The adaptive threshold operates over multiple timescales to regulate neural activity under the varied arrival timings of input in a highly interconnected multilayer network with feedforward and feedback. The competitive column architecture is demonstrated on a large-scale (54,000 neurons and 18 million synapses), invariant model of border ownership. The model is trained on four simple, fixed-scale shapes: two squares, one rectangle, and one symmetric L-shape. Tested on 1899 synthetic shapes of varying scale and complexity, the model correctly assigned border ownership with 74% accuracy. The model's abilities were also illustrated on contours of objects taken from natural images. Combined with conflict learning, the competitive column and ambiguity give a better intuitive understanding of how feedback, modulation, and inhibition may interact in the brain to influence activation and learning.