Are Flat Minima an Illusion?

Flat minima are widely believed to generalise better than sharp ones. I present a theoretical and empirical case that this belief is wrong. On the theoretical side, I prove that weakness, the measure-theoretic volume of compatible completions in the learner's embodied language, is reparameterisation-invariant, minimax-optimal, and tracked by the PAC-Bayes bound. Sharpness is none of these things. On the empirical side, I show that the large-batch generalisation advantage vanishes as a function of training set size. On MNIST, large-batch networks generalise better than small-batch networks when training data is scarce, and the advantage becomes negligible when data is abundant. A quantity that correlates, anticorrelates, or does not correlate depending on the data regime is not a cause. It is a confound. I confirm reparameterisation non-invariance of the Hessian trace on trained networks (up to 99x change with zero change in function or generalisation). I construct a formal vocabulary for frozen-partition ReLU networks and measure the pair proxy via linear feasibility on 100 networks. Weakness outperforms sharpness at model selection (ρ = +0.374, p = 0.00012 vs ρ = -0.226, p = 0.024).