Unrestricted numbers of finite-state aggregate gates

The exact capacity is one bit per binary gate plus the sum of the logarithms of the special register cardinalities. The fixed hard family requires capacity greater than (1 - ε)n eventually, without any sparsity condition. Register sizes bounded by r ≥ 2 give a gate coefficient of 1 / log₂ r.

The polarity module also exposes the hard family's two-sided rectangle and sumset disperser properties for further geometric arguments. No stronger whole-basis coefficient follows from those properties alone in this module.