Cutwidth lower bounds with finite commutative aggregate gates

The fixed polynomial-time hard family retains its unconditional coefficient above 4.56249 for mixed circuits with sublinear total aggregate budget. The public theorem is Aggregate.sourceReductionHardFamily_eventually_lt_size_gaussian; the budget is Algebraic.Aggregate.budget. AND, OR, and modular examples are provided by Algebraic.Aggregate. All ordinary operations are arbitrary binary Boolean functions. Without a sparsity condition, sourceReductionHardFamily_eventually_lt_realCapacity gives a leading coefficient of one for the exact state capacity; uniformly bounded state spaces turn it into a whole-basis gate lower bound. For signed unbounded AND/OR/XOR gates, affine pairing and biased-message counting improve the unrestricted whole-basis coefficient; Geometry exports the successive bounds.

Joint compression removes the per-occurrence charge when all local special-register contributions factor through a finite commutative monoid M. Its budget is Compressed.budget M = 2 * ceil(log₂ |M|): one joint guess and one accumulator. Compressed.sourceReductionHardFamily_eventually_lt_size_gaussian retains the same coefficient for sublinear compressed budget, allowing arbitrary special-to-special wiring. For a prime modulus, Compressed.Modular supplies the factorization automatically through the contribution matrix's column space, with exactly p ^ rank states.