[Autoplectic]

Summary

Generators beyond the lifted search, a linear-programming test for generator states, support-based lower bounds, and a driver that brackets the generative complexity.

• sofic.generators.coupled_generators
  • coupled_search / coupled_generative_model: minimize state entropy over all k-state generators (state law, beliefs about S+ and S-, and a free kernel) subject to the edge law and past/future consistency; augmented Lagrangian with analytic gradients.
  • memory_lift / memory_lift_search: state W_t = (V_t, V_{t+1}) with V_t ~ nu(. | B_t), so consecutive states share randomness.
  • generator_kernel: for a fixed channel the generator equations are linear in the kernel; a linear program returns the smallest violation and the kernel.
  • polish_separating_channel: re-solve a separating variable's decomposition exactly (dit's optima satisfy it only to a tolerance, enough to fail generator_kernel).
  • bilevel_search (experimental).
• sofic.generators.generator_bounds: support_type_bound, a lower bound on G[S+ : S-] (hence C_g) from the rectangular supports of separating values.
• sofic.generators.generative_search.generative_model: lower bounds, the persistent-crypticity check, polished exact-common-information optima tested with the kernel program, then ε-machines, memory lifts, deterministic and lifted channels, and the coupled search seeded with everything found; returns the best verified generator and a bracket [lower, upper] on C_g.

Results

process	lifted search	this PR	bracket on C_g
golden mean	0.9183	0.9183	certified
(2,inf) run-length-limited	1.5	1.5	[1.4056, 1.5]
Z_t = Z_{t-2} xor Bern(0.1)	1.9911	1.9822 = 2h(4/9)	upper bound

On all 142 binary unifilar 3-state topologies (43 searchable with positive persistent crypticity), the driver certifies C_g in 9, beats the lifted search in 7 of 25 (by up to 0.22 bits) and finds generators below C_mu in 15.

Test plan

• pytest tests/test_coupled_generators.py tests/test_lifted_common_information.py (35 passed)
• ruff check, ruff format --check, ty check on the new modules