generative_gap, and describe the lifted search as restricted to lifted generators - #31

Follow-up to #29.

generative_gap (missed #29's merge)

generative_gap(measure="exact" | "wyner"), also BidirectionalEpsilonMachine.generative_gap(), returns GenerativeGap(lower, upper, gap, model): the unconstrained common information of (S+, S-) against its value over lifted generator states. For the exact measure G[S+:S-] <= C_g <= H[G], so gap bounds C_g - G[S+:S-] from above. On the (2,∞) process: exact 1.406 → 1.5, Wyner 1.129 → 1.311; zero for first-order Markov processes. This commit was pushed to #29's branch after it was merged, so it is cherry-picked here unchanged.

Lifted generators are a proper subclass of generators

#29 described the constrained optimizers as searching "generator states". They search lifted generator states: the HMM built by applying p(g | s+, s-) at both ends of every bidirectional edge draws G_0 and G_1 independently given the edge, and a general generator may couple them.

Counterexample: for Z_t = Z_{t-2} xor Bernoulli(0.1) (two interleaved first-order chains), the product of the chains' minimal generators has state G_t = (U_t, U_{t+1}), sharing U_{t+1} with G_{t+1}.

So it is a generator state that the lifted search cannot reach; sofic reports 1.991–2.0 there. This PR corrects the docs, docstrings and CHANGELOG:

• exact and Wyner values are upper bounds over all generators (still valid bounds on C_g: every lifted model is a genuine generator), not minima over all generators;
• generative_gap can be positive when a non-lifted generator attains the unconstrained value;
• the functional search is unaffected: for a deterministic channel the coupling is forced, so it is exact over all deterministic generator states.

No behavior change beyond generative_gap. Searching coupled kernels (optimizing p(w_0, w_1 | edge) with fixed marginals) would close the gap and is left for a follow-up.

More starts for the lifted search

Besides S+, S-, (S+, S-), the dit optimum and random channels, the continuation now also starts from:

• the dit optimum blended with random channels (strengths 0.05 and 0.2, two copies each);
• time-sharing mixtures of the dit optimum with each ε-machine channel (weights 0.5 and 0.8; disjoint state labels keep the mixture separating).

On the 3-state process 012301 (G = 1.370, ε-machines 1.584) this finds the 1.488 generator in two of three seeds instead of one; on 020213 in one of three. Remaining seed dependence comes mostly from dit's unconstrained optimum varying between runs. Cost: the lifted test files take about 4.6 min instead of 1.6 min locally.

Tests

tests/test_lifted_common_information.py and tests/test_minimal_generative_model.py: 64 passed locally; ruff clean.