Uses max-plus semiring algebra:

Selects K active experts from N total based on input affinity.

Hypersimplex capacity: C(N,K) distinct routing regions.

Auxiliary loss ensures uniform expert utilization:

L_balance = N × Σ(f_i × P_i)

Where f_i = fraction of tokens routed to expert i, P_i = average routing probability for expert i.

Sparse routing achieves exponential combinatorial depth without proportional compute cost. <write_to_file> docs/architecture/forward-forward.md

Each layer learns independently using local goodness metrics. No gradient chain required.

Tropical inner product: goodness = max(x_i × w_i)

Hebbian rule (gradient-free):

w_new = w_old + lr × (goodness_pos - goodness_neg) × input

Layers train independently — fully parallelizable across SYCL work groups.

• No vanishing gradients
• Local learning (no global coordination)
• Hardware-friendly (integer operations)
• Natural fit for ternary state space

SYCL provides hardware-agnostic parallelism for GPU and multi-core CPU acceleration.

queue.submit([&](handler& h) {
    h.parallel_for(range{n}, [=](id<1> i) {
        // Parallel tableau row update
    });
});

cmake -DUSE_SYCL=ON ..
cmake --build .

Requires Intel oneAPI or DPC++ toolchain.

• [Stabilizer Tableau](Stabilizer Tableau) — operations being parallelized
• [Build Guide](Guides-SYCL Setup.md) — SYCL environment setup