nx_mt_r2_gate.nx

nx_mt_r2_gate.nx -- GATE for MT-R2: CROSS-TOKEN CONTEXT enables REORDERING. The first genuinely "machine translation" phenomenon: an output token that depends on MORE than its own aligned input token. Task = REVERSE+map: source [a,b] -> target [tmap[b], tmap[a]] (the canonical MT reordering, e.g. EN adjective-noun "red house" -> ES noun-adjective "casa roja"). A per-token aligned map (MT-R1 / nx_mt_core) provably CANNOT do this; a cross-token model can. This rung proves the CAPABILITY + the measured necessity contrast; the parameter-free fnet_mix and attention are the SCALABLE realizations of this same cross-token mixing, wired at R3+ when we move to real arrays/vocab. MODEL (cross-token, tape-trainable -- linear, so pure add/mul through nx_autograd): two learned heads W[h] (h=0,1), each [T x 2S], over the CONCATENATED one-hot input [onehot(a) ; onehot(b)]. head h, output j logit = W[h,j,a] + W[h,j,S+b] (the two active concat dims) ; argmax_j. head 0 target = onehot(tmap[b]) (reordered!), head 1 target = onehot(tmap[a]). Full-batch GD, MSE, zero-init. The loss is convex; lr chosen for the bipartite Hessian (degree ~ pairs/token). Vocab S=T=4 (small -> clean convex convergence + fast): EN 0 hello 1 family 2 love 3 good ; ES 0 familia 1 bueno 2 hola 3 amor ; tmap=[2,0,3,1]. TRAIN = 13 of the 16 (a,b) pairs; HELD-OUT = 3 pairs {(0,3),(3,0),(1,2)} (combinations never trained). FOUR GATES: A LEARNS reordering: per-token acc == 2*train_pairs, loss decreased to < half. B GENERALIZES: per-token acc == 2*held_pairs on pairs NEVER trained. C BIT-EXACT: retrain -> identical W bits. D PER-TOKEN MODEL CANNOT (measured necessity contrast + liar-kill): train MT-R1's per-token shared map (nx_mt_core) on the SAME reverse task -> its acc < 2*train_pairs. Proves the cross-token capacity is NECESSARY, not decorative. genealogy_id: rumelhart_1986_backprop (realized_in nx_autograd) + lee_2021_fnet (the mixing motive) lineage_id: sovereign_neural_mt_r2_crosstoken_reorder_v1 license_tier: ORIGINAL