nx_fnet_grad_gate.nx

nx_fnet_grad_gate.nx -- GATE for FNET-002: the FNet mixer's backward pass (VJP). Proves, by RUNNING, that the mixer is SELF-ADJOINT, so dL/dx = fnet_mix(dL/dy) -- backprop is one forward mix, zero stored activations, zero parameters. Three independent proofs (n=8, d=4 -> a 32-dim operator): G1 SYMMETRY extract the operator matrix M by impulse responses (M[:,k] = fnet_mix(S*e_k)); the real-linear FNet operator must be SYMMETRIC (A = A^T) because it is the real part of a Kronecker product of symmetric DFT matrices. Measure max|M[i,j]-M[j,i]|. G2 ADJOINT TEST the relationship backprop relies on: <A x, g> == <x, A* g> with A* = A. Measure |<Ax,g> - <x,Ag>| -- zero (to rounding) IFF using fnet_mix as the backward is the correct VJP. G3 GRADCHECK finite-difference the scalar loss L(x)=sum a*fnet_mix(x): central differences vs the analytic backward fnet_mix_backward(a). Max |fd - analytic|. All three are robust to the Q14 fixed-point rounding (tolerances are a small % of signal, actuals reported). Evidence -> knowledge/status/fnet_grad.log (FNETGRADGATE authored=organ ... verdict=GREEN). license_tier: ORIGINAL