Lattice Monodromy Residual Block

Use a compact subset of hidden channels, for example 16 or 32 channels in a Transformer residual stream or the state vector of an SSM. Choose a small integer lattice basis of dimension d and a symmetric Gram matrix B; for an MVP use a diagonal or block-diagonal B with entries in {+1,-1}, then choose integer vectors e and f representing the fiber and flux directions. Precompute the d-by-d matrix T from the formula Tγ=γ+0.5(f^T Bγ)e-0.5(e^T Bγ)f+3(e^T Bγ)e. If half-integer coefficients occur, use a scaled basis or store T in float32 after checking its condition number and inverse. At layer k, take z=h[..., :d], apply z'=T^k z with k modulo a chosen period, write h[..., :d]=z', and then execute the usual normalized residual block. Pseudocode: initialize e,f,B; construct T; set P[0]=I and P[k+1]=T P[k]; for each layer k, z=h[...,0:d]; z=P[k mod p] @ z; h[...,0:d]=z; h=h+MLP(LN(h)). Precompute powers rather than exponentiating per batch. The mathematical object is the fixed monodromy matrix; empirical choices are d, period p, scaling, and insertion point. First test a 6-layer Transformer on CIFAR-10 or a small language-model corpus against the same model with identity and random orthogonal transforms. Measure validation accuracy, gradient-norm variance, activation norms, and loss at equal FLOPs. A positive signal is equal or better accuracy with lower gradient variance and stable norms when the monodromy cycle is repeated at greater depth.