Cyclotomic-Quotient Phase Embedding

(1) Integration point: implement a frozen or lightly trainable embedding layer for categorical IDs, finite-state observations, graph node labels, or synthetic modular-arithmetic tokens. Map each ID to an element x in F_q using a fixed primitive-polynomial representation. The layer outputs cosine and sine phase channels generated from the retained cyclotomic representatives, then sends them to an MLP, Transformer token projection, or graph-NN node encoder.

(2) Pseudocode:

construct F_q with q=p**r and enumerate field elements x
compute C(l) modulo q-1 for l=0,...,q-2
R = one canonical representative from each C(l)
initialize theta[l] for l in R
for minibatch of IDs:
    x = field_encode(ID)
    z = zero_field_element
    for l in R:
        z += theta[l] * field_pow(x, l)
    t = (2*pi/p) * field_trace(z)
    e = concat(cos(t), sin(t))
    loss = task_loss(network(e), target)
    backpropagate(loss)
    update theta and downstream weights

For a stable first implementation, precompute the feature table for every x in F_q and freeze it, learning only a downstream projection. If trainable coefficients are required, use a real soft relaxation for each finite-field coefficient or learn a continuous coefficient vector shared across all exponents in the same coset; do not allocate independent parameters to orbit members. Handle exponent q-1 separately because x^(q-1) is not the same function at x=0 as the exponent-zero monomial.

(3) Computed directly from the paper's mathematics: field multiplication and exponentiation, the trace, additive-character phases, cyclotomic cosets, and the Frobenius identity proving orbit redundancy. Estimated empirically: the representative count |R|, feature-matrix rank, duplicate-column count in the unreduced construction, gradient covariance, and downstream accuracy.

(4) First cheap experiment: choose q=2^8 or q=3^4, generate modular-addition and modular-multiplication classification datasets, and compare an unconstrained character-polynomial feature layer using every exponent with the cyclotomic-representative layer and a standard learned embedding. Use the same small MLP, optimizer, training steps, and output width. Success means at least 20% fewer phase parameters, zero exact duplicate feature directions after quotienting, a lower gradient-covariance condition number, and no more than 1% absolute test-accuracy loss; faster early loss descent at equal parameter count would provide additional evidence.