Tie neural parameters across feature channels according to the Schur multiplicity pattern of the fermionic coinvariant representation. In the two-fermion case, use one learned parameter block for each Schur degree instead of independently parameterizing every ordered pair of fermionic channels, eliminating redundant copies while preserving the relevant GL2 channel symmetry.
Replace explicitly stored independent random positional features with deterministic multiplicative phase features generated from a small table of random phases indexed by primes. Restrict positions to integers whose prime factors exceed a slowly growing threshold, because the paper's central-limit result says periodic weighted sums over these rough integers recover Gaussian behavior despite strong multiplicative dependence.
Add a centro-affine Dirichlet penalty to a neural module whose inputs or outputs lie on a sphere, such as normalized embeddings or attention directions. The penalty measures intrinsic variation under an unconditional convex-body metric while projecting out the constant and coordinate-affine modes excluded by the theorem.
Represent selected hidden features as z = sqrt(N) exp(i theta), with a persistent phase and an explicitly stochastic amplitude. Regularize the ratio between coherent power |E[z]|^2 and total power E[|z|^2] toward the condensate prediction pi/4, while optionally matching higher amplitude moments.
Insert a fixed polynomial mixer before an MLP or retrieval index for tuples of discrete features. The mixer maps n+2 bounded scalar feature codes to one or several expanded scalar codes, and the paper's theorem guarantees that its image cannot collapse below order |A|^n when all coordinates come from a finite alphabet A. Use multiple independent permutations or coefficient choices to obtain a vector representation while retaining the deterministic algebraic structure.