Convex-lifted additive codebook

(1) Integration point: use the construction as the representation layer for a finite scalar codebook, tokenizer IDs, quantized latent values, or compositional retrieval keys. For N ordered codes, set t_j=2j/(N−1)−1 and replace the ordinary scalar representation t_j with z_j=(t_j,t_j²). Concatenate z_j to the standard embedding, or apply a learned matrix P∈R^{d×2} initialized with orthogonal columns. Keep P fixed in one ablation and trainable in another. The energy guarantee applies directly before arbitrary nonlinear transformations, so collision measurements should be made on z_j or on an invertible linear transform.
(2) Pseudocode: create t=linspace(−1,1,N), z=stack([t,t*t],axis=−1). For each sampled triple (j1,j2,j3), compute s=z[j1]+z[j2]+z[j3]. Sort exact rational or quantized sums and count duplicate pairs to estimate J_3. For continuous floating-point experiments, normalize s and count pairs satisfying ||s_a−s_b||₂<τ. Feed z, or Pz, into the downstream model and train with the ordinary task loss. Optionally add λ exp(−||s_a−s_b||₂²/τ²) over distinct sampled triples as a differentiable near-collision penalty when t or P is trainable.
(3) The theorem supplies the strict-convexity construction and the near-cubic exact-energy target. The implementation must empirically estimate approximate energy because finite precision, projection, normalization, and noise are not covered by the theorem.
(4) First experiment: use N=256 codes in a synthetic task whose labels depend on unordered triples. Compare a learned scalar embedding, random two-dimensional embeddings, and fixed (t,t²) embeddings at matched dimension and parameter count. Measure the number of unique triple sums, the number of pairs within thresholds τ∈{10^−3,10^−2,10^−1}, noisy top-1 triple decoding accuracy, and validation accuracy. Success means fewer near-collisions and higher compositional retrieval accuracy at equal dimension; failure is indistinguishable collision statistics or no downstream gain.