{
 "artifacts": null,
 "category": "architecture",
 "description": "Insert a projection-space mixer that combines several fixed or learned directions using reciprocal correlations with the current feature, then normalize the result. The exact construction has a universal beta law for its squared input-output cosine, so it can create controlled angular diversity while remaining deterministic and independent of the chosen direction dictionary.",
 "formulas_latex": [
  "$$g_{a,w}(U)=\\sum_{j=1}^{k}w_j\\frac{a_j}{a_j^{\\top}U},\\qquad N_{a,w}(U)=\\frac{g_{a,w}(U)}{\\|g_{a,w}(U)\\|}.$$",
  "$$U^{\\top}g_{a,w}(U)=1,\\qquad \\{U^{\\top}N_{a,w}(U)\\}^{2}=\\frac{1}{\\|g_{a,w}(U)\\|^{2}}.$$",
  "$$\\{U^{\\top}N_{a,w}(U)\\}^{2}\\sim\\operatorname{Beta}\\!\\left(\\frac12,\\frac{p-1}{2}\\right).$$",
  "$$\\widetilde g(x)=\\sum_{j=1}^{k}w_j a_j r_{\\varepsilon}(a_j^{\\top}u),\\quad u=\\frac{x}{\\|x\\|+\\delta},\\quad r_{\\varepsilon}(t)=\\frac{\\operatorname{sign}(t)}{\\max(|t|,\\varepsilon)},\\quad y=\\|x\\|\\frac{\\widetilde g(x)}{\\|\\widetilde g(x)\\|+\\delta}.$$"
 ],
 "id": 2980,
 "implementation": "1. Integration point: apply the mixer to a normalized embedding or transformer residual vector x in R^p, preferably in a projection head or after an MLP rather than directly inside every attention block. Maintain k prototype vectors A in R^{k by p}; initialize them as random unit vectors, and parameterize weights with w=softmax(theta) so they remain nonnegative and sum to one. 2. Pseudocode: compute u=x/(norm(x)+delta); t=A@u; t_clip=sign(t)*clamp(abs(t),min=eps); coefficients=w/t_clip; g=sum_j coefficients[j]*A[j]; y=norm(x)*g/(norm(g)+delta); return residual_scale*y+(1-residual_scale)*x. 3. The paper supplies the exact simplex reciprocal construction and target law. Empirically measure the fraction of clipped projections, the distribution of q=(u dot normalize(g))^2, and its Wasserstein or Kolmogorov distance to Beta(1/2,(p-1)/2). Sweep eps in {1e-2,1e-3,1e-4} and optionally stop prototype gradients when clipping exceeds 5 percent. 4. First experiment: use a 2-layer MLP on CIFAR-10 or a small ViT with a 128-dimensional penultimate embedding. Compare the reciprocal mixer against an ordinary learned linear projection, a normalized random-prototype mixer, and an identity residual at equal parameter count. Test accuracy, calibration, representation uniformity, gradient-norm variance, and throughput. A positive result is improved test accuracy or linear-probe performance at unchanged FLOPs, with q close to the predicted beta law and no more than 10 percent throughput loss.",
 "math_summary": "For nonzero vectors a_1,...,a_k in R^p, nonnegative weights w_1,...,w_k with sum_j w_j=1, and U uniform on the sphere S^{p-1}, define g_{a,w}(U)=sum_j w_j a_j/(a_j^T U) away from the hyperplanes a_j^T U=0. Since U^T g_{a,w}(U)=sum_j w_j=1, its normalized output N_{a,w}(U)=g_{a,w}(U)/||g_{a,w}(U)|| has squared incidence cosine {U^T N_{a,w}(U)}^2=1/||g_{a,w}(U)||^2. The paper's central identity states that this random variable is exactly Beta(1/2,(p-1)/2), independently of k, the directions, their rank, and the simplex weights. In a neural layer, x/||x|| plays U, the a_j are learned or fixed unit prototype directions, and the beta identity motivates preserving the reciprocal mixer. The singular reciprocal must be replaced in code by a clipped reciprocal, so the implemented layer is an approximation to the exact theorem.",
 "math_tags": [
  "probability",
  "geometry",
  "linear-algebra"
 ],
 "ml_areas": [
  "embedding",
  "mlp",
  "transformer",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2609.00603",
  "arxiv_url": "https://arxiv.org/abs/2609.00603",
  "summary_what_math_gives_to_ml": "The paper gives an unusually strong distributional invariance for a deterministic reciprocal-projection construction: any simplex-weighted collection of nonzero directions produces an output whose squared cosine with the input is exactly Beta(1/2,(p-1)/2), regardless of the directions, rank, or overcompleteness. This can be transferred into neural networks as a geometry-aware angular mixing layer or as a calibration target for representation maps, providing controlled angular diversity without injecting random output directions. The main engineering risk is the pole at a_j^T U=0, so practical implementations must use clipping or a smooth reciprocal and explicitly measure how much the exact beta law degrades.",
  "title": "Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "generalization"
 ],
 "title": "Reciprocal Beta-Angle Mixer",
 "url": "https://synthcore.org/idea/2980/reciprocal-beta-angle-mixer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}