{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace raw polynomial features in a scalar MLP expansion with endpoint-weighted orthonormal Jacobi features. The paper's envelope gives a degree- and parameter-aware scale for each feature, preventing high-degree terms or endpoint behavior from dominating gradients while preserving a richer approximation basis than low-degree monomials.",
 "formulas_latex": [
  "$$w_{\\alpha,\\beta}(x)=(1-x)^\\alpha(1+x)^\\beta,\\qquad \\int_{-1}^{1}p_n^{(\\alpha,\\beta)}(x)p_m^{(\\alpha,\\beta)}(x)w_{\\alpha,\\beta}(x)\\,dx=\\delta_{nm}.$$",
  "$$(1-x)^{\\alpha+1/2}(1+x)^{\\beta+1/2}\\left|p_n^{(\\alpha,\\beta)}(x)\\right|^2\\leq C\\max\\left\\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\\right\\},\\qquad S=\\alpha+\\beta+1.$$",
  "$$q_n(x)=(1-x)^{\\alpha/2+1/4}(1+x)^{\\beta/2+1/4}p_n^{(\\alpha,\\beta)}(x),\\qquad |q_n(x)|\\leq \\sqrt{C B_n},\\quad B_n=\\max\\left\\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\\right\\}.$$",
  "$$h_n^{(\\alpha,\\beta)}=\\int_{-1}^{1}\\left[P_n^{(\\alpha,\\beta)}(x)\\right]^2w_{\\alpha,\\beta}(x)\\,dx=\\frac{2^{\\alpha+\\beta+1}}{2n+\\alpha+\\beta+1}\\frac{\\Gamma(n+\\alpha+1)\\Gamma(n+\\beta+1)}{\\Gamma(n+1)\\Gamma(n+\\alpha+\\beta+1)},\\qquad p_n^{(\\alpha,\\beta)}(x)=P_n^{(\\alpha,\\beta)}(x)/\\sqrt{h_n^{(\\alpha,\\beta)}}.$$"
 ],
 "id": 2883,
 "implementation": "Integrate the construction at the scalar-feature expansion inside an MLP, immediately after a linear projection and a bounded map into [-1,1]. For an input tensor u, compute x=tanh(linear(h)). Choose fixed alpha,beta \u003e= -1/2 and maximum degree N. Precompute S=alpha+beta+1 and B[n]=max(1,S^(1/3),S^(1/2)*(n+1)^(-1/6)). Evaluate the orthonormal Jacobi polynomials p[n](x) with a stable three-term recurrence or a trusted special-function routine. Form q[n]=(1-x)^(alpha/2+1/4)*(1+x)^(beta/2+1/4)*p[n], then normalize phi[n]=q[n]/sqrt(B[n]+epsilon). Compute each output channel as y[j]=sum over n of a[j,n]*phi[n], followed by the ordinary residual connection and normalization. In pseudocode: x=tanh(linear(h)); for n=0,...,N: p[n]=jacobi_orthonormal(n,alpha,beta,x); q[n]=(1-x)^(alpha/2+0.25)*(1+x)^(beta/2+0.25)*p[n]; phi[n]=q[n]/sqrt(B[n]+epsilon); y=sum_n a[n]*phi[n]. The theorem determines the functional form of B[n]; estimate the unknown constant C by evaluating max over a dense grid of q[n](x)^2/B[n], then multiply the normalization by a safety factor of 2 to 4. First test on CIFAR-10 using a four-layer width-256 MLP, comparing ReLU, raw Jacobi features, and normalized Jacobi features at equal parameter count. Record loss versus training FLOPs, gradient-norm percentiles, activation maxima, and validation accuracy. The method succeeds if it reduces gradient spikes and either improves loss descent or accuracy at degrees N=8 or 16, particularly when inputs are concentrated near x=1 or x=-1.",
 "math_summary": "For alpha,beta \u003e= -1/2, define the Jacobi weight w_{alpha,beta}(x)=(1-x)^alpha(1+x)^beta on [-1,1], and let p_n^{(alpha,beta)} be degree-n polynomials orthonormal under the weighted inner product integral from -1 to 1 of p_n(x)p_m(x)w_{alpha,beta}(x) dx = delta_nm. The paper proves the uniform estimate (1-x)^{alpha+1/2}(1+x)^{beta+1/2}|p_n^{(alpha,beta)}(x)|^2 \u003c= C B_n, where S=alpha+beta+1 and B_n=max{1,S^{1/3},S^{1/2}(n+1)^{-1/6)}. Define q_n(x)=(1-x)^{alpha/2+1/4}(1+x)^{beta/2+1/4}p_n^{(alpha,beta)}(x). The theorem implies |q_n(x)| \u003c= sqrt(C B_n) uniformly on [-1,1]. The universal constant C is absorbed into an empirical safety factor. The S^{1/3} term must remain because the paper's Bessel turning-point argument shows that it cannot generally be omitted.",
 "math_tags": [
  "approximation-theory",
  "functional-analysis",
  "spectral-theory"
 ],
 "ml_areas": [
  "mlp",
  "initialization",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.30304",
  "arxiv_url": "https://arxiv.org/abs/2608.30304",
  "summary_what_math_gives_to_ml": "The paper proves a sharp, degree- and parameter-sensitive envelope for weighted orthonormal Jacobi polynomials, including the non-removable intermediate growth factor S^{1/3}. This can be transferred to neural networks as a principled bounded polynomial feature basis: instead of feeding raw high-degree Jacobi polynomials into an MLP, use endpoint-weighted functions with theorem-based degree normalization. The bound provides an explicit scale for coefficient initialization and regularization, potentially enabling higher-degree polynomial layers with fewer activation and gradient explosions.",
  "title": "The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Krasikov-Normalized Jacobi Feature Layer",
 "url": "https://synthcore.org/idea/2883/krasikov-normalized-jacobi-feature-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}