{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a dense second-order interaction layer on paired binary channels by a low-degree Boolean feature map obeying the quadratic-form cocycle from the paper. The layer uses only linear and pairwise features, but ties them through a learned or fixed symplectic form, reducing the number of independent interaction parameters and enforcing invariance under basis changes that preserve the pairing.",
 "formulas_latex": [
  "$$q(x+y)=q(x)+q(y)+\\widehat{\\iota}(x,y),\\qquad x,y\\in H=\\mathbb{F}_2^{2g}.$$",
  "$$\\overline{x+y}=\\overline{x}+\\overline{y}+\\widehat{\\iota}(x,y).$$",
  "$$\\dim \\operatorname{B}_n(g)=\\sum_{k=0}^{n}\\binom{2g}{k},\\qquad \\dim \\operatorname{B}_2(g)=1+2g+\\frac{2g(2g-1)}{2}=2g^2+g+1.$$",
  "$$\\mathcal{L}_{\\mathrm{symp}}=\\mathbb{E}_{U\\sim\\mathcal{G}}\\left[\\left\\|F(Uz)-\\rho(U)F(z)\\right\\|_2^2\\right],\\qquad U^\\top J U=J.$$"
 ],
 "id": 191,
 "implementation": "Integrate this as a replacement for the first dense pairwise-interaction block in a small MLP or graph neural network. Let the incoming tensor be \\(z\\in\\mathbb{R}^{B\\times d}\\), with \\(d=2g\\) channels grouped into \\(g\\) pairs. First compute a linear term \\(u=W_1z\\). For each pair \\((a_i,b_i)\\), compute the symplectic interaction \\(s_i=z_{a_i}z_{b_i}\\); optionally compute cross-pair terms only through tied coefficients determined by the fixed matrix \\(J\\), rather than learning an independent coefficient for every ordered pair. Concatenate \\([1,z,s]\\), apply normalization and a projection \\(W_2\\), and add the result residually to the original block. A practical pseudocode version is: `z = input; u = W1 @ z; s_i = z[:,a_i]*z[:,b_i] for i=1..g; phi = concat(ones(B,1), z, s); out = W2 @ phi + bias; y = z + gate*tanh(out)`. For a stronger version, binarize or softly binarize channels with \\(p=\\operatorname{sigmoid}(z/\\tau)\\), evaluate sampled quadratic forms \\(q_r\\) using \\(q_r(x+y)=q_r(x)+q_r(y)+x^\\top Jy\\pmod 2\\), and add the displayed \\(\\mathcal{L}_{\\mathrm{symp}}\\) term. Generate \\(U\\) from cheap symplectic basis operations: swap two coordinate pairs, swap members within a pair with sign adjustment, and apply elementary shears; these satisfy \\(U^\\top JU=J\\). The paper supplies the exact cocycle and dimension count; the real-valued relaxation, gate, temperature \\(\\tau\\), and sampled transformation distribution are engineering choices. The first experiment should use a two-layer MLP on CIFAR-10 embeddings or a molecular graph network, with \\(d=16\\) or \\(32\\), comparing a dense quadratic layer, a standard MLP, and this tied layer at equal hidden width. Measure validation accuracy, parameter count, and training loss per FLOP. A positive result is equal-or-better accuracy with at least 25% fewer interaction parameters, or faster convergence under random symplectic channel permutations.",
 "math_summary": "The paper defines a symplectic vector space \\(H=\\mathbb{F}_2^{2g}\\) with intersection form \\(\\widehat{\\iota}:H\\times H\\to\\mathbb{F}_2\\), and a quadratic form \\(q:H\\to\\mathbb{F}_2\\) satisfying \\(q(x+y)=q(x)+q(y)+\\widehat{\\iota}(x,y)\\). For a coordinate symplectic basis \\(S=\\{a_1,b_1,\\ldots,a_g,b_g\\}\\), the values \\(q(a_i)=\\lambda_i\\) and \\(q(b_i)=\\lambda_i'\\) determine \\(q\\) uniquely. The paper's Boolean algebra uses formal generators \\(\\overline{x}\\) with the relation \\(\\overline{x+y}=\\overline{x}+\\overline{y}+\\widehat{\\iota}(x,y)\\), and its degree-\\(n\\) subspace \\(\\operatorname{B}_n(g)\\) has dimension \\(\\sum_{k=0}^n\\binom{2g}{k}\\); in particular \\(\\dim \\operatorname{B}_2(g)=1+2g+\\binom{2g}{2}\\). We adapt this to real activations by pairing \\(d=2g\\) channels, choosing a skew-symmetric block matrix \\(J\\) with \\(J_{a_i b_i}=1\\), \\(J_{b_i a_i}=-1\\), and using the cocycle term \\(x^\\top J y\\) to constrain second-order interactions. Over \\(\\mathbb{F}_2\\), the quadratic relation is exact; over real activations, it becomes a parameter-sharing and augmentation-equivariance constraint.",
 "math_tags": [
  "algebra",
  "representation-theory",
  "combinatorics",
  "geometry"
 ],
 "ml_areas": [
  "embedding",
  "graph-nn",
  "mlp",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.26001",
  "arxiv_url": "https://arxiv.org/abs/2608.26001",
  "summary_what_math_gives_to_ml": "The paper develops a finite-dimensional Boolean-function algebra built from quadratic refinements of a symplectic vector space over \\(\\mathbb{F}_2\\), together with an explicit degree filtration and an \\(\\operatorname{Sp}_{2g}(\\mathbb{F}_2)\\)-action. The transferable asset is not the mapping-class-group application itself, but the combination of paired coordinates, a nontrivial bilinear cocycle, and symmetry-preserving low-degree feature spaces. This suggests a compact equivariant module for binary or discretized representations: represent interactions using the paper's quadratic relation rather than unconstrained pairwise features, and train with random symplectic augmentations. The idea is most plausible for models whose inputs already have natural paired channels, such as graph edge orientations, error-correcting codes, spin variables, or learned binary latent codes.",
  "title": "The kernel of the Birman-Craggs-Johnson homomorphism",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "memory",
  "generalization",
  "accuracy"
 ],
 "title": "Symplectic quadratic interaction layer",
 "url": "https://synthcore.org/idea/191/symplectic-quadratic-interaction-layer",
 "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
   }
  }
 }
}