{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a standard recurrent state update or finite-order SSM filter with a causal relative-history operator using a weakly singular kernel k(s)=s^{p-1}m(s), where 0\u003cp\u003c1. The resulting layer retains information over a power-law range of timescales and introduces tunable frequency-dependent phase and attenuation, while remaining implementable through a small bank of exponentially decaying states.",
 "download_zip": "https://synthcore.org/download/1119",
 "formulas_latex": [
  "$$\\eta^{t}(s)=u(t)-u(t-s),\\qquad s\\geq 0,$$",
  "$$\\mathfrak{m}(s,x)=s^{p-1}m(s,x),\\qquad 0\u003cp\u003c1,$$",
  "$$C_p=\\Gamma(p)e^{\\mathrm{i}\\pi p/2},$$",
  "$$R(t)=\\int_0^\\infty k(s)\\bigl(x(t)-x(t-s)\\bigr)\\,\\mathrm{d}s\\approx \\left(\\sum_{j=1}^{J}\\frac{w_j}{\\lambda_j}\\right)x(t)-\\sum_{j=1}^{J}w_jq_j(t),\\qquad \\dot q_j(t)=x(t)-\\lambda_jq_j(t).$$"
 ],
 "id": 2868,
 "implementation": "Integrate the layer at the recurrent or SSM block, immediately after the input projection and before the usual gated update or residual output. For a sequence x[1:T] with channel dimension d, choose J=8--32 positive decay rates lambda_j logarithmically spaced between 1/T_context and 1, or between the target longest timescale and the Nyquist-scale decay. Maintain q[j,d]. At timestep t with spacing Delta, update each mode using the exact discretization q_j \u003c- exp(-lambda_j Delta) q_j + (1-exp(-lambda_j Delta))/lambda_j times x_t. Compute r_t = a x_t - sum_j w_j q_j, with a=sum_j w_j/lambda_j. Feed the concatenation of x_t and r_t into a learned projection or gated transition, for example h_t=tanh(W_x x_t + W_r r_t + A h_{t-1}+b). Parameterize lambda_j=softplus(theta_j)+epsilon and w_j=softplus(phi_j) so every mode remains stable and positive. Initialize lambda_j logarithmically and initialize weights proportional to lambda_j^p over logarithmic bins, which approximates k(s)=s^{p-1}; optionally learn p through p=0.05+0.90 sigmoid(p_raw). For a complex-valued SSM, initialize the memory projection phase using C_p=Gamma(p) exp(i pi p/2); for a real network, represent its real and imaginary parts with two channels. The paper provides the fractional exponent and phase structure, while the exponential quadrature weights and rates are learned or fitted empirically. First test a 1-layer 256-hidden-unit model on copy/retrieval and Long Range Arena Pathfinder or ListOps against a GRU and diagonal SSM at equal parameter count and FLOPs. Measure loss versus sequence length, long-context accuracy, and stability under extrapolated context. Success means slower accuracy decay beyond the training length or equal accuracy with fewer recurrent states than the baselines.",
 "math_summary": "The paper defines the relative-history variable eta^t(s)=u(t)-u(t-s), where u(t) is the current signal and s is lag, and uses a memory kernel mathfrak{m}(s)=s^{p-1}m(s), with 0\u003cp\u003c1 and nonnegative envelope m(s). For a neural sequence x(t), define R(t)=integral from 0 to infinity of k(s)(x(t)-x(t-s)) ds, where k(s)=s^{p-1}m(s). Approximate the singular kernel by positive exponential modes k(s) approximately equal to sum over j of w_j exp(-lambda_j s), with w_j\u003e0 and lambda_j\u003e0. Define q_j(t)=integral from 0 to infinity of exp(-lambda_j s)x(t-s) ds. Then q_j satisfies the stable state equation dot q_j=x-lambda_j q_j, and R(t) is approximately (sum_j w_j/lambda_j)x(t)-sum_j w_j q_j(t). The paper identifies the high-frequency complex coefficient C_p=Gamma(p) exp(i pi p/2), where Gamma is the Gamma function; this coefficient supplies the characteristic phase of the fractional response. Positive weights and decay rates guarantee fading-memory stability, while logarithmically spaced rates approximate a power law over many frequencies.",
 "math_tags": [
  "dynamical-systems",
  "functional-analysis",
  "approximation-theory",
  "numerical-analysis"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model"
 ],
 "paper": {
  "arxiv_id": "2608.30138",
  "arxiv_url": "https://arxiv.org/abs/2608.30138",
  "summary_what_math_gives_to_ml": "The paper gives a constructive fractional-memory mechanism whose singular kernel produces non-integer frequency scaling, attenuation, and phase shift rather than the single-timescale decay of ordinary recurrent units. This suggests a causal neural layer with a positive power-law memory kernel, implemented efficiently by a bank of exponential states, providing long context with controlled stability and logarithmic memory cost. The most promising transfer is not the full geometric-optics proof, but the explicit relation between the exponent p, the complex coefficient C_p, and frequency-dependent memory response, which can guide initialization and regularization of a recurrent or state-space module.",
  "title": "High frequency wave propagation for the viscoelastic wave equation with singular memory",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "memory",
  "sample-efficiency",
  "stability"
 ],
 "title": "Fractional Memory State-Space Layer",
 "url": "https://synthcore.org/idea/2868/fractional-memory-state-space-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "queued_mechanism",
  "status_label": "Queued — mechanism check",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}