{
 "artifacts": null,
 "category": "architecture",
 "description": "Augment an RNN or state-space model with a finite-state binary-context module whose transitions are those of a de Bruijn graph, while a signed transition channel records a quadratic parity function of the recent context. The exact finite-memory branch preserves cancellation-sensitive parity features that a continuous hidden state may forget, and a learned readout can combine it with the ordinary neural state.",
 "formulas_latex": [
  "$$s'=(s_2,\\ldots,s_M,a),\\qquad D_{s,s'}=\\mathbf{1}\\{s'=(s_2,\\ldots,s_M,a)\\}.$$",
  "$$q_T(s,a)=\\left[\\sum_{t\\in T}u_{M+2-t}u_{M+1}\\right]_2,\\quad u=(s_1,\\ldots,s_M,a),\\qquad B_{s,s'}=\\mathbf{1}\\{s'=\\operatorname{shift}(s,a)\\}(-1)^{q_T(s,a)}.$$",
  "$$\\zeta_R(z)=\\frac{1}{\\det(I-zR)},\\qquad \\log\\zeta_R(z)=\\sum_{n=1}^{\\infty}\\frac{\\operatorname{tr}(R^n)}{n}z^n.$$",
  "$$h_{k+1}=\\tanh\\!\\left(W_hh_k+W_xx_k+U\\phi_B(s_k)\\right),\\qquad \\phi_B(s_k)=B^\\top e_{s_k}.$$"
 ],
 "id": 3005,
 "implementation": "Integration point: add a small exact automaton branch to an RNN or SSM before its nonlinear recurrent update. For binary or discretized inputs, maintain a one-hot context vector p_k in R^(2^M). For real inputs, begin with a straight-through threshold a_k=1{v^T x_k\u003e0}; later replace it with a soft bit probability. At each step, compute the legal successor s'=(s_2,...,s_M,a), update p_{k+1}=P(a_k)p_k, compute the sign r_k(s)=(-1)^{q_T(s,a_k)}, and update the signed feature b_{k+1}=P(a_k)diag(r_k)b_k. Concatenate p_k or a learned low-dimensional projection of it with the ordinary hidden state h_k, then apply the standard neural update and train end-to-end. The exact transition and sign construction come from the paper; input binarization, projection, and neural readout are adaptations. Use sparse successor operations rather than dense matrices. First experiment: compare a vanilla GRU with a GRU-plus-parity-memory on synthetic length-256 sequences labeled by XORs of input pairs at offsets T, then test Copy Memory and permuted sequential-MNIST. Sweep M in {3,5,7}. The predicted signature is that parity-task accuracy remains nearly constant with sequence length up to the exact memory order, while the GRU baseline decays. The signed feature must equal the exact product of transition signs, with numerical error below 1e-6. Its empirical periodic response should also peak near phases associated with eigenvalues of B.",
 "math_summary": "Let the memory order be M\u003e=1, let the discrete context state be s=(s_1,...,s_M) in {0,1}^M, and let the next input bit be a in {0,1}. The de Bruijn transition is s'=(s_2,...,s_M,a). For a set of quadratic offsets T contained in {2,...,M+1}, define q_T(s,a)=[sum over t in T of u_{M+2-t}u_{M+1}]_2 modulo 2, where u=(s_1,...,s_M,a). The signed transfer matrix is B_{s,s'}=1{s'=shift(s,a)}(-1)^{q_T(s,a)}, while the unsigned matrix is D_{s,s'}=1{s'=shift(s,a)}. The paper's Fourier decomposition shows that an auxiliary parity coordinate produces the blocks D and B. For any finite matrix R, the zeta identity is zeta_R(z)=1/det(I-zR), and log zeta_R(z)=sum_{n\u003e=1} tr(R^n)z^n/n, so eigenvalues and signed closed-walk counts govern long-run responses.",
 "math_tags": [
  "dynamical-systems",
  "spectral-theory",
  "linear-algebra",
  "combinatorics"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "embedding"
 ],
 "paper": {
  "arxiv_id": "2609.01399",
  "arxiv_url": "https://arxiv.org/abs/2609.01399",
  "summary_what_math_gives_to_ml": "The paper provides a constructive symbolic-dynamics mechanism: a quadratic rotation-symmetric Boolean rule induces a signed binary de Bruijn transfer matrix, and Fourier transformation in an auxiliary parity coordinate block-diagonalizes the finite-type shift into unsigned and signed components. Its dynamical zeta function is exactly the reciprocal characteristic determinant, so periodic-orbit counts are encoded by traces of transfer-matrix powers and the eigenvalues determine recurrence behavior. This can transfer to neural sequence models as a finite-memory parity/automaton module and as a spectral-zeta monitor or regularizer for recurrent dynamics. The transfer is most credible for RNNs, state-space models, and discrete latent sequence architectures rather than generic feedforward networks.",
  "title": "The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Signed de Bruijn parity memory",
 "url": "https://synthcore.org/idea/3005/signed-de-bruijn-parity-memory",
 "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
   }
  }
 }
}