{
 "artifacts": null,
 "category": "memory",
 "description": "Replace a growing stream of cached token or event representations with one normalized exponentially updated anchor. Use the anchor as a global context vector or as a compressed key/value memory; the state retains recent information while avoiding the full movement cost of repeatedly attaching to the previous token.",
 "formulas_latex": [
  "$$x_0=p_0,\\qquad x_i=\\gamma x_{i-1}+(1-\\gamma)p_i,\\qquad 0\\leq\\gamma\\leq1.$$",
  "$$x_i=\\gamma^i p_0+(1-\\gamma)\\sum_{j=1}^{i}\\gamma^{i-j}p_j.$$",
  "$$C_N=\\sum_{i=1}^{N}\\left\\|p_i-x_{i-1}\\right\\|^{\\alpha},\\qquad \\limsup_{N\\to\\infty}\\frac{C_N}{N}\\leq\\left(\\frac{2}{1+\\gamma}\\right)^{\\alpha}\\quad(0\u003c\\alpha\\leq3).$$",
  "$$\\gamma_N^*=1-N^{-1/2}+\\frac{1}{2}N^{-1}+O(N^{-3/2})\\qquad(\\alpha=2).$$"
 ],
 "id": 169,
 "implementation": "Integrate the method into a streaming Transformer at the layer that produces per-token representations. Project each representation h_t into a memory dimension d and normalize it as p_t=h_t/(||h_t||_2+epsilon), so the unit-ball assumption is approximately satisfied. Maintain one state vector x_t per sequence, layer, or attention head. Before processing token t, use x_{t-1} as a global context token, or create one compressed key/value pair k_t=W_k x_{t-1} and v_t=W_v x_{t-1}. After the token is processed, update the state with x_t=gamma x_{t-1}+(1-gamma)p_t. Pseudocode is: initialize x=p_0; for each token, compute p=normalize(project(h)); compute context=f(x); compute the network output using h and context; then set x=gamma*x+(1-gamma)*p. For a context horizon N, initialize gamma=1-1/sqrt(N)+0.5/N using the paper's alpha=2 expansion, and compare this with gamma values 0.9, 0.99, and 0.999. The exact mathematical computations are the recurrence, geometric weights, and worst-case proxy B(gamma,alpha)=(2/(1+gamma))^alpha. Estimate empirically the embedding radius, average attachment cost, perplexity, and gradient variance. Start with a small causal Transformer on WikiText-2 or enwik8, comparing full KV caching, last-token memory, mean pooling, and this one-vector memory. Measure peak KV memory, tokens per second, validation perplexity versus context length, and worst-decile attachment cost. Success requires a large memory reduction with less perplexity degradation than last-token compression at equal parameter count.",
 "math_summary": "The paper defines points p_0,p_1,...,p_N in the Euclidean unit ball B_2^d(1), and the state recursion x_0=p_0 and x_i=gamma x_{i-1}+(1-gamma)p_i, where gamma is in [0,1]. Expanding the recursion gives x_i=gamma^i p_0+(1-gamma) sum_{j=1}^i gamma^{i-j}p_j, so x_i is a convex combination of observed points and remains in B_2^d(1). The insertion cost is C_N=sum_{i=1}^N ||p_i-x_{i-1}||^alpha, where alpha\u003e0. For arbitrary input sequences and 0\u003calpha\u003c=3, the paper states that the largest asymptotic mean cost is (2/(1+gamma))^alpha, strictly below the path value 2^alpha when gamma\u003e0. For independent uniform points, the minimizing constant parameter obeys 1-gamma_N^*=Theta(N^{-1/2}); for alpha=2, gamma_N^*=1-N^{-1/2}+1/(2N)+O(N^{-3/2}). In the neural adaptation, p_i is a normalized token embedding and x_i is the compressed memory vector.",
 "math_tags": [
  "geometry",
  "probability",
  "dynamical-systems"
 ],
 "ml_areas": [
  "transformer",
  "ssm",
  "kv-cache",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.25298",
  "arxiv_url": "https://arxiv.org/abs/2608.25298",
  "summary_what_math_gives_to_ml": "The paper studies a one-vector online state that interpolates between retaining only the previous observation and retaining a persistent anchor. Its transferable asset is the explicit geometric control of attachment distances: exponential memory remains a convex combination of bounded observations and improves the worst-case movement scale while using constant working memory. This suggests compressing streaming neural representations into a single prototype used as global context, retrieval state, or a routing anchor. The paper also provides a horizon-dependent tuning law, including an explicit asymptotic expansion for squared distance, making the proposal directly testable.",
  "title": "Sequential Euclidean tree construction with exponential memory: distributional performance and worst-case guarantees",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 4,
  "usefulness": 6
 },
 "solves": [
  "memory",
  "stability",
  "scalability"
 ],
 "title": "Bounded Exponential Anchor Memory",
 "url": "https://synthcore.org/idea/169/bounded-exponential-anchor-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
   }
  }
 }
}