{
  "abstract": "We study the edge-power cost of the labelled tree generated by the $γ$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$. Retaining $x_i$ subdivides the insertion segment into a spine edge and a leaf edge. The memory parameter $γ$ controls how long earlier points influence subsequent attachment points. We minimize the sum of the $α$-powers of the edge lengths under independent uniform input and arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $α=1$. Its continuous extension is minimized at the boundary for $0<α\\leq1$, while every global minimizer is interior for $α>1$. We determine the finite optimizer in the joint window $α_N=1+\\varepsilon_N$, $\\varepsilon_N\\log N\\toλ$. Below an explicit threshold it lies on the $N^{-1/2}$ scale, at the threshold its scale is $\\sqrt{\\log N/(N\\log\\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies the governing correction, and differentiated estimates prove eventual uniqueness. At $α=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. For arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per point are explicit for $0<α\\leq3$. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to $2\\log2/\\logα$. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27777",
  "author_total": 1,
  "authors": [
    {
      "name": "Pedro M. M. de Castro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1xyhemqz7e1y5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27777",
  "fulltext_url": null,
  "id": "1xyhemqz7e1y5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27777",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7204810967",
  "pdf_url": "https://arxiv.org/pdf/2608.27777",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1xyhemqz7e1y5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.27777",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27777"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions",
  "title_text": "Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/1xyhemqz7e1y5",
  "venue": "arXiv",
  "versions": []
}