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  "slug": "braided-memory-register-a-composite-architecture-unifying-ultrametric-temporal-a",
  "title": "Braided Memory Register: A Composite Architecture Unifying Ultrametric, Temporal, and Associative Memory",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Memory architectures for modern AI systems are fragmented across hierarchical similarity indexes, append‑only logs, and relational graphs, causing latency, storage duplication, and provenance inconsistencies. This paper introduces the Braided Memory Register (BMR), an abstract data structure that unifies an ultrametric tree, a hash‑chained temporal spine, and a sparse associative cross‑link graph, all governed by version‑control semantics. Formal definitions are given, and analytic derivations yield (i) logarithmic ultrametric descent cost, (ii) a retrieval‑time bound of O(log N + k + d_max), (iii) a storage overhead of ≈ 2× for a balanced binary tree, (iv) a birthday‑collision probability of 4.3×10^-60 for a 256‑bit chain over 10⁹ commits, and (v) a speedup of 4.2×10⁴ over linear scan for N=10⁶ memories. Concrete numerical illustrations grounded in existing associative‑memory literature demonstrate that the BMR can be instantiated within realistic compute and storage budgets. The result changes practice by offering a single formal object that simultaneously guarantees similarity recall, temporal order, and verifiable provenance, enabling engineers to replace three disparate pipelines with a unified “quipu‑net” architecture.",
  "abstract_markdown": "Memory architectures for modern AI systems are fragmented across hierarchical similarity indexes, append‑only logs, and relational graphs, causing latency, storage duplication, and provenance inconsistencies. This paper introduces the **Braided Memory Register (BMR)**, an abstract data structure that unifies an ultrametric tree, a hash‑chained temporal spine, and a sparse associative cross‑link graph, all governed by version‑control semantics. Formal definitions are given, and analytic derivations yield (i) logarithmic ultrametric descent cost, (ii) a retrieval‑time bound of $O(\\log N + k + d_{\\max})$, (iii) a storage overhead of $\\approx$ 2× for a balanced binary tree, (iv) a birthday‑collision probability of $4.3\\times10^{-60}$ for a 256‑bit chain over $10^{9}$ commits, and (v) a speedup of $4.2\\times10^{4}$ over linear scan for $N=10^{6}$ memories. Concrete numerical illustrations grounded in existing associative‑memory literature demonstrate that the BMR can be instantiated within realistic compute and storage budgets. The result changes practice by offering a single formal object that simultaneously guarantees similarity recall, temporal order, and verifiable provenance, enabling engineers to replace three disparate pipelines with a unified “quipu‑net” architecture.",
  "date": "2026-10-21",
  "first_published": "2026-10-21",
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  "url": "https://papers.qnfo.org/papers/braided-memory-register-a-composite-architecture-unifying-ultrametric-temporal-a",
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