{
  "abstract": "A phase that reaches a fault-tolerant processor in additive pieces can be remembered until the last piece arrives, or executed on arrival: the first option costs classical memory carried across rounds, the second costs magic states committed before the phase is known. We determine the exchange rate $α$, committed $T$ gates per bit of memory forgone, for ancilla-free coordinatewise Clifford+$T$ compilation. The Ramanujan bound gives $α\\ge2$ with explicit constants, the square-root barrier of the spectral method. An elementary determinant method, using that quaternion numerators lie on spheres in both real embeddings of $\\mathbb{Q}(\\sqrt2)$, counts words near any rotation coset below that barrier. With a height dichotomy for the sphere sections it produces and a fibration over rational projections, it gives every $α<5/2$ unconditionally and uniformly over frames, by an exact SMT check of its continuum case analysis; $5/2$ is where this method stops. At Clifford-framed cosets the volume law holds up to subexponential factors: all but a vanishing fraction of $z$-rotations need $T$-count $(3-o(1))\\log_2(1/\\varepsilon)$, and processes whose committed pieces are near Clifford-framed $z$-rotations, including per-rotation pipelines, have $α\\ge3-o(1)$, which a fractional-passthrough family attains under the Ross-Selinger typical-cost hypothesis. Under an equidistribution conjecture supported by exhaustive enumeration to $T$-count 22, $α=3$ in general and memory should be shed in whole rotations. The bounds hold even when the phases cancel to the identity; side information enters through a conditional entropy; probabilistic mixing halves the costs but not the rate. With clean ancillas and a phase-gradient catalyst, table lookups batched over coordinates drive the rate to $O(1/\\log\\log(1/\\varepsilon))$, so the constant-rate law is specific to coordinatewise synthesis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37368",
  "author_total": 3,
  "authors": [
    {
      "name": "Jinze Yang"
    },
    {
      "name": "Yangyang Li"
    },
    {
      "name": "Xiu-Hao Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b536bgz507ga6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37368",
  "fulltext_url": null,
  "id": "b536bgz507ga6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37368",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214892849",
  "pdf_url": "https://arxiv.org/pdf/2609.37368",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b536bgz507ga6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:48Z",
      "native_id": "2609.37368",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37368"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A Memory-Magic Exchange Law in Streaming Clifford+T Compilation",
  "title_text": "A Memory-Magic Exchange Law in Streaming Clifford+T Compilation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/b536bgz507ga6",
  "venue": "arXiv",
  "versions": []
}