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The price of unreachable perfection: dimension laws at every regular scale

Seth Douglas and Nidhal Mghirbi · version 1.0.1 · Zenodo, · 24 pages · doi:10.5281/zenodo.23774171 · PDF

In plain wordsFor games that finite quantum systems can win with probability approaching one but never surely, the dimension needed as the error shrinks can be made to follow, on a doubly logarithmic scale and up to lower-order terms, any sufficiently regular growth law, not only power laws; computability alone is not enough.

The result

As the paper states it, in briefThe growth of the least dimension can be prescribed at every regular scale, not only along power laws. At every asymptotically flat scale the normalized endpoints are exactly the (Σ20,Π20) pairs, and a computable target can still lie outside every game’s profile.

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Novelty and credit

Novelty
New, with named precedents in form: Bartholdi and Erschler; Kassabov and Pak; Hochman and Meyerovitch; Meyerovitch; Sapir, Birget and Rips; Brady and Bridson. The base-scale classifications it restates are those of The uncomputable price of precision.

Versionstwo, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23754198, which always resolves to the newest.
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1.0.110.5281/zenodo.23774171Correction: Packaging: version 1.0.0 omitted the recorded check outputs listed in its checksum manifests, and one manifest entry was stale. The paper is unchanged. PDF hosted here.
1.0.010.5281/zenodo.23754199

Paper, code and links

PDF

papers/unreachable-perfection-v1.0.1.pdf, version 1.0.1, 602,266 bytes, 24 pages.

SHA-256 049e8e32b49b36da097b04a0a6be90280124c0e0238f7ebc9d305deb348ab0e9

Note

Version 1.0.1 changes packaging only

Corrections

  1. Version 1.0.1. Packaging: version 1.0.0 omitted the recorded check outputs listed in its checksum manifests, and one manifest entry was stale. The paper is unchanged.doi:10.5281/zenodo.23774171

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Cite

@misc{douglas2026unreachableperfection,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {The price of unreachable perfection: dimension laws at every regular scale},
  year    = {2026},
  version = {1.0.1},
  doi     = {10.5281/zenodo.23774171},
  url     = {https://doi.org/10.5281/zenodo.23774171}
}

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