bibkey: hurwitz1891irrational authors: Adolf Hurwitz year: 1891 title: Ueber die angenaeherte Darstellung der Irrationalzahlen durch rationale Brueche doi: 10.1007/BF01206656 claim: Every irrational has infinitely many rational approximants at the sharp reciprocal-square-root-five scale, with the golden continued-fraction class extremal. strata_touched:

• D5/S0/Tower/Hardness/RationalSpectrum license: citation-only triage: anchor

Ueber die angenaeherte Darstellung der Irrationalzahlen durch rationale Brueche

Hurwitz’s paper is the classical source for the sharp Diophantine-approximation constant 1 / sqrt(5). In normalized regular-continued-fraction coordinates, every irrational orbit has lower-limit approximation coefficient at most this constant, and the continued-fraction class with an all-one tail attains it.

The repository declaration expresses the same extremum as a least element of the set of upper bounds of the hardness spectrum. This is the order-correct reading of the source atom’s phrase “the bottom of the supremum structure”: the hardness values themselves have sharp supremum 1 / sqrt(5).

Search log

• 2026-08-17: Searched D5 declarations first. The exact local declarations D5.S1.Depth.golden_hurwitz_bound and D5.S3.AnalyticClosure.GoldenApproximationConstant.golden_fibonacci_approximation_constant_tendsto cover a golden-ratio lower bound and the Fibonacci attainment limit, respectively. Neither supplies the universal sharp upper bound, and their S1/S3 modules cannot be imported upward into the required S0 destination.
• 2026-08-17: Queried Crossref directly by DOI. The resolver returned Adolf Hurwitz, the article title, June 1891, Mathematische Annalen volume 39, pages 279-284, and DOI 10.1007/BF01206656.
• 2026-08-17: Queried the public summary of Hurwitz’s number-theory theorem to confirm the irrational rational-approximation attribution. The Springer PDF endpoint returned an access-check HTML page, so no claim of inspecting the article PDF is made.
• 2026-08-17: Searched pinned Mathlib v4.31.0, Loogle, LeanSearch, and GitHub Lean code for the sharp theorem and badly-approximable formulation. Only Dirichlet’s constant-one theorem, Legendre’s constant-one-half criterion, golden-ratio identities, and Liouville-style results were found. No exact reusable sharp Hurwitz theorem was found, so the repository declaration uses a direct normalized continued-fraction proof.

Verified locator

• DOI: https://doi.org/10.1007/BF01206656