AWhat changed
Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant.
Scientific scope
Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant. The universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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