SWhat changed
A deterministic sequence can spread around a circle as evenly as interval lengths predict. Weyl’s exponential-sum criterion tests this by asking whether each nonzero integer frequency cancels out on average. His 1916 paper applies the method to polynomial sequences with an irrational nonconstant coefficient and to multidimensional settings under the stated independence conditions. Even distribution is a long-run frequency statement; it does not make the sequence independent, random or perfectly balanced after a fixed number of terms.
Scientific scope
A deterministic sequence can spread around a circle as evenly as interval lengths predict. Weyl’s exponential-sum criterion tests this by asking whether each nonzero integer frequency cancels out on average. His 1916 paper applies the method to polynomial sequences with an irrational nonconstant coefficient and to multidimensional settings under the stated independence conditions. Even distribution is a long-run frequency statement; it does not make the sequence independent, random or perfectly balanced after a fixed number of terms. A qualitative asymptotic theorem for each sequence or fixed polynomial under the stated irrationality conditions, not a uniform numerical discrepancy rate over all irrational coefficients or a finite-sample randomness guarantee. An irrational constant term alone is insufficient. The scalar linear case predates this paper and is credited jointly in the original to Bohl, Sierpiński and Weyl; Hardy–Littlewood's earlier and parallel polynomial work is explicitly retained. Later metric theorems and the geometric appendix lie outside this record.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. A qualitative asymptotic theorem for each sequence or fixed polynomial under the stated irrationality conditions, not a uniform numerical discrepancy rate over all irrational coefficients or a finite-sample randomness guarantee. An irrational constant term alone is insufficient. The scalar linear case predates this paper and is credited jointly in the original to Bohl, Sierpiński and Weyl; Hardy–Littlewood's earlier and parallel polynomial work is explicitly retained. Later metric theorems and the geometric appendix lie outside this record.
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 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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