The Golden Norm Hears the Dedekind Zeta: A Pre-Registered Detection of ζ_{ℚ(√5)}(s) Zeros via the Indefinite Form p² − 5q²
Abstract
We report a pre-registered empirical detection of the zeros of the Dedekind zeta function ζ_{ℚ(√5)}(s) = ζ(s)·L(s, χ₅) using a family of functions built from an equidistributed phase sequence and the indefinite Galois norm N(a) = p² − 5q² on ℝ(√5). Four adversarial falsification tests were pre-registered with commit SHAs prior to data analysis. At N=10000 non-trivial Riemann zeros, the overall test statistic reaches z = −22.91 against the same-density uniform-random null (discovery threshold is |z| ≥ 5). The √N scaling from N=100 to N=10000 is confirmed to within 11% of prediction. Three further pre-registered tests identify the mechanism. The golden ratio φ is NOT specific to the effect: alternative equidistributed phases using √2, e, and π all produce |z| > 5; √2 is stronger than φ at |z| = 9.38. The golden norm, however, is essential: {nφ} evaluated with the circular norm fails (|z| = 2.63, below threshold), while {n√2} evaluated with the golden norm passes (|z| = 9.38). Finally, a Dedekind partition test confirms the mechanism: both factors of ζ_{ℚ(√5)} are detected, with z = −8.14 against Riemann zeros and z = −6.18 against L(s, χ₅) zeros. The surviving claim is structural, not phase-specific: the indefinite form N is the measurement instrument that the Dedekind zeta of ℚ(√5) responds to, and any sufficiently dense equidistributed probe suffices to realise the detection. This paper explicitly retracts three claims from v1.3: that the golden-angle phase {nφ} is essential (refuted), that the "100/100 matched" framing of v1.3 §3.1.1 represents near-perfect correspondence (at W=1.0 same-density random minima already achieve 89%), and that Re(s) = 1/2 and Re² = 5·Im² are the same constraint in different coordinates (refuted by direct numerical test at individual zeros). The v2.0 paper is narrower in scope and sharper in claim than v1.3: one mechanism, one instrument, four falsification tests that could have refuted it, none of which did.
Fields of Study & Subjects
Advanced Mathematical Theories and Applications
Analytic Number Theory Research
Advanced Mathematical Identities
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 12 April 2026
- Document type
- Preprint