golden-zeta

golden-zeta

Computational companion to Paper 150 v2.0 — The Golden Norm Hears the Dedekind Zeta DOI: 10.5281/zenodo.19162130 (all-versions) Author: Clifford Keeble · ORCID 0009-0003-6828-2155

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Summary

The Dedekind factorisation ζ_ℚ(√5)(s) = ζ(s) · L(s, χ₅) (Dedekind, 1871) makes the Riemann zeros one factor of the Dedekind zeta function of the smallest non-trivial real quadratic field. This directory tests the hypothesis that those zeros are detectable as statistical minima of a Fejér-weighted Dirichlet sum under the golden-field norm N(a + bφ) = a² + ab − b² on ℤ[φ].

At N = 10,000 Riemann zeros, the test statistic reaches z = −22.91 against a same-density uniform-random null. Discovery threshold is |z| ≥ 5. The investigation was executed under a pre-registered four-test falsification protocol on 12 April 2026; all four tests passed, with mechanism isolated to the field norm rather than to φ specifically.

The four tests and what they showed

Test	Question	Result
1. Primary	Does the golden-norm detector separate Riemann zeros from random?	z = −22.91 at N = 10,000
2. φ-specificity	Is φ the essential phase?	No. √2 phase gives |z| = 9.38; e and π both exceed |z| > 6. Any equidistributed irrational works.
3. Norm decoupling	Is the golden field norm essential?	Yes. φ-phase with circular norm fails (|z| = 2.63, sub-threshold). √2-phase with golden norm passes (|z| = 9.38). The norm carries the effect.
4. Dedekind partition	Are both factors of ζℚ(√5) independently detected?	Yes. Riemann zeros at |z| = 8.14, L(χ5) zeros at |z| = 6.18.

The headline is not that φ is special. The headline is that the indefinite form p² − 5q² of ℚ(√5) is an empirical Dedekind probe. Any sufficiently dense equidistributed phase sequence excites it.

Pre-registration audit trail

Two commits establish the protocol before any analysis was run:

• 21f74ed — primary protocol (view commit) Committed 2026-04-12 08:24:28 UTC

• c2b9d0d — compound addendum (Tests 2, 3, 4) (view commit) Committed 2026-04-12 08:49:09 UTC

Every dependent analysis commit is timestamped 12+ minutes after its pre-registration parent. Linear history, no rewrites. The chronology is inspectable.

This matters because the headline test statistic is only meaningful against a protocol that was fixed before the analyses were run. That protocol is in pre_registration_N1000.md and pre_registration_compound.md at the SHAs above.

Directory contents

Pre-registration

• pre_registration_N1000.md — primary protocol (commit 21f74ed)
• pre_registration_compound.md — compound addendum with four falsification tests (commit c2b9d0d)

Analysis

• run_N1000.py — primary test at N = 1000
• run_N10000.py — primary test at N = 10000 (headline result)
• run_alt_alpha.py — Test 2 (φ-specificity)
• run_norm_decoupling.py — Test 3 (norm essentiality)
• run_dedekind_partition.py — Test 4 (both factors of Dedekind product)

Forensic data (per-zero output tables)

• per_zero_1000.csv, per_zero_N10000.csv — primary-test outputs
• per_zero_sqrt2.csv, per_zero_e.csv, per_zero_pi.csv — alternative-phase probes
• per_zero_circular_norm.csv, per_zero_sqrt2_goldenNorm.csv — norm-decoupling probes
• per_zero_Lchi5.csv — L(χ₅) detection table

Narrative findings

• findings_N1000.md
• findings_compound.md

Reproduction

git clone https://github.com/CliffordKeeble/bootstrap-universe.git
cd bootstrap-universe/golden-zeta
pip install -r requirements.txt
python run_N1000.py              # primary test, smaller N, faster turnaround
python run_N10000.py             # headline test — longer runtime; see note
python run_alt_alpha.py          # Test 2
python run_norm_decoupling.py    # Test 3
python run_dedekind_partition.py # Test 4

Runtime note. The primary result at N = 1000 completes in minutes on a modern laptop. The headline result at N = 10,000 is substantially longer because mpmath.zetazero computes each Riemann zero from scratch; expect this to run into several hours depending on hardware. For verification of the protocol and mechanism rather than the headline scale, run_N1000.py is the recommended entry point.

Riemann zero data is computed on the fly via mpmath; no external dataset download is required. LMFDB zeros are cited in the paper for cross-reference but are not a runtime dependency.

Licence

MIT. See ../LICENSE.

Correspondence

Dr. Clifford Keeble, PhD — ORCID 0009-0003-6828-2155 academia.edu/CliffordKeeble Woodbridge, UK

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