"Fibonacci, Lucas and Φ structures in neutrino physics - Structural connections and testable predictions within the Geier programme: An analytical review with reproducible mathematical and numerical results" by Stefan Geier et al.
"Fibonacci, Lucas and Φ structures in neutrino physics - Structural connections and testable predictions within the Geier programme: An analytical review with reproducible mathematical and numerical results"
by Stefan Geier et al.
"Occurrences
of the Golden Ratio Φ in Neutrino Physics?"
A summary
based on: S. A.
Geier et al., Fibonacci, Lucas and 
structures in neutrino physics: Structural
connections and testable predictions within the Geier programme (21 October 2026; DOI: 10.13140/RG.2.2.31802.96962):
- Recurrence Matrix Eigenvalues: In neutrino mass model
building, the golden ratio
first arises as the dominant eigenvalue
of the Fibonacci recurrence matrix 
, which possesses eigenvalues and
, generates Fibonacci entries 
, and yields Lucas traces 
(Geier et al., Jul 2026a, Jul 2026b;
Kajiyama, Raidal & Strumia, 2007).
- Fixed-Point Equilibration: The recurrence ratio map
induced by 
has
as
its attracting fixed point with derivative 
, providing an exact mathematical
realization of Geier’s equilibrium concept for flavour-space operators
(Geier et al., Jul 2026a, Jul 2026b).
- Solar Mass Block and Eigenvalue
Ratio: In the
Kajiyama–Raidal–Strumia neutrino mass texture, embedding into the
solar mass block 
yields signed eigenvalues 
and
, fixing the physical solar neutrino mass
ratio to 
, the solar mass-squared splitting to 
, and the basis-invariant squared-mass
traces to even Lucas numbers 
(Kajiyama, Raidal & Strumia, 2007).
- Light Normal-Ordering Mass
Spectrum:
Combining the Fibonacci-block prediction with the JUNO–NuFIT normal-ordering
oscillation benchmark selects a conditional light neutrino spectrum of
and
a total mass sum of 
, which lies below the historical PDG
cosmological benchmark of 
and
the illustrative 
DESI DR2–CMB threshold (Olive, 2024;
Abdul-Karim et al., 2025; Elbers et al., 2025; JUNO Collaboration, 2026;
Esteban et al., 2026).
- Alternative Golden Mass-Ratio
Hypotheses:
Beyond the
solar block, defines direct pairwise mass-ratio
constraints—specifically 
, 
, and 
—that select heavier testable
normal-ordering spectra with total mass sums of 
, 
, and 
, respectively, whereas inverted ordering
(
) excludes both and
solar ratios (Geier et al., Oct 2026).
- Golden Solar Mixing Angles (GR1
and GR2): In
the PMNS lepton mixing matrix, specifies two classic leading-order
solar-angle prescriptions—GR1 (
, giving 
and
) and GR2 (
, giving 
and
)—which differ from the JUNO benchmark (
) by 
and
standard errors and quantify the required
charged-lepton or symmetry-breaking corrections (Kajiyama, Raidal &
Strumia, 2007; Adulpravitchai, Blum & Rodejohann, 2009; Gehrlein et
al., 2015; Puyam & Singh, 2025; JUNO Collaboration, 2026).
- Icosahedral
Symmetry and the 
Modular Phase Step: In rotational icosahedral () flavour-symmetry models and Geier’s
modular-angle framework, and
emerge as the matrix traces of 
(
) and 
(
) three-dimensional rotations and connect
directly to Geier’s 
modular phase step through the exact
geometric identity 
(Everett & Stuart, 2009; Feruglio
& Paris, 2011; Geier et al., Dec 2025, Mar 2026, Jul 2026b).
References
- Abdul-Karim, M. et al. (DESI Collaboration) (2025). DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints. Physical Review D 112, 083515. DOI: 10.1103/tr6y-kpc6. arXiv:2503.14738v3.
- Adulpravitchai, A., Blum, A. & Rodejohann, W. (2009). Golden Ratio Prediction for Solar Neutrino Mixing. New Journal of Physics 11, 063026. DOI: 10.1088/1367-2630/11/6/063026. arXiv:0903.0531.
- Elbers, W. et al. (DESI Collaboration) (2025). Constraints on Neutrino Physics from DESI DR2 BAO and DR1 Full Shape. Accepted in Physical Review D. DOI: 10.48550/arXiv.2503.14744. arXiv:2503.14744v3.
- Esteban, I., Gonzalez-Garcia, M. C., Maltoni, M., Martinez-Soler, I., Pinheiro, J. P. & Schwetz, T. (2026). Lessons from the first JUNO results. Journal of High Energy Physics 04, 089. DOI: 10.1007/JHEP04(2026)089. arXiv:2601.09791v2.
- Everett, L. L. & Stuart, A.
J. (2009).
Icosahedral () Family Symmetry and the Golden Ratio
Prediction for Solar Neutrino Mixing. Physical Review D 79,
085005. DOI: 10.1103/PhysRevD.79.085005. arXiv:0812.1057.
- Feruglio, F. & Paris, A.
(2011). The
Golden Ratio Prediction for the Solar Angle from a Natural Model with Flavour Symmetry. Journal of
High Energy Physics 03, 101. DOI: 10.1007/JHEP03(2011)101.
arXiv:1101.0393.
- Gehrlein,
J., Oppermann, J. P., Schäfer, D. & Spinrath, M. (2015). An
Golden Ratio Flavour Model. Nuclear
Physics B 890, 539–568. DOI: 10.1016/j.nuclphysb.2014.11.023.
arXiv:1410.2057. - Geier,
S. et al. (Dec 2025). GEIER’s
rule as a candidate “angular quantum”
across tetrapyrrolic macrocycles, strained polyhedra and viral capsids
(Part 1). ResearchGate Preprint. DOI:
10.13140/RG.2.2.13182.86089. - Geier,
S. A. et al. (Mar
2026). Quantum
Rotation Operators, Crystallographic
-Fold Symmetry, and GEIER’s Rule (Part 5.1): Radian-consistent
formulation, commensurability, and example SALCs for a (
) phase-step. ResearchGate
Preprint. DOI: 10.13140/RG.2.2.20923.07201. - Geier, S. A. et al. (Jul 2026a). GEIER’s equations and structuralism according to Jean PIAGET: A preliminary approach (From numerical nearness to Fibonacci–Lucas transformation orbits, equilibration and falsifiable cross-domain bridge laws). ResearchGate Preprint. DOI: 10.13140/RG.2.2.17051.43041.
- Geier, S. A. et al. (Jul 2026b). GEIER’s equations through Jean Piaget’s transformation structuralism: A preliminary approach (including culture): The Fibonacci–Lucas–golden family as a proposed common formal spine from nuclide state spaces to culture. ResearchGate Preprint. DOI: 10.13140/RG.2.2.29923.31520.
- Geier,
S. A., Geier, C., Geier, S., Geier, C., Geier, K., Blättermann-Goldstein,
N. & Geier-Noehl, M. (Oct 2026). Fibonacci, Lucas and structures in neutrino physics:
Structural connections and testable predictions within the Geier
programme. ISTS & LMU Munich Research Manuscript
(Version 0.0.0.0, 21 October 2026).
- JUNO Collaboration (2026). Measurement of reactor neutrino oscillation with the first JUNO data. Nature 654, 343–348. DOI: 10.1038/s41586-026-10538-z.
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- Olive, K. A. (2024). Sum of Neutrino Masses (Revised August 2023). In S. Navas et al. (Particle Data Group), Review of Particle Physics, Physical Review D 110, 030001. DOI: 10.1103/PhysRevD.110.030001.
- Puyam, V. & Singh, N. N.
(2025). symmetry and deviation from Golden Ratio
mixing with charged lepton flavor violation. Nuclear Physics B
1011, 116800. DOI: 10.1016/j.nuclphysb.2025.116800. arXiv:2308.05944.
Yours Stefan Geier, Haidholzen
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