A Note on A φ-Hierarchy in Earth’s Degree-2 LOVE Numbers and its Relationship to GEIER’s α–e–Φ–√(π/5) Equations (LOVE Numbers and GEIER’s Equations, Part 1). ResearchGate (2025)

 

A Note on A φ-Hierarchy in Earth’s Degree-2 LOVE Numbers and its Relationship to GEIER’s α–e–Φ–√(π/5) Equations (LOVE Numbers and GEIER’s Equations, Part 1). ResearchGate Preprint (2025)

Geier and colleagues advance an ambitious and unusually integrative hypothesis: that Earth’s degree-2 solid-Earth tidal response admits a compact parametrization in terms of the reciprocal golden ratio φ and the π/5 kernel that underlies it, and that this same geometric scaffold interfaces naturally with a broader α–e–Φ–√(π/5) constant-family proposed in their earlier work.¹

From a geophysical editorial standpoint, the manuscript’s main strength is that it anchors its numerical claims to operational standards and mainstream references rather than to cherry-picked single estimates. In particular, the IERS Conventions (2010) provide the canonical nominal degree-2 Love/Shida numbers used throughout modern geodesy (h₂≈0.6078, l₂≈0.0847; k₂≈0.29525 in the geopotential formulation), and the manuscript’s primary comparisons are framed explicitly against these values and their known frequency dependence.^2,3,4

Within that frame, the paper makes a clear, testable statement: the proposed mapping h₂≈φ and k₂≈φ/2 holds at the few-percent level for nominal IERS values and for PREM-like Earth models, and remains directionally consistent when expressed through standard derived combinations (e.g., gravimetric and tilt factors). This is precisely the sort of phenomenological ansatz with falsification hooks that can be useful—provided that it is presented as a constrained parametrization rather than as a derived law.^1,2,5

The manuscript further argues that a multipole perspective on the tide-generating potential can tighten the h₂–φ proximity: because the lunar forcing is quadrupole-dominated (the next multipole being suppressed by a factor of order a/R), a quadrupole-only observed constant should be slightly reduced relative to a hypothetical full-multipole constant. Levine’s discussion captures this scaling succinctly, and the IERS conventions themselves implement degree separation (including explicit degree-3 terms) rather than folding higher degrees into degree-2 response coefficients.^6,2,4

In that limited and carefully defined sense, the Earth’s Love-number phenomenology provides supportive corroboration for the central mathematical motif emphasized by Geier et al.: a recurrent π/5–φ geometry can be used to organize the order of magnitude and relative scale of the dominant tidal response parameters, and to motivate a compact set of candidate relations that invite sharper testing. The work’s value is therefore less in asserting a finished mechanism, and more in offering a structured set of hypotheses that can now be confronted with frequency-dependent complex Love numbers, higher degrees (n>2), and independent estimation pipelines.^1,2,4,7

I would encourage my coauthors to preserve their generous cross-disciplinary reach while tightening the boundary between demonstrated geodetic facts and conjectural constant-family extensions. A particularly compelling next step—already anticipated in the manuscript—is an out-of-sample validation against the IERS frequency-dependent (complex) displacement response in the diurnal band, where resonant behaviour is well established and where a constant-family ansatz should be expected to fail unless it explicitly incorporates the underlying dynamics.^4,7

Overall, the paper is noteworthy for its clarity of numerical definitions, its engagement with established geodetic standards, and its willingness to state falsifiable predictions. With careful presentation, the manuscript could stimulate productive discussion at the boundary between classical tidal theory and modern constant-family speculation—particularly by clarifying which aspects of the α–e–Φ–√(π/5) programme are genuinely constrained by the Earth-tide data and which remain open.^1,2,6 

Stefan Geier, Haidholzen

References

1. Geier, S. A. et al. A φ-Hierarchy in Earth’s Degree-2 LOVE Numbers and its Relationship to GEIER’s α–e–Φ–√(π/5) Equations (LOVE Numbers and GEIER’s Equations, Part 1). ResearchGate Preprint (2025) (version 0.0.0.3). https://doi.org/10.13140/RG.2.2.24310.87362

2. Petit, G. & Luzum, B. (eds.) IERS Conventions (2010). IERS Technical Note No. 36 (IERS Conventions Centre, 2010).

3. IERS Conventions (2010), Chapter 6: Geopotential. In IERS Technical Note No. 36 (2010).

4. IERS Conventions Centre. Displacement of reference points (Chapter 7). In IERS Conventions (2010) (version 1 Feb 2018).

5. Dziewoński, A. M. & Anderson, D. L. Preliminary reference Earth model. Phys. Earth Planet. Inter. 25, 297–356 (1981). https://doi.org/10.1016/0031-9201(81)90046-7

6. Levine, J. The earth tides. Phys. Teach. 20, 588–595 (1982). https://doi.org/10.1119/1.2341161

7. Mathews, P. M., Dehant, V. & Gipson, J. M. Tidal station displacement. J. Geophys. Res. 102(B9), 20469–20477 (1997). https://doi.org/10.1029/97JB01515