Eigenfrequencies and the Solar System Resonance Cycle

Classical secular theory describes planetary orbital evolution through eigenfrequencies g_j (eccentricity) and s_j (inclination), determined by linearising gravitational perturbation equations or by direct numerical integration. The Holistic model derives all planetary periods as integer divisors of the Solar System Resonance Cycle (8H = 2,682,536 yr). This page maps the two frameworks side by side and notes where they agree, where they differ, and what each framing emphasises.

How secular theory determines eigenfrequencies

1. Secular eigendecomposition (Lagrange–Laplace): linearises the planetary perturbation equations in eccentricity and inclination, averages over orbital periods, and diagonalises the resulting constant-coefficient secular matrix to yield g_j (eccentricity) and s_j (inclination) eigenfrequencies with their planet-by-planet eigenvectors.

Historical lineage:

• Brouwer & Van Woerkom (1950) — first comprehensive secular theory by hand calculation; covered all major planets and Pluto.
• Bretagnon (1974) — refinement using early electronic computers; the values used by Berger (1978) and most Milankovitch references.
• Laskar (1988, 2004) — modern values from numerical integration combined with secular theory for higher accuracy.

2. Direct numerical integration: integrates the full Newtonian (and relativistic) equations of motion for millions of years and Fourier-analyses the resulting eccentricity and inclination time series. Peak frequencies in the spectrum are identified as the eigenmodes — capturing non-linear effects (chaotic dynamics, resonances) that secular theory cannot.

Eigenfrequencies are typically quoted to 4–5 significant figures, with uncertainties in the last digit reflecting truncation of the secular series, choice of reference epoch, and inclusion of post-Newtonian corrections. Values from different research groups (Bretagnon, Laskar, Standish, Brouwer) agree at the 0.1–1% level for the dominant terms.

The eigenmodes are real; only the planet attribution differs

Before the tables: the eigenmodes g_j, s_j are mathematical objects — eigenvalues of the Laplace–Lagrange secular perturbation matrix that captures gravitational coupling between all eight planets. Their numerical values (g₁ ≈ 5.5965 ″/yr, g₂ ≈ 7.4555 ″/yr, …) come from planetary masses and semi-major axes. They exist in any framework that solves the same physics. Both Berger 1978 and the Holistic model accept the eigenmodes. What differs between frameworks is physical attribution:

Notation on this page: the “Planet association” / “(Jupiter)” / “(Mercury)” labels in the tables below are Berger’s convention — included because that’s what the literature uses. The Holistic model does not equate these to its own planet-specific cycles.

All 32 L1 lattice integers have explicit Holistic-model attributions. Every LR04 lattice integer has both (a) a Berger / secular-theory label (g_j+k, s_j+k, g_j−g_k, s_j−s_k) and (b) an explicit Holistic-model multi-planet beat from PLANET_CYCLES. The full ranked table — top-1 attribution + alternatives per integer — is at doc 93 — L1 attribution reference . The 3-step status tally on doc 93 reports zero exact agreements between Berger and the Holistic model’s top-1 attribution across all 32 components: every integer names a different planet, a different mechanism class, or has no Berger prediction at all.

For empirical climate evidence that the eigenmodes are real but their beats and combinations — not single-planet attributions — are what shows up in paleoclimate spectra, see Climate Formula.

Published eigenfrequencies (Laskar 2004)

Eccentricity eigenmodes:

Inclination eigenmodes (all retrograde or near-zero):

(Bretagnon 1974 values differ by 0.1–0.5% from these. Berger planet labels are the literature convention — see framework comparison above.)

Six period types per planet (model)

Each planet has up to six distinct long-period cycles in the model, all expressible as integer divisors of 8H. The complete 8 × 6 period table (in both 8H/N form and years) is canonical at Fundamental Cycles §The full 8 × 6 period table. Earth’s row, used in the comparisons below:

Venus is the sole planet with prograde axial precession — its obliquity is ~177° (essentially upside-down rotation), so the standard cos(obliquity) factor flips sign (Cottereau & Souchay 2009 ). Uranus would also be prograde (obliquity ~98°) if its axial precession were not effectively frozen.

Direct comparisons

Berger climatic precession peaks

The Berger (1978) climatic-precession spectrum (g_j + k beats) maps directly to integer fractions of 8H within <0.4% — including Saturn:

Berger names each peak after a single planet (g_j + k); the model derives the same lattice peaks via multi-planet beats from PLANET_CYCLES — typically Earth.Axial combined with one or two Jupiter/Saturn elements. The two frameworks agree on which periods exist and disagree on which planet drives each beat. The 3 peaks marked ”— (not in canonical L1)” sit between adjacent lattice integers and the model does not recognise them as distinct lines. Full structural decomposition (n = 104 + δ_j) and Fibonacci analysis: Supporting Evidence §9.

g_j eigenmode periods vs model H/n divisors

Each eigenfrequency’s period (= 1,296,000 / g_j) against the model’s nearest H/n divisor:

Mercury (g₁), Mars (g₄), and Saturn (g₆) — the three planets where the model’s ecliptic divisor falls within 10% of the bare eigenfrequency — are the three planets where the dominant g_j is the strongest contributor to the planet’s perihelion precession with limited mode mixing. For the others:

• Earth: bare g₃ (75 kyr) does not match any H/n divisor directly. Earth’s full perihelion precession (~112k) does match H/3 cleanly — because Earth’s perihelion is a sum of g₃ plus substantial mixing from g₅ (Jupiter) and other modes, and H/n is calibrated to the full sum, not to g₃ alone.
• Outer planets (Jupiter, Uranus, Neptune): bare g_j periods are 300–2000 kyr while their model ecliptic divisors fall in the 67–670 kyr range. The model’s outer-planet ecliptic frame is defined differently than the bare g_j eigenmode.

The model’s H/n divisors are therefore calibrated to full per-planet perihelion rates with mode-mixing built in, not to bare eigenfrequency periods.

Per-planet theoretical perihelion precession (vs model)

Earth is the strongest case: classical theoretical perihelion (~112k) matches model ICRF (H/3 = 111,772) to 0.06%, and the classical climatic-precession spectrum mean (~21k) matches model ecliptic (H/16 = 20,957) to within 1%. Inner planets (Mercury, Mars) match in the ecliptic frame to 5–8%. Outer planets (Jupiter, Saturn, Uranus, Neptune) show divergence between the model’s ecliptic perihelion and the classical theoretical sidereal perihelion — the model’s outer-planet ecliptic frame uses a different definition.

Per-planet ascending node regression

The published s_j eigenfrequencies vs the model’s ascending-node integers:

The s_j and the model’s ascending-node integer divisors describe related but distinct quantities: s_j is the eigenfrequency of one secular eigenmode, while the model’s integer N gives a period (8H / N) at which the planet’s ascending node on the invariable plane completes a full regression in the J2000-fixed frame. Earth alone matches at 2.5% (because Earth’s ecliptic precession is by definition the s₃-related rate).

The ascending-node integers are: Mercury 9 = 3², Mars 64 = 8², Jupiter/Saturn 36 = 4×9, Uranus 11 (prime), Neptune 3 = F₄, Venus 1 (= 8H, full Solar System Resonance Cycle). They do not match the Laplace–Lagrange s_j exactly — they are an alternative integer assignment that produces a tighter JPL trend fit while preserving the Jupiter–Saturn lockstep.

Three universal identities

Working with the Resonance Cycle integer divisors, three identities connect the six cycle types per planet.

Identity 1: Frame conversion (ICRF ↔ ecliptic)

N_ICRF = 104 − N_ecl    (for prograde ecliptic planets)

104 = 8 × 13 is Earth’s axial precession integer N. Three special cases:

• Earth breaks the sum rule (N_ecl = 128 > 104, giving a prograde ICRF perihelion — see Why Earth Is Unique).
• Saturn is retrograde ecliptic, so N_ICRF = 104 + |N_ecl| = 168.
• Venus is also retrograde ecliptic, so N_ICRF = 104 + |N_ecl| = 110.

Identity 2: Eccentricity cycle as integer beat

N_ecc = |N_axial ± N_ICRF|

Opposite directions (one prograde, one retrograde) → add. Same direction → subtract.

For Uranus and Neptune (frozen axial), the beat reduces to the ICRF rate alone — a structural definition of “frozen axial precession” at the resonance-cycle scale.

Identity 3: Obliquity decomposition

N_ecl = N_obliq + N_eclPrec

For Jupiter and Saturn this is a Fibonacci-anchor identity using their ecliptic-perihelion Fibonacci anchors (H/5, H/8): the decomposition is consecutive-Fibonacci in H-denominators (Jupiter 5 = 2 + 3; Saturn 8 = 3 + 5), and the residual is a real model period (H/3, H/5). On the 8H lattice their actual secular ecliptic-perihelion periods are 8H/39 and 8H/65 (Law 6); those do not decompose into Fibonacci-clean residuals — the clean decomposition is a property of the Fibonacci lattice, not of the secular dynamics. Identities 1 and 2 survive the shift to 8H-lattice secular periods; identity 3 is Fibonacci-anchor specific.

Mirror pair: Mercury ↔ Uranus share N_eclPrec = 8 (period = H).

Lattice stability — Test C series

The direct numeric comparisons above show that the model’s 8H/n divisors line up with Laskar’s eigenmode beats at the modern epoch. Two pre-registered tests on the published Laskar 2004 nominal solution (INSOLN.LA2004.BTL.ASC, −51 Myr to 0) ask a sharper question: does the 8H lattice act as a preserved spectral manifold in LA2004 itself, and does the alignment persist back through the Cenozoic?

Test C-Invariant — Is the 8H lattice a preserved spectral manifold?

Method. For each sliding 5-Myr window of LA2004 Earth eccentricity and obliquity, compute the Welch PSD; sum the power inside ± bands around each of the 32 L1 integers; divide by total power over periods 10–500 kyr. Compare to 200 random 32-integer lattice selections in n ∈ [5, 200].

Result — obliquity is a strict invariant on the 8H lattice. At ±7.5 % bands the on-lattice fraction reaches 100 % — every Joule of LA2004 obliquity power lies on L1, with zero off-lattice content across the entire 50-Myr backward extension. Random 32-integer baselines capture only ~68 % of the obliquity power on average, so L1’s capture is structurally specific to the framework’s integer choice rather than a trivial consequence of having many bands.

Result — eccentricity is a dominant attractor on the lattice, but not strict. At ±15 % bands the on-lattice fraction reaches 73.8 %. L1 captures 2.5× the power of random 32-integer controls (at ±5 % bands: L1 = 63.8 % vs random average 25.0 %; none of 200 controls beat L1, p < 1/200). The residual ~26 % of eccentricity power lies in genuinely off-lattice frequencies and is consistent with documented inner-planet secular chaos in Laskar’s solution (Laskar 1989 , 1994, 2009), with Mercury’s eccentricity diffusion the main contributor.

The asymmetry has a structural explanation. The Earth-Saturn axis identified by Config #7 and Law 6 (the k+g₆ stability sub-lattice; canonical at L1 Attribution) is specifically a spin / obliquity coupling axis, and obliquity is precisely the channel where the 8H lattice is strictly conserved. The mechanism behind that conservation is action-angle closure: 8H is a real closed-orbit period in the obliquity sector of the secular dynamics, while Mercury-chaos-driven perturbation breaks closure in the eccentricity vector (canonical: Physical Origin §4). Scripts: scripts/l1_invariant_test.py.

Test C-50 — Lattice persistence across −−50 Myr

The question Test C-50 is designed to settle: if the 8H lattice match persists back 50 Myr in LA2004, the framework’s integers act as a KAM-frozen lattice invariant across the Cenozoic; if the match collapses, the framework would reduce to a modern parameterization of slowly-evolving Laskar dynamics. The test distinguishes the two views directly.

Method. Slide 5-Myr non-overlapping windows across LA2004 −50 to 0 Myr; compute MTM spectra of Earth eccentricity (LA2004 column 2) and obliquity (column 3); identify the top-12 spectral peaks per window; match each peak to the appropriate L1 sub-band (eccentricity-band integers n ≤ 53; obliquity-band integers n ≥ 65). Lattice positions are computed at each window center via the proper-physics formula — 8H(t)/n where H(t) follows from the Farhat 2022 Moon-distance polynomial combined with angular-momentum conservation, anchored at modern H = 335.317 kyr (see Expanding Resonance for the full derivation). H(t) is about 1 % smaller at −50 Myr than at the modern epoch, so every lattice position 8H(t)/n is correspondingly shorter in deep time.

Result — the canonical Milankovitch beats are dynamically stable on the lattice. The four canonical Milankovitch eigenmode beats — Berger’s 95-kyr eccentricity (n = 28) and the three obliquity beats (n = 65, 66, 68) — all stay within 4 % of their proper-physics-predicted positions across the entire 50-Myr backward extension. The drift figure below is the maximum percentage by which the observed LA2004 spectral peak strays from the predicted lattice position, measured across all ten 5-Myr windows:

Match-fraction by epoch (median % of LA2004 peaks within 5 % of an L1 integer): 83 % modern (>−10 Myr), 75 % mid (−30 to −10 Myr), 71 % deep (< −30 Myr). A modest 13 pp decline across 50 Myr — not collapse. The 405-kyr g₂−g₅ cycle dominates the eccentricity spectrum in every deep-time window — a successful prediction of the framework’s L1 / L2 separation, since 405 kyr is deliberately off the L1 integer lattice and lives in the L2 carbon-cycle thermostat layer (Climate Formula).

Precession sidebands (n ≥ 113) drift more. The k+g₅ (n = 113, 23.7 kyr) and k+g₂ (n = 120, 22.4 kyr) climatic-precession sidebands drift 16–24 % across the same window. This is physically expected: inner-planet chaos affects high-harmonic precession beats more strongly than low-harmonic obliquity beats (Laskar 1989 , 2011). The framework’s precession-band integers are stable at the modern epoch and drift with the chaotic dynamics in deep time.

Scope qualification. LA2004 is itself a numerical N-body solution with model assumptions and acknowledged chaos-driven divergence past ~40–50 Myr. The claim is therefore “the 8H lattice agrees with LA2004’s spectrum and is stable in LA2004” — not “the 8H lattice matches the true solar-system orbital spectrum over 50 Myr.” That stronger claim would require cross-validation against La2010a–d and La2011 (which diverge from LA2004 past ~40 Myr) and testing against radiometrically or magnetostratigraphically dated deep-time proxies. What Test C-50 defensibly establishes is that the framework is consensus-aligned with the working reference solution used to build every astronomical age model in the Cenozoic. Scripts: scripts/l1_vs_laskar_published_50myr.py.

Conclusions

1. The correspondence is real, not coincidental — Berger spectrum matches integer 8H/n within <0.4%; Earth’s perihelion in both frames within <1%; inner-planet theoretical rates within 5–8%; Saturn’s g₆+k ≈ ICRF perihelion within 3%.

2. The match is empirical, not derivational — H/n divisors are calibrated to full perihelion rates (with mode mixing built in). Bare g_j eigenfrequencies don’t match cleanly for half the planets. The model captures observables, not the secular eigenstructure beneath them.

3. The three structural identities are the strongest non-numerical claim — N_ICRF + N_ecl = 104, N_ecc = |N_axial − N_ICRF|, and N_ecl = N_obliq + N_eclPrec hold across all planets via simple integer arithmetic, tying together quantities that secular theory treats as independent eigenvalues. (The first two hold at both attribution levels — Fibonacci anchors and 8H-lattice secular periods; the obliquity decomposition is Fibonacci-anchor specific.)

4. Earth is the validation case — its perihelion matches secular theory + GR in both frames simultaneously (sidereal ↔ H/3, ecliptic ↔ H/16) to <1%. No other planet shows this two-frame consistency.

5. Divergences are honest, not failures — outer planets show 50%+ divergence on theoretical perihelion, but this reflects different frame definitions in the model (see Fundamental Cycles), not an error. The Berger spectrum match works because it uses the moving-equinox frame where definitions align. The bridge is axial precession: k = H/13 (integer 104) is where the two frameworks agree exactly, by definition — every Berger climatic-precession peak then adds a planet-specific eigenfrequency contribution δ_j to give n = 104 + δ_j.

6. The framework is falsifiable — if the H/n hierarchy reflects real underlying structure, refined numerical integrations should pull published g_j toward rational multiples of 1/H. If they don’t, the framework is empirical-only. Either outcome is testable.

In one sentence: classical secular theory and the Holistic model describe the same orbital observables to <1% in the strongest cases, but they disagree on what is fundamental — eigenmodes versus integer divisors of a single fundamental period — and the model’s most distinctive claim is three integer identities that secular eigendecomposition does not expose.

Open questions

1. Do the published eigenfrequencies converge on H/n rational values as numerical accuracy improves? If yes, that is evidence for the model’s structure; if not, the H/n framework is a coincidence. Either outcome is testable.

2. Do the model’s ascending-node integers describe the same quantity as Laplace–Lagrange s_j? Strictly, they don’t — the integers were re-fitted (2026-04-09) to match JPL J2000-fixed-frame ascending-node trends to <2″/century per planet while preserving the Jupiter–Saturn lockstep. This produces a tighter empirical fit at the cost of departing from the secular eigendecomposition.

3. Does Earth’s two-frame match generalise to other planets? Earth’s perihelion in both frames matches secular theory to <1%; whether other planets’ two-frame matches are as strong is unproven.

4. Are the planetary eigenfrequencies g_j derivable from Fibonacci-organised planetary masses? Secular theory takes masses as input; whether the masses themselves follow Fibonacci ratios — and whether that would force g_j into the integer divisors observed — is unproven.

5. Why do specific divisors N appear? Mercury N=9 (3²), Mars N=63 (7×9), Jupiter/Saturn N=36 (4×9) are clean small factors but not Fibonacci. The mix of Fibonacci and non-Fibonacci integers in the per-planet table is unexplained.

6. Why does Earth alone have a 1× tier in the Fundamental Cycle hierarchy? All seven other planets sit at 8H — only Earth’s three integer divisors share enough common structure to reduce to H. See Fundamental Cycles.

These remain testable predictions and structural claims rather than derivations from first principles.

See also

• Fundamental Cycles — Earth Fundamental Cycle (H) and Solar System Resonance Cycle (8H), with per-planet derivations and the full 8 × 6 period table
• Fibonacci Laws — the six laws and their derivation
• Supporting Evidence §9 — Berger (1978) spectral comparison (obliquity + climatic precession)
• Scientific Background §5 — Milankovitch / Laskar comparison and the model’s reinterpretation
• Climate Formula — companion empirical document with the canonical 3-layer formula (32-integer L1 lattice + 3-line L2 carbon thermostat + 6-step L3 transitions), per-regime ridge fit on LR04 / CENOGRID / EPICA / CenCO2PIP, per-planet contribution analysis