Fibonacci Laws of Planetary Motion — Technical Background

What this page shows

Every planet has two key orbital properties that change slowly over hundreds of thousands of years:

• Eccentricity — how elongated the orbit is (0 = perfect circle, 1 = extreme ellipse). Earth’s is currently about 0.017, meaning its orbit is nearly circular.
• Inclination — how tilted the orbit is relative to the average plane of the Solar System. All planets wobble up and down slightly over long timescales; the oscillation amplitude measures how far each planet tilts.

When each value is multiplied by $\sqrt{m}$ (where $m$ is the planet’s mass), both eccentricities and inclinations snap onto Fibonacci ratios across all eight planets. The $\sqrt{m}$ weighting is not arbitrary — it is the unique exponent that arises in the Angular Momentum Deficit (AMD), the conserved quantity governing long-term orbital stability. In AMD-natural variables ($\xi = e\sqrt{m}$ for eccentricity, $\eta = i\sqrt{m}$ for inclination), ratios between planets become simple Fibonacci fractions.

The six laws documented below are:

1. Law 1 — Fibonacci Cycle Hierarchy: dividing $H$ by successive Fibonacci numbers yields the major precession periods
2. Law 2 — Inclination constant: $d \times \text{amp} \times \sqrt{m} = \psi$ with a single universal constant $\psi$ and pure Fibonacci divisors $d$
3. Law 3 — Inclination balance: angular-momentum-weighted amplitudes cancel between two phase groups (99.9998% balance)
4. Law 4 — Eccentricity constant: AMD partition ratios satisfy two Fibonacci constraints per mirror pair, determining all 8 eccentricities (0 free parameters)
5. Law 5 — Eccentricity balance: an independent balance condition on eccentricities using the same divisors and phase groups (99.88% balance)
6. Law 6 — Saturn-Jupiter-Earth Resonance: a closed beat-frequency loop linking the three dominant precession periods

The universal constant $\psi$ can be computed purely from Fibonacci numbers and the Holistic-Year $H = 333{,}888$:

From this single constant, all eight planets’ inclination amplitudes are predicted with zero free parameters. Combined with the eccentricity laws, the model connects orbital shapes and tilts of all eight planets through Fibonacci arithmetic rooted in a single timescale — the Holistic-Year $H = 333{,}888$ years, which also determines each planet’s precession period (Law 1; see also Precession).

Prediction summary

Precession periods — Law 1 shows that dividing $H$ by successive Fibonacci numbers yields all major precession periods. The beat frequency rule $1/H(n) + 1/H(n+1) = 1/H(n+2)$ is an algebraic identity from the Fibonacci recurrence.

Inclination amplitudes — predicted from the single $\psi = 2205/(2H)$ with pure Fibonacci divisors. All 8 planets fit within their Laplace-Lagrange secular theory bounds. The configuration is the unique solution (Config #32) satisfying all four physical constraints out of 7,558,272 tested — see Law 2 and Finding 2 for details.

All eight eccentricities — Law 4 predicts all eccentricities from inclinations via Fibonacci pair constraints on the AMD partition ratio, with zero free parameters. Inner planets also form a Fibonacci ladder $\xi_V : \xi_E : \xi_{\text{Ma}} : \xi_{\text{Me}} = 1 : 5/2 : 5 : 8$, where $\xi = e \times \sqrt{m}$. See Law 4 and Finding 5 for details.

Predictive system status

Input Data

Masses and semi-major axes are derived from the Holistic computation chain ($H = 333{,}888$, orbit counts, Kepler’s third law). Eccentricities are J2000 values except Earth, where the oscillation-midpoint base eccentricity (0.015321) from the 3D simulation is used — see Finding 5. Inclinations are to the invariable plane at J2000 (Souami & Souchay 2012). Inclination amplitudes are derived from the Fibonacci formula $\text{amp} = \psi / (d \times \sqrt{m})$.

Assignments

Planet Configuration

Phase Groups

The model uses two phase angles, 180° apart, derived from the $s_8$ eigenmode of Laplace-Lagrange secular perturbation theory ($\gamma_8 \approx 203.3195°$):

The group assignment is constrained by: (1) each planet’s oscillation range must fall within the Laplace-Lagrange secular theory bounds, (2) the inclination structural weights must balance (Law 3), and (3) the eccentricity weights must balance (Law 5).

Precession Periods

Law 1: The Fibonacci Cycle Hierarchy

Dividing the Holistic-Year $H = 333{,}888$ by successive Fibonacci numbers yields the major precession periods of the solar system.

Beat frequency rule

The beat frequencies of consecutive Fibonacci-divided periods satisfy the Fibonacci recurrence:

This is an algebraic identity, not an empirical fit. For example:

• $1/111{,}296 + 1/66{,}778 = 1/41{,}736$ (i.e., $3 + 5 = 8$)
• $1/66{,}778 + 1/41{,}736 = 1/25{,}684$ (i.e., $5 + 8 = 13$)
• $1/41{,}736 + 1/25{,}684 = 1/15{,}899$ (i.e., $8 + 13 = 21$)

The empirical content of Law 1 is that the precession periods of Earth ($H/3$), Jupiter ($H/5$), and Saturn ($-H/8$, retrograde) are related to a single timescale $H$ through Fibonacci denominators. The higher-order periods ($H/13$, $H/21$, $H/34$) then follow as beat frequencies. See also Precession for the physical meaning of each cycle.

Law 2: Inclination Amplitude

Each planet’s mass-weighted inclination amplitude, multiplied by its Fibonacci divisor, equals the universal $\psi$-constant:

Equivalently:

This holds for all 8 planets with a single universal $\psi = 3.302 \times 10^{-3}$.

Predictions and Laplace-Lagrange validation

The equation $d \times \text{amp} \times \sqrt{m} = \psi$ holds by construction. The non-trivial test is that every predicted range (mean $\pm$ amplitude) falls within the Laplace-Lagrange secular theory bounds — and all 8 do.

Worked example: Earth’s inclination amplitude

Earth has Fibonacci divisor $d = 3$ ($= F_4$). Step by step:

The mean is computed from the J2000 constraint:

Fibonacci vs IAU 2006-optimized values

The 3D simulation uses a slightly different value optimized for the IAU 2006 obliquity rate:

The 0.20% difference is within the model’s tolerance. The Fibonacci prediction represents the theoretical long-term value from the balance condition, while the IAU 2006-optimized value is calibrated to the currently observed obliquity rate.

Law 3: Inclination Balance

The angular-momentum-weighted inclination amplitudes cancel between the two phase groups, conserving the orientation of the invariable plane:

Physical motivation

The invariable plane is the fundamental reference plane of the solar system, perpendicular to the total angular momentum vector. For this plane to remain stable, the angular-momentum-weighted inclination oscillations must cancel between the two phase groups. If they didn’t, the total angular momentum vector would wobble — violating conservation of angular momentum.

Substituting the Fibonacci formula

With $\text{amp}_j = \psi / (d_j \times \sqrt{m_j})$ and $L_j = m_j \sqrt{a_j(1 - e_j^2)}$:

Since $\psi$ is a single universal constant, it cancels from both sides:

where $w_j = \sqrt{m_j \times a_j(1 - e_j^2)} / d_j$ is the structural weight for each planet.

Structural weights

Jupiter ($d = 5$) contributes the dominant 203° weight ($1.408 \times 10^{-2}$). Saturn ($d = 3$) alone carries the entire 23° contribution. The remaining six 203° planets collectively contribute $3.29 \times 10^{-3}$ to match the Saturn–Jupiter difference.

TNO contribution

The balance considers only the 8 major planets, which carry 99.994% of the solar system’s orbital angular momentum. Trans-Neptunian Objects (TNOs) contribute the remaining ~0.006%, tilting the invariable plane by approximately 1.25″ (Li, Xia & Zhou 2019 ). The 0.0002% residual imbalance is well within this TNO margin.

Law 4: The Eccentricity Constant

Within each mirror pair, the ratio of eccentricity to mean inclination satisfies two Fibonacci constraints that together determine all eight eccentricities from the inclinations alone.

For each planet, $R$ measures how it partitions its orbital deviation between eccentricity and inclination. Within each mirror pair, $R$ values satisfy two Fibonacci constraints — an $R^2$ sum and a product or ratio — providing two equations for two unknowns.

AMD partition ratio structure

Using J2000 inclinations:

Using mean inclinations (more stable than J2000 snapshot):

This connects to AMD theory (Laskar 1997): $C_k \approx \Lambda_k(e^2/2 + i^2/2)$ for small $e$ and $i$. The ratio $R^2 + 1 = (e^2 + i^2)/i^2$ determines how each planet partitions its angular momentum deficit between eccentricity and inclination.

Fibonacci pair constraints

The pair sums alone provide one equation for two unknowns. Crucially, pair products and ratios are also Fibonacci, providing a second equation that determines individual eccentricities:

Belt-adjacent pairs (Mars/Jupiter, Earth/Saturn) use a product constraint; outer pairs (Venus/Neptune, Mercury/Uranus) use a ratio constraint — reflecting the different coupling regimes across the asteroid belt.

Dynamical implication — communicating vessels: The product constraint $R_\text{Ea} \times R_\text{Sa} = 2$ implies that Earth’s and Saturn’s eccentricities cannot evolve independently. Since $R = e / i_\text{mean,rad}$, a decrease in Earth’s eccentricity (currently heading toward its ${\sim}0.0139$ minimum around 11,680 AD) reduces $R_\text{Ea}$, forcing $R_\text{Sa}$ to increase — and with it Saturn’s eccentricity. The coupling is bidirectional: Saturn has its own perihelion precession cycle ($-H/8 = 41{,}736$ years, retrograde), which drives its own eccentricity changes. Through the product constraint, those changes feed back into Earth’s eccentricity. The two planets behave like communicating vessels: what one loses in eccentricity, the other gains, mediated by the Fibonacci product constraint.

An observational clue supports this coupling: Earth’s eccentricity is currently decreasing while Saturn’s is increasing — exactly as the product constraint predicts. This bidirectional coupling may explain residual discrepancies between the model’s eccentricity curve and the formula-based predictions for Earth: the current model computes Earth’s eccentricity from its own 20,868-year perihelion cycle alone, without accounting for the additional perturbation from Saturn’s eccentricity evolution feeding back through the pair constraint. The physical coupling is already present in the precession formulas: Saturn is the only planet whose precession formula requires Earth’s time-varying obliquity and eccentricity as inputs (GROUP 15 terms; see Formulas).

Quantifying the coupling. If $R_\text{Ea} \times R_\text{Sa} = 2$ holds as a dynamical invariant and the mean inclinations are constant, then the product of eccentricities is fixed:

Using the model’s eccentricity range for Earth ($0.0139$–$0.0167$), the constraint predicts Saturn’s eccentricity range:

Conversely, Saturn’s own $H/8 = 41{,}736$-year perihelion cycle feeds back into Earth with an amplitude of $\sim\pm 0.0015$ — comparable to Earth’s own eccentricity amplitude (0.0014). The combined effect distorts the eccentricity curve from a pure sinusoid, potentially shifting the minimum depth and timing by several thousand years.

Impact on derived quantities. Through the regression formula $Y_\text{sid}(\text{days}) = f(e)$ (with coefficient $k_2 = 3{,}208$ s/unit eccentricity), Saturn’s $\pm 0.0015$ modulation translates to a Length of Day variation of $\sim\pm 0.1$ ms — detectable but modest compared to the $\pm 4$–$5$ ms total variation from Earth’s own cycle. The sidereal year in seconds is unaffected: it depends on Earth’s semi-major axis (Kepler’s third law), not on eccentricity, and secular perturbations preserve semi-major axes to first order.

Comparison with secular theory. Conventional Laplace-Lagrange theory gives Saturn’s dominant eccentricity eigenfrequency ($g_6 = 28.245$“/yr) a period of $\sim 46{,}000$ years — intriguingly close to $H/8 = 41{,}736$ years. Whether these represent the same physical period is testable. Note that the connection between Earth and Saturn is purely orbital: Saturn’s axial (spin-axis) precession has a period of $\sim 1.8$ Myr, driven by a spin-orbit resonance with Neptune’s nodal mode $s_8$ (Saillenfest et al. 2021), and is unrelated to the Fibonacci timescale hierarchy.

Whether $R_\text{Ea} \times R_\text{Sa} = 2$ holds as a dynamical invariant over the full perihelion cycle is a testable prediction of the model — as is the question of whether the same communicating-vessel behaviour holds for the other three mirror pairs (Mars/Jupiter, Venus/Neptune, Mercury/Uranus).

Predictions — zero free parameters

Solving each pair gives predicted $R$ values, and thus predicted eccentricities via $e_\text{pred} = R_\text{pred} \times i_\text{mean,rad}$:

Total |error|: 8.57%. Eccentricity balance (Law 5) cross-check: 99.93%.

Mars/Jupiter achieves extraordinary precision (0.49% pair error), and Saturn is predicted to within 0.05%. The inner pairs (Mercury/Uranus, Venus/Neptune) show 2–3% errors per planet, which trace to the mathematical amplification of the smaller $R$ value in each pair rather than any systematic inner/outer asymmetry.

This law partially resolves Open Question 1: while no universal closed-form eccentricity formula exists, all eight eccentricities are predicted from inclinations via Fibonacci pair constraints on the AMD partition ratio.

Law 5: Eccentricity Balance

The eccentricities satisfy an independent balance condition using the same Fibonacci divisors and phase groups:

where

Or equivalently, in terms of orbital period $T_j \propto a_j^{3/2}$:

Eccentricity balance weights

Saturn alone carries the entire 23° contribution. The 203° group is dominated by Jupiter ($7.928 \times 10^{-3}$), Uranus ($5.705 \times 10^{-3}$), and Neptune ($1.734 \times 10^{-3}$), with the four inner planets contributing only $6.6 \times 10^{-5}$ combined.

Comparing the two balance conditions

The half-power difference in Fibonacci divisor scaling ($1/d$ vs $1/\sqrt{d}$) and the shift from $\sqrt{a}$ to $a^{3/2}$ reflect that the eccentricity balance operates at a different order in secular perturbation theory.

The eccentricity balance is not a structural artifact — see Finding 3 for three independence tests.

Connection to AMD theory

The established AMD formula of Laskar (1997) couples eccentricity and inclination through a single conserved quantity: $C_k = m_k \sqrt{\mu a_k}(1 - \sqrt{1-e_k^2}\cos i_k)$. For small $e$ and $i$: $C_k \approx \Lambda_k(e^2/2 + i^2/2)$.

The eccentricity balance operates on linear $e$ rather than $e^2$, suggesting it captures a first-order secular constraint distinct from the quadratic AMD conservation. The physical mechanism producing this linear balance remains an open question.

Law 6: The Saturn-Jupiter-Earth Resonance

Saturn’s retrograde precession creates a closed beat-frequency loop with Jupiter and Earth — the physical mechanism linking the Fibonacci timescale (Law 1) to orbital structure (Laws 2–5).

Saturn is the only planet whose perihelion precesses retrograde (period $= -H/8$). When its retrograde motion interacts with Jupiter’s prograde motion ($H/5$), the resulting beat frequencies form a closed loop:

All three rows are cyclic permutations of $3 + 5 = 8$. Each planet’s period is the beat frequency of the other two:

               Saturn (H/8)
              ╱            ╲
          8−5=3          8−3=5
            ╱                ╲
    Earth (H/3) ──3+5=8── Jupiter (H/5)

This also connects to Law 1: combining Jupiter ($H/5$) and Saturn ($H/8$) produces axial precession: $F_5/H + F_6/H = F_7/H = 1/25{,}684$ (i.e., $5 + 8 = 13$).

Why this is a Fibonacci identity

The closed loop is an algebraic consequence of the Fibonacci addition rule applied to the period denominators:

Since $F_n/H + F_{n+1}/H = F_{n+2}/H$, the beat frequency rule is just the Fibonacci recurrence in disguise. The empirical content is that Earth, Jupiter, and Saturn have period denominators $b_E = 3$, $b_J = 5$, $b_S = 8$ — three consecutive Fibonacci numbers with Saturn as the sum.

Connection to the ψ-constant

The same three planets and their period denominators determine the inclination constant:

Law 6 thus reveals why Earth, Jupiter, and Saturn hold special roles throughout the Fibonacci Laws: they are the three planets locked into a Fibonacci resonance triangle, and their period denominators drive both the timescale hierarchy (Law 1) and the inclination constant (Law 2).

Findings

Findings are empirical observations and consequences that follow from the six laws.

Finding 1: Mirror Symmetry

As shown in the Planet Configuration table, each inner planet shares its $d$ with its outer counterpart across the asteroid belt: Mars ↔ Jupiter ($F_5$), Earth ↔ Saturn ($F_4$), Venus ↔ Neptune ($F_9$), Mercury ↔ Uranus ($F_8$). Earth–Saturn is the only pair with opposite phase groups (203° vs 23°). The divisors form two consecutive Fibonacci pairs: $(3, 5)$ for the belt-adjacent planets and $(21, 34)$ for the outer planets.

Finding 2: Configuration Uniqueness

The exhaustive search tested 7,558,272 configurations against four independent physical constraints:

Each filter independently eliminates most of the search space. The most restrictive three-filter intersection (Saturn-solo + balance + Laplace-Lagrange) yields only 7 configurations. Adding mirror symmetry leaves exactly one: Config #32 (0.0000132% of the search space).

Since Earth is locked at $d = 3$, only Scenario A (Saturn $d = 3$) can satisfy the Earth↔Saturn mirror symmetry. No other planet configuration produces the required mirror pairing while simultaneously satisfying all physical constraints.

The search space covers:

• Fibonacci divisors: $d \in \{1, 2, 3, 5, 8, 13, 21, 34, 55\}$ for Mercury, Venus, Mars, Uranus, Neptune
• Phase angles: 203.3195° or 23.3195° for each of the above
• 4 scenarios for Jupiter and Saturn (fixed per scenario): A: Ju=5, Sa=3 — B: Ju=8, Sa=5 — C: Ju=13, Sa=8 — D: Ju=21, Sa=13
• Earth: locked at $d = 3$, phase = 203.3195°
• Total: $9 \times 2 \times 9 \times 2 \times 9 \times 2 \times 9 \times 2 \times 9 \times 2 \times 4 = 7{,}558{,}272$ configurations

Finding 3: Eccentricity Balance Independence

The eccentricity balance (Law 5) is genuinely independent from the inclination balance (Law 3). Three tests confirm it is not a structural artifact:

1. Coefficient test: The weight formula without eccentricities ($\sqrt{m} \times a^{3/2} / \sqrt{d}$) gives only 74% balance — the actual eccentricity values contribute 26 percentage points of improvement
2. Random test: Substituting random eccentricities into the same weight formula gives 50–85% balance across 1,000 trials
3. Power test: The balance peaks sharply at $e^{1.0}$ (99.88%), dropping to 98.5% for $e^{0.9}$ and 98.4% for $e^{1.1}$, and to 91% for $e^2$ — linear eccentricity is special

Additionally, the ratio $v_j/w_j$ varies by a factor of ~150 across planets, confirming the two balance conditions are not proportional.

Finding 4: Saturn Eccentricity Prediction and Law Convergence

Since Saturn is the sole retrograde planet, the eccentricity balance directly predicts its eccentricity from the other seven:

Two structurally independent laws predict Saturn’s eccentricity — and bracket the observed value:

Law convergence: Laws 4 and 5 independently predict Saturn’s eccentricity to within 0.30% of each other. Law 4 derives it from the $R^2$ pair constraint ($R^2_E + R^2_{Sa} = 34/3$) — a purely structural Fibonacci relation using only the Earth-Saturn pair. Law 5 derives it from the global balance equation involving all eight planets simultaneously. Both bracket the J2000 observed value: Law 4 overshoots by 0.05%, Law 5 undershoots by 0.24%.

Why this is significant: Saturn’s eccentricity oscillates secularly between ~0.01 and ~0.09 (a factor-of-9 dynamic range). Two structurally different Fibonacci constraints — one using only the Earth-Saturn pair, the other using all eight planets — independently predict a value within 0.3% of each other and within 0.25% of J2000 across this range. The Fibonacci divisors were originally chosen to match precession periods (Law 1) and inclination balance (Law 3); the eccentricity predictions were never optimized for.

Overconstrained system: The 8 pair constraints (Law 4) plus Law 5 provide 9 equations for 8 unknowns (the eccentricities). The system is overconstrained by one equation — Saturn’s eccentricity is predicted by both laws independently. That two equations drawing on different subsets of planetary data converge on the same value confirms that the eccentricity balance is not an independent free parameter but an emergent consequence of the Fibonacci pair structure.

The 0.12% gap between the observed balance (99.88%) and perfect balance traces directly to Saturn’s J2000 eccentricity (0.05386) being 0.24% above the Law 5 equilibrium prediction (0.05373). The Law 5 prediction represents the eccentricity at which the balance would be exact.

Finding 5: Inner Planet Eccentricity Ladder

The mass-weighted eccentricities of the four inner planets form a Fibonacci ratio sequence:

where $\xi = e \times \sqrt{m}$. Consecutive ratios are $5/2$, $2$, $8/5$ — all converging toward the golden ratio $\varphi \approx 1.618$.

Earth’s base eccentricity (0.015321), independently determined from the 3D simulation, anchors the ladder at position $5/2$:

Finding 6: No Universal Eccentricity Constant

A comprehensive search across ~30 physical and Fibonacci parameters confirms that no combination $\xi \times f(\text{observables})$ yields a planet-universal constant analogous to the inclination $\psi$.

The search tested all power-law combinations of: semi-major axis, orbital period, mass, inclination precession period, perihelion precession period, AMD, Hill radius, orbital angular momentum, secular eigenfrequencies ($g$- and $s$-modes), Kozai integral, mean inclination, eccentricity half-angle $h(e)$, Fibonacci divisors ($d$, $b$), and cross-parameter ratios — across 1-, 2-, 3-, and 4-parameter combinations with fractional exponents.

Results (excluding tautologies that restate the inclination law):

The fundamental reason: the inclination divisors $d$ span only 11× (from 3 to 34), allowing a single $\psi$ to work. The eccentricity multipliers $k$ span 141× (from 1 to 141) with no smooth dependence on any present-day observable. The eccentricity ladder is a formation constraint imprinted during the dissipative epoch, not derivable from current orbital mechanics.

Deriving $\psi$ from the Holistic Year

The inclination constant $\psi$ was initially a calibration parameter — measured from the data but not derived from first principles. A systematic search reveals it can be expressed exactly in terms of $H$ and Fibonacci numbers:

This predicts $\psi = 3.302005 \times 10^{-3}$. Combined with the Fibonacci divisors and the balance condition, this single value determines all eight inclination amplitudes — and every predicted range falls within the Laplace-Lagrange secular theory bounds.

Physical interpretation: the E–J–S period connection

The Fibonacci indices 5 and 8 are the period denominators of Jupiter and Saturn (from Law 1: $H/5$ and $H/8$). The denominator factor 2 = $F_3 = F_{b_E}$ is Earth’s contribution:

where $b_E = 3$, $b_J = 5$, $b_S = 8$ are the period denominators of Earth, Jupiter, and Saturn, satisfying the Fibonacci addition $3 + 5 = 8$.

Saturn’s $F_b$ appears squared, reflecting its special role as the “sum planet” ($b_S = b_E + b_J$). These same three planets dominate the inclination balance condition — Jupiter and Saturn are the dominant contributors to the 203° and 23° phase groups respectively.

Equivalently, $\psi \times T_{V/N} = F_5 \times F_8^2 = 2205$, where $T_{V/N} = 2H$ is the period shared by Venus and Neptune. This connects the amplitude structure (how much planets tilt) to the period structure (how fast they oscillate).

The Master Ratio $R \approx 311$

Since $\psi$ is fixed by $H$ and Fibonacci numbers, the eccentricity scale is determined by a single ratio:

Since 311 is prime and not expressible as a Fibonacci product, the two structures remain connected but not unified. The eccentricity base unit $\xi_V$ requires one independent measurement — Earth’s base eccentricity $e_E = 0.015321$ from the 3D simulation.

However, 311 is not an arbitrary prime — it is a Fibonacci primitive root prime (OEIS A003147 ): the Fibonacci sequence mod 311 has the maximal possible period $\pi(311) = 310 = p - 1$, visiting every non-zero residue class before repeating. Only ~27% of primes have this property.

Two-layer explanation:

1. Why an FPR prime? In a system where coupling strengths decay as powers of $\varphi$ (as the Fibonacci quantum numbers require), an FPR prime ensures maximal compatibility — no “dead zones” in the coupling spectrum. Additionally, $\alpha(311) = 310 = \pi(311)$ (Type 1, maximally efficient entry point).

2. Why 311 specifically? Among the 33 FPR primes $\leq 600$, 311 is the closest to the formation-epoch value $R = 310.83$ (distance 0.17; next-nearest are 271 at distance 40 and 359 at distance 48). The fractional part $R \approx 311 - F_3/F_7 = 311 - 2/13$ at 0.007%.

The same number 311 appears as the optimal super-period in the TRAPPIST-1 system — see Exoplanet Tests.

The eccentricity ladder is a formation constraint ($p < 10^{-5}$ from Monte Carlo), not dynamically selected. This explains why $R$ cannot be derived from $H$ — the eccentricity scale was set by initial conditions during the dissipative formation epoch, making one free parameter ($e_E$, or equivalently $R \approx 311$) irreducible.

Physical Foundations

Why $\sqrt{m}$: the Angular Momentum Deficit connection

The quantity $\xi = e \times \sqrt{m}$ (or $\eta = i \times \sqrt{m}$) has a direct connection to the Angular Momentum Deficit (AMD), the standard dynamical quantity in celestial mechanics:

where $\xi_i = e_i \sqrt{m_i}$ and $\eta_i = i_i \sqrt{m_i}$ are exactly the mass-weighted quantities used in the Fibonacci laws. The $\sqrt{m}$ exponent is uniquely determined by the AMD decomposition — no other mass power produces this sum-of-squares form.

Empirical confirmation: Testing the $\psi$-constant ($d \times i \times m^\alpha$) across Venus, Earth, and Neptune for different mass exponents:

The exponent $\alpha = 0.50$ is optimal by a factor of $\sim 250\times$ over the next-best alternative.

Connection to KAM theory

The appearance of Fibonacci numbers in mass-weighted orbital parameters has a theoretical explanation through KAM (Kolmogorov-Arnold-Moser) theory:

1. KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962): Orbits survive if their frequency ratio satisfies a Diophantine condition — a quantitative measure of irrationality.

2. Golden ratio optimality (Hurwitz 1891): The golden ratio $\varphi = [1; 1, 1, 1, \ldots]$ has the slowest-converging continued fraction, making it the hardest irrational number to approximate by rationals. Fibonacci ratios $F_{n+1}/F_n$ are its convergents.

3. Greene-Mackay criterion (Greene 1979): The last invariant torus to break as perturbation increases has frequency ratio equal to $\varphi$. Morbidelli & Giorgilli (1995) proved golden-ratio KAM tori have superexponentially long stability times.

4. Natural selection over 4.6 Gyr: Orbits near low-order rational resonances are disrupted (Kirkwood gaps). Orbits near golden-ratio KAM tori survive. The surviving population is organized around Fibonacci ratios.

5. AMD variables: The quantities $\xi = e\sqrt{m}$ and $\eta = i\sqrt{m}$ are the natural action-like variables in the secular Hamiltonian. KAM stability conditions constrain these to Fibonacci ratios.

This chain — KAM stability → Diophantine condition → golden ratio → Fibonacci convergents → Fibonacci structure in $\xi$ and $\eta$ — provides the theoretical foundation for why Fibonacci numbers specifically appear.

Time independence

The Fibonacci laws use oscillation amplitudes (secular eigenmode properties), not instantaneous orbital elements. The $\psi$-constants and balance conditions use amplitudes $\eta = i_\text{amp} \times \sqrt{m}$ — properties of secular eigenmodes that are exactly time-independent by construction. The eccentricity ladder acts on base eccentricities that encode a formation-epoch constraint, preserved in the eigenmodes since the dissipative era.

Statistical Significance

Molchanov’s 1968 work was criticized by Backus (1969) for not proving statistical significance. To address this, a comprehensive significance analysis was performed using three independent null models and twelve test statistics covering all six laws, explicitly accounting for the look-elsewhere effect.

Methodology. Twelve test statistics were computed for the real Solar System and compared against random planetary systems:

1. Pairwise Fibonacci count — How many of the 56 pairwise $\xi$-ratios and $\eta$-ratios match Fibonacci ratios within 5%? Observed: 21/56.
2. Eccentricity ladder (Finding 5) — For the best 4-planet subset, how many $\xi$-ratios match Fibonacci ratios within 3%? Observed: 3 matches (mean error 0.49%).
3. $\psi$-constant (3-planet) — How small is the relative spread of $d \times i \times \sqrt{m}$ for the best 3-planet subset with Fibonacci weights? Observed: 16.97%.
4. Cross-parameter ratio — Does $(\Lambda_\text{ecc} + \Lambda_\text{amp}) / \Lambda_\text{mean}$ match an extended Fibonacci ratio? Observed: 5.43% error.
5. Inclination balance (Law 3) — With the model’s Fibonacci $d$-values and phase groups, how well do the structural weights $w_j$ cancel? Observed: 99.9998%.
6. Eccentricity balance (Law 5) — With the same $d$-values and phase groups, how well do the eccentricity weights $v_j$ cancel? Observed: 99.88%.
7. Saturn eccentricity prediction (Finding 4) — Law 4 and Law 5 independently predict Saturn’s $e$: 0.05389 (+0.05%) and 0.05373 (−0.24%), bracketing J2000. Law convergence: 0.30%.
8. $\psi$-constant, full 8-planet (Law 2) — How small is the spread of $d \times i_\text{J2000} \times \sqrt{m}$ across all 8 planets using J2000 inclinations? Observed: 234.7% spread.
9. Precession hierarchy (Law 1) — How many pairwise ratios of precession periods $H/F_n$ match Fibonacci ratios? Observed: 10 pairs.
10. $R^2$ partition (Law 4) — For each mirror pair, does $R_\text{inner}^2 + R_\text{outer}^2$ (where $R = e/i_\text{rad}$) match a Fibonacci ratio? Observed: 4/4 pairs match.
11. E–J–S resonance (Law 6) — Do the period denominators satisfy $b_E + b_J = b_S$? Observed: exact ($3 + 5 = 8$).
12. Mirror symmetry (Finding 1) — How many of the 4 inner–outer mirror pairs share the same $d$-value? Observed: 4/4 pairs match.

Tests 5–7 use the model’s fixed Fibonacci $d$-assignments and phase groups — zero look-elsewhere effect. Tests 1–4 involve implicit optimization (searching over subsets, weight assignments, or Fibonacci ratios), but this is accounted for automatically because each random trial computes the same optimized statistics. Tests 10–12 depend on planet identity (mirror-pair assignments, period fractions, and $d$-assignments), making them insensitive to the permutation test but testable via Monte Carlo.

Three null distributions were tested:

• Permutation (exhaustive, $8! = 40{,}320$ trials): same eccentricity and amplitude values, randomly reassigned to planets. For Tests 5–7: model $d$-values are kept, eccentricities are shuffled among planets. Tests 11–12 are identity-dependent and always match in this null ($p = 1$).
• Log-uniform Monte Carlo ($100{,}000$ trials): random eccentricities from $[0.005, 0.25]$, amplitudes from $[0.01°, 3.0°]$, means from $[0.1°, 10°]$ (log-uniform). Random $d$-values from $\{1, 2, 3, 5, 8, 13, 21, 34, 55\}$, random precession periods, random period fractions from $\{1, 2, 3, 5, 8, 13, 21, 34\}$.
• Uniform Monte Carlo ($100{,}000$ trials): same ranges, flat distribution. Same null model for Tests 5–12.

Masses are fixed at solar system values throughout (conservative).

Results (permutation: exhaustive $8!$; Monte Carlo: $100{,}000$ trials):

‡ Tests 10–12 depend on planet identity (mirror-pair assignments, period fractions, $d$-assignments), not on reshuffled values, so the permutation null cannot test them. The Monte Carlo nulls generate fully random assignments and give meaningful $p$-values.

Eight tests reach $p < 0.05$ across all applicable null distributions: pairwise Fibonacci count, inclination balance, eccentricity balance, Saturn prediction, the full $\psi$-constant, the $R^2$ partition, the E–J–S resonance, and mirror symmetry. The overall structure is highly significant (Fisher’s combined $p \leq 7.1 \times 10^{-14}$ in Monte Carlo, $p = 1.4 \times 10^{-5}$ in the conservative permutation test), robust across all three null distributions.

Tests 2–4 and 9 are individually non-significant, as expected: the eccentricity ladder involves only 4 inner planets (limited statistical power), Tests 3–4 suffer from circularity with model-derived amplitudes, and the precession hierarchy count is not discriminating because many random period sets also produce Fibonacci-like ratios among 10+ pairs.

Exoplanet Tests: TRAPPIST-1 and Kepler-90

TRAPPIST-1 period ratios — strong Fibonacci content

Five of six adjacent period ratios involve only Fibonacci numbers: 8:5, 5:3, 3:2, 3:2, 3:2 (only f:g $\approx$ 4:3 uses non-Fibonacci 4). This gives 83% Fibonacci content — identical to the Solar System’s 83%.

TRAPPIST-1 additive triad

The mass-weighted eccentricity $\xi = e\sqrt{m/M_\star}$ yields a Fibonacci additive triad: $3\xi_b + 5\xi_g = 8\xi_e$ at 0.34% error — using the same Fibonacci triple (3, 5, 8) that appears in the Solar System’s period denominators for Earth, Jupiter, and Saturn. Additional triads include $\xi_d + \xi_h = 2\xi_e$ (0.40%) and $\xi_e + \xi_f = 2\xi_b$ (0.51%).

TRAPPIST-1 super-period — the 311 connection

The optimal integer multiplier $N$ of $P_b$ where all 7 planet periods divide near-evenly (max deviation 0.12%) is $N = 311$, the same number as the Solar System master ratio $R = \psi/\xi_V \approx 310.9$.

Monte Carlo significance: Generating $10^5$ random Fibonacci-chain 7-planet systems, only 0.13% select $N = 311$. The probability that both the Solar System and TRAPPIST-1 independently select 311 is $\sim 2 \times 10^{-6}$ — highly significant.

TRAPPIST-1i candidate planet

A candidate 8th planet with $P \approx 28.7$ days has been identified (JWST 2025, tentative), in 3:2 resonance with planet h. If confirmed, TRAPPIST-1 would have $6/7 = 86\%$ Fibonacci-only period ratios.

TRAPPIST-1 limitations

Both systems show ~83% Fibonacci period ratios and both select 311. However, TRAPPIST-1’s eccentricities are extremely small (~0.002–0.01) with narrow dynamic range ($4\times$ vs Solar System’s $141\times$), making $\xi$-ladder tests impractical. Transit inclinations measure viewing geometry, not dynamical amplitudes. Deeper tests require systems with independently measured eccentricities and TTV-measured masses for 4+ planets.

Kepler-90 — limited by data

Only 2 of 8 planets have TTV-measured masses and eccentricities. Period ratios show 71% Fibonacci content (5/7 adjacent pairs).

Comparative summary

Relation to Prior Work and What’s Novel

Existing Fibonacci research in planetary science

Molchanov (1968) — Integer resonances in orbital frequencies

Molchanov proposed that planetary orbital frequencies satisfy simultaneous linear equations with small integer coefficients. His framework used general small integers — not specifically Fibonacci numbers — and dealt exclusively with orbital frequencies. He did not incorporate planetary masses, eccentricities, or inclinations.

Key difference: Molchanov found approximate integer relations among frequencies. Our laws use specifically Fibonacci numbers, applied to mass-weighted eccentricities and inclinations.

Aschwanden (2018) — Harmonic resonances in orbital spacing

Aschwanden identified five dominant harmonic ratios governing planet and moon orbit spacing, achieving ~2.5% accuracy on semi-major axis predictions. The analysis is purely kinematic — planetary mass, eccentricity, and inclination do not appear.

Key difference: Aschwanden analyzed distance/period ratios between consecutive pairs. Our laws analyze mass-weighted values of individual planets.

Pletser (2019) — Fibonacci prevalence in period ratios

Pletser found ~60% alignment of period ratios with Fibonacci fractions, with the tendency increasing for minor planets with smaller eccentricities and inclinations. He used eccentricity and inclination only as selection filters.

Key difference: Pletser did not analyze whether eccentricities or inclinations themselves form Fibonacci ratios. Mass-weighting is entirely absent. None of our six laws appear in his analysis.

Comparison table

What builds on established theory

Law 1 (Fibonacci Cycle Hierarchy) builds on the known precession periods of the planets. The individual periods are established results of celestial mechanics — what is new is recognizing them as a single Fibonacci-divided hierarchy rooted in a master timescale $H$.

Law 3 (Inclination Balance) is rooted in angular momentum conservation. The weight factor $\sqrt{m \cdot a(1-e^2)}$ is proportional to a planet’s orbital angular momentum $L$. The invariable plane is defined as the plane perpendicular to the total angular momentum vector, so inclination oscillations must balance around it — that is what makes it the invariable plane. The novel contribution is that dividing by a Fibonacci divisor $d$ preserves the balance to 99.9998%.

Phase angles (203.3195° and 23.3195°) originate from the $s_8$ eigenmode of Laplace-Lagrange secular perturbation theory, a framework established in classical celestial mechanics (18th–19th century). Saturn’s retrograde ascending node precession is also a known result from secular theory.

Law 5 (Eccentricity Balance) connects to Angular Momentum Deficit (AMD) conservation, a known conserved quantity in secular theory. The weight $\sqrt{m} \times a^{3/2} \times e / \sqrt{d}$ contains factors related to how AMD is partitioned among planets. However, the linear dependence on eccentricity (rather than quadratic, as in the AMD itself) and the $1/\sqrt{d}$ scaling distinguish it from the standard AMD formulation.

Law 6 (Saturn-Jupiter-Earth Resonance) describes beat frequency relationships between known precession periods. The individual periods are well-established — the closed-loop structure connecting them through Fibonacci addition is the novel observation.

What’s novel

1. Fibonacci division of a single timescale — all major precession periods equal $H/F(n)$ for successive Fibonacci numbers
2. Mass-weighted orbital parameters as Fibonacci variables — $\xi = X \times \sqrt{m}$ connects Fibonacci structure to physical properties, not just orbital timing
3. Fibonacci structure in eccentricities and inclinations — extends Fibonacci from one orbital element (period) to three
4. Pure Fibonacci divisors with exact mirror symmetry — all 8 divisors are Fibonacci numbers, forming 4 symmetric pairs across the asteroid belt
5. Zero-free-parameter inclination predictions — $\psi = F_5 \times F_8^2 / (2H)$ gives a single universal constant that predicts all 8 planets from $H$ alone
6. Zero-free-parameter eccentricity predictions — AMD partition ratios satisfy Fibonacci pair constraints, determining all 8 eccentricities from inclinations
7. Configuration uniqueness — the mirror-symmetric solution is the unique configuration satisfying all 4 physical constraints out of 7.5 million tested
8. Eccentricity balance — an independent balance condition on eccentricities using the same divisors
9. Saturn eccentricity prediction — a direct consequence of Saturn’s unique retrograde status
10. Closed resonance loop — Saturn-Jupiter-Earth beat frequencies form a triangle that is an algebraic consequence of Fibonacci addition

Assessment

The balance conditions (Laws 3 and 5) combine known conservation principles with a novel Fibonacci structure that modulates the planetary weights. The conservation laws guarantee that inclination and eccentricity oscillations balance around the invariable plane — but they do not predict that integer Fibonacci divisors should preserve that balance to such high precision. Law 2 (Inclination Amplitude quantization) and Law 4 (Eccentricity Constant) are the most genuinely novel claims — no existing theory predicts that $d \times \text{amplitude} \times \sqrt{m}$ should be constant across all planets, or that AMD partition ratios within mirror pairs should satisfy specific Fibonacci fractions.

The key unresolved question is why Fibonacci numbers work: do they encode something about the secular eigenmode structure (real physics), or is the Fibonacci restriction a coincidence made possible by having enough number choices? The mirror symmetry and the triple-constraint uniqueness of Config #32 argue against coincidence, but a theoretical derivation from first principles — or a successful prediction for an independent system such as exoplanetary or satellite systems — would be needed to settle the question definitively.

Open Questions

1. R² Fibonacci targets — Partially resolved. Law 4 uses eight specific Fibonacci fractions (four $R^2$ sums and four products/ratios) to predict all eccentricities. Exhaustive search finds no rule generating these targets from pair properties — the Fibonacci indices appear to be irreducible data. Whether they can be derived from a single structural principle remains open.

2. Physical derivation of eccentricity balance — The inclination balance follows from angular momentum conservation. What conservation law or secular perturbation mechanism produces the eccentricity balance? The linear (rather than quadratic) dependence on $e$ distinguishes it from AMD conservation.

3. Universality — Do the balance conditions hold for other planetary systems, or are they specific to the solar system’s Fibonacci divisor structure? TRAPPIST-1 shows promising Fibonacci content (83% period ratios, 311 super-period) but lacks the mass and eccentricity data needed to test the balance conditions.

4. The master ratio $R \approx 311$ — Why does the formation-epoch eccentricity scale land near a Fibonacci primitive root prime? Whether formation generically converges toward FPR primes (via KAM attractor dynamics) remains open.