Fibonacci Laws of Planetary Motion

The Holistic Universe Model assigns each planet a precession period that is a Fibonacci fraction of the Holistic-Year H = 333,888 years (see Precession). This page documents three independent Fibonacci laws that extend beyond periods to connect base eccentricities and inclination amplitudes across all eight planets.

The central quantity is the mass-weighted parameter:

where $X$ is a planet’s base eccentricity or inclination oscillation amplitude, and $m$ is its mass in solar units (JPL DE440). The Fibonacci laws constrain how $\xi$ values relate across planets.

Input Data

All masses are from JPL DE440. J2000 eccentricities are from the NASA Planetary Fact Sheet. Inclination amplitudes are derived from the 3D model: each planet’s amplitude was calibrated so that the inclination change across its oscillation period matches the observed J2000 inclination rate of change. They are not directly measured values but model-derived quantities constrained by observation. See Invariable Plane and Ascending Node Calibration for the full derivation.

*Predicted by Fibonacci laws (see below). †J2000 ≈ base (near oscillation midpoint). ‡Saturn’s nodal precession is retrograde.

Only Earth’s base eccentricity (0.015321) is independently determined from the 3D simulation. All other base values are either predicted by the laws below or approximated from J2000 values.

Period Assignments

Every planet’s inclination oscillation period is a Fibonacci fraction of H:

where $a$ and $b$ are Fibonacci numbers (or sums of Fibonacci numbers for Mercury).

Law 1: Inner Planet Eccentricity Ladder

Statement: The mass-weighted eccentricity $\xi = e_{\text{base}} \times \sqrt{m}$ for the four inner planets forms a Fibonacci ladder:

Earth’s base eccentricity (0.015321), independently determined from the 3D simulation, anchors the ladder at position $5/2$. All other inner planet base eccentricities are predicted:

where $k \in \{1, 5, 8\}$ is the planet’s Fibonacci multiplier in the ladder.

Verification

Interpretation

• Earth anchors the ladder — its base eccentricity (0.015321) is independently determined from the 3D simulation
• Venus and Mars (both near their oscillation midpoints, J2000 ≈ base) match at sub-0.2% — essentially exact
• Mercury is off by 1.41%, suggesting its base eccentricity ($\approx 0.2085$) is above J2000 — consistent with Mercury currently being below its oscillation midpoint
• The multipliers $\{1, 5/2, 5, 8\}$ are all Fibonacci numbers or ratios of consecutive Fibonacci numbers
• Consecutive ratios: 5/2, 2, 8/5 — all Fibonacci ratios, converging toward $\varphi = 1.618...$

Additional inner planet eccentricity relationships

From the ladder, direct pairwise identities follow:

The Mars–Venus identity at 0.08% is the tightest Fibonacci relationship found in the entire eccentricity dataset.

Law 2: Inclination $\psi$-Constant

Statement: The Fibonacci-weighted inclination amplitude $\psi = d \times i \times \sqrt{m}$ is constant across Venus, Earth, Mars, and Neptune:

where $d = \{2, 3, 13/5, 5\}$ are all Fibonacci numbers or ratios of Fibonacci numbers.

Verification

Mean $\psi = 3.2995 \times 10^{-3}$. Spread across all four planets: 0.38%.

Pairwise Fibonacci ratios

The equivalent pairwise statement — the ratio $i \times \sqrt{m}$ between any two of these planets is a Fibonacci ratio:

Mars connects to the $\psi$-constant through its weight $d = 13/5$ rather than through a direct pairwise Fibonacci ratio with the other Law 2 planets.

Predictions

Calibrating from Earth ($\psi = 3 \times 0.634 \times \sqrt{m_E}$), the law predicts:

Multiple $\psi$-levels

The extended search revealed that the $\psi$-constant is not unique — there are multiple levels, and their ratio is itself a Fibonacci ratio:

The ratio $3/2$ is a Fibonacci ratio (consecutive Fibonacci numbers). Venus serves as the bridge: it participates in both levels with different weights ($d=2$ in $\psi_1$, $d=3$ in $\psi_2$), and the weight ratio $3/2$ directly gives the $\psi$-level ratio.

Uranus provides independent confirmation — it was not used in determining $\psi_1$, yet $8\eta_U$ falls within 0.72% of $\psi_2 = (3/2)\psi_1$.

Law 3: Giant Planet Fibonacci Triad (3 + 5 = 8)

Statement: For the Earth–Jupiter–Saturn triad, the Fibonacci additive identity holds:

where $X$ is the base eccentricity or inclination amplitude, and $\{3, 5, 8\}$ are consecutive Fibonacci numbers satisfying $3 + 5 = 8$.

Verification

The eccentricity error is larger because J2000 values are snapshots, not base (oscillation midpoint) values. Jupiter and Saturn’s base eccentricities are not yet independently determined — the 3.72% gap indicates their bases differ from J2000 by a few percent.

Simplified form (Jupiter–Saturn dominance)

Earth’s contribution to the eccentricity LHS is only ~1% (because $m_E \ll m_J$). Dropping it gives the dominant relationship:

This directly predicts the ratio of Jupiter and Saturn base eccentricities from their masses alone:

The J2000 ratio is 0.8985 — off by 2.6%, which should narrow once base eccentricities replace J2000 snapshots.

Physical interpretation: weights as period divisors

The Fibonacci weights 8 in Law 3 are not arbitrary — they are the same numbers that define each planet’s precession period as a fraction of the Holistic-Year:

This correspondence is exact: the weight assigned to each planet in Law 3 equals the denominator of its period fraction. This allows Law 3 to be rewritten in a physically revealing form.

Since $T = H/d$ where $d$ is the weight, multiplying by $d$ is equivalent to dividing by $T$ and multiplying by $H$. The Law 3 identity:

becomes:

The quantity $X\sqrt{m}/T$ is a mass-weighted amplitude rate — the rate at which mass-weighted eccentricity or inclination oscillates per unit time. Law 3 therefore states that the amplitude rates of Earth and Jupiter sum to equal Saturn’s amplitude rate.

Why Earth belongs in Law 3

The amplitude-rate form reveals why Earth — despite its small mass compared to Jupiter and Saturn — cannot be dropped from the inclination version:

• For eccentricity: Earth’s amplitude rate is ~1% of the total, because $e_E \sqrt{m_E}$ is small. Jupiter dominates, and the simplified $5 \times e_J \sqrt{m_J} \approx 8 \times e_S \sqrt{m_S}$ holds to 2.6%.
• For inclination: Earth’s amplitude rate contributes ~15% of the LHS. This is because Earth’s inclination amplitude (±0.634°) is much larger relative to Jupiter’s (±0.123°) than its eccentricity is relative to Jupiter’s.

The physical reason Earth’s inclination amplitude is so large is that Earth’s orbital plane is being driven by Jupiter and Saturn. Jupiter dominates ecliptic precession — it causes the ascending node of Earth’s orbit to precess around the invariable plane. Saturn dominates the obliquity cycle — it modulates the interaction between Earth’s axial and inclination tilts (see Obliquity & Inclination).

Earth’s inclination amplitude is not an independent degree of freedom but a gravitational response to forcing by the two giant planets. The amplitude-rate form of Law 3 makes this explicit: Earth’s response rate plus Jupiter’s own rate equals Saturn’s rate. The system is closed — the three planets form a coupled oscillator where the total mass-weighted amplitude rate is conserved.

Additional Fibonacci Relationships

Beyond the three main laws, the systematic search found additional Fibonacci identities connecting remaining planets:

Inclination identities

The Mars–Jupiter pair constitutes an independent $\psi$-level ($\psi_3 = \eta_J = 3\eta_M \approx 3.81 \times 10^{-3}$, spread 0.44%) that is not a Fibonacci multiple of $\psi_1$ or $\psi_2$.

Eccentricity identities

The eccentricity data also reveals $\psi$-constant structure:

• Inner quartet: $(8/3)\xi_{Me} = 13\xi_V = 8\xi_E = (13/3)\xi_M$ — spread 1.38%. All four inner planets share a single mass-weighted eccentricity constant.
• Outer triplet: $(1/3)\xi_S = \xi_U = 5\xi_N$ — spread 2.82%. Three outer planets (excluding Jupiter) form a second constant.

Fibonacci connection network

A graph-theoretic analysis reveals the full structure: two planets are connected if their $\eta$ or $\xi$ ratio matches a Fibonacci ratio within 5%. The maximal cliques (fully connected subgroups) are:

Inclination network (at 5% threshold):

• 4-planet clique: Venus, Earth, Saturn, Neptune
• 3-planet clique: Mars, Jupiter, Uranus
• Mercury is isolated (no inclination connections)

Eccentricity network (at 5% threshold):

• 4-planet clique: Mercury, Venus, Earth, Mars
• 3-planet clique: Jupiter, Saturn, Uranus

The Solar System thus splits into two complementary groups: the inner planets are connected by eccentricity, and the outer planets by inclination, with Venus, Earth, and Saturn bridging both networks.

Coverage

These relationships, combined with the three main laws, connect all 8 planets through Fibonacci identities in at least one parameter. Only Mercury’s inclination amplitude (±0.012°) lacks a clean Fibonacci relationship — its very small amplitude and mass make it an outlier.

Predicted Base Eccentricities

Using the three laws, base eccentricities can be predicted for all planets:

Predicted Inclination Amplitudes

All predictions are within 1% of model values. The ”$i$ (model)” column contains inclination amplitudes derived from the 3D model, calibrated to reproduce the observed J2000 inclination rate of change for each planet — not direct observations.

Summary

Three independent Fibonacci laws connect the orbital parameters of all eight planets:

A deeper structural analysis reveals that the weights $d$ in these laws decompose as $d = b \times F$ where $b$ is the oscillation period denominator and $F$ is a second Fibonacci ratio — the coupling quantum number. The Fibonacci index of $F$ forms a mirror-symmetric sequence across the asteroid belt, governed by the selection rule $\text{idx}(F) = 2k - 4$ where $k$ is the ordinal distance from the belt. All $F$ values obey an even-index constraint and approximate even powers of the golden ratio ($F \approx \varphi^{2n}$). This symmetry is structural, not gravitational — mirror pairs are not dominated by each other’s perturbations. See Two Fibonacci Quantum Numbers for the full analysis.

Together with the additional pairwise identities (Mars–Jupiter, Venus–Uranus, Uranus–Neptune) and the multiple $\psi$-levels ($\psi_2/\psi_1 = 3/2$), these laws constrain base eccentricities and inclination amplitudes for all 8 planets through the quantity $X \times \sqrt{m}$ with Fibonacci weights. The Solar System’s Fibonacci structure splits into two complementary networks: inner planets connected by eccentricity, outer planets connected by inclination, with Venus, Earth, and Saturn bridging both.

Structural Analysis: Two Fibonacci Quantum Numbers

The Fibonacci weights $d$ in Laws 2 and 3 are not arbitrary assignments — they decompose into a product of two independent Fibonacci quantities, each with a clear physical origin.

The decomposition $d = b \times F$

Every planet’s inclination oscillation period is $T = H \times a/b$ where $b$ is the period denominator (a Fibonacci number). The key discovery is that the Fibonacci weight $d$ always decomposes as:

where $F = d/b$ is itself an exact ratio of Fibonacci numbers. This holds for all seven non-Mercury planets with zero error — it is an algebraic identity, not an approximation.

Each planet therefore carries two independent Fibonacci quantum numbers:

• $b$ — the period quantum number, setting the oscillation timescale ($T = H \times a/b$)
• $F$ — the coupling quantum number, setting the amplitude coupling strength

Complete quantum number table

Every entry in the $F$ column is an exact Fibonacci ratio: numerator and denominator are both Fibonacci numbers ($1, 2, 3, 5, 8$). This was verified algebraically — the decomposition is not a numerical fit.

Mirror symmetry across the asteroid belt

The Fibonacci index of $F$ (defined as $\text{idx}(F) = \text{idx}(\text{numerator}) - \text{idx}(\text{denominator})$ using the Fibonacci sequence position $F_1=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13$) reveals a remarkable mirror pattern:

Reading from the asteroid belt outward in both directions:

Mars and Jupiter — the two planets flanking the asteroid belt — share the same Fibonacci quantum number $F = 1/5$. Earth and Saturn, each one step further from the belt, share $F = 1$. The pattern continues outward: Venus at $\text{idx}(F) = +2$ is mirrored by Uranus, and Neptune at $\text{idx}(F) = +4$ extends the outer sequence with no inner counterpart.

The even-index constraint

A deeper investigation reveals that the mirror pattern is not arbitrary — it obeys a strict parity rule: all coupling quantum numbers $F$ have even Fibonacci index. Writing $F = F_n / F_m$ where $F_n$ and $F_m$ are Fibonacci numbers at sequence positions $n$ and $m$, the constraint is:

This means $n$ and $m$ always have the same parity — both odd-indexed or both even-indexed. The Fibonacci numbers split into two subsequences by index parity:

Each $\psi$-group draws both numerator and denominator of $F$ from the same subsequence. This constraint excludes exactly half of all possible Fibonacci ratios — values like $F = 3, 1/3, 8, 1/8$ (which would require mixing odd and even subsequences) never appear as coupling constants.

Golden ratio power structure

The even-index constraint has a striking consequence. The odd-indexed Fibonacci numbers $\{1, 2, 5, 13, 34, \ldots\}$ satisfy:

This means every coupling quantum number approximates an even power of the golden ratio:

The approximation is only rough for small Fibonacci numbers, but the ordering is exact: $F$ increases monotonically as $\varphi^{\text{idx}(F)}$. More precisely, $F = F_{2k+1} / F_{2j+1}$ — a ratio of odd-indexed Fibonacci numbers — which converges to $\varphi^{2(k-j)}$ for large indices.

The step pattern $-4, 0, +2, +4$ in Fibonacci index units therefore corresponds to $-2, 0, +1, +2$ in golden-ratio-power units.

Selection rule for the coupling quantum number

The coupling quantum number $F$ follows a simple formula based on each planet’s ordinal distance $k$ from the asteroid belt (counting outward in both directions):

The formula is exact for all seven non-Mercury planets. It reveals a two-phase structure:

• Phase 1 — Denominator collapse ($k = 1 \to 2$, step $+4$): The coupling ratio $F$ transitions from $1/5 = F_1/F_5$ to $1/1 = F_1/F_1$. The denominator drops from $F_5$ to $F_1$ while the numerator stays fixed. This is the transition from the weakly-coupled belt boundary to the fully-coupled middle.

• Phase 2 — Numerator ascent ($k = 2 \to 3 \to 4$, steps $+2$ each): The coupling ratio climbs from $1$ to $2$ to $5$ — the numerator advances through $F_1 \to F_3 \to F_5$ while the denominator remains $F_1$. Each step is one position in the odd-indexed Fibonacci subsequence.

The asteroid belt acts as a structural node: coupling strength increases as $\sim \varphi^{2k}$ with distance from this node, analogous to exponential decay from a boundary in wave mechanics.

Nature of the mirror symmetry

The mirror symmetry is a structural property of the Fibonacci quantum number assignment — it does not arise from direct gravitational interaction between the paired planets.

Evidence against a gravitational origin:

• Gravitational dominance: In Laplace-Lagrange secular perturbation theory, Jupiter dominates the precession of all inner planets (Venus, Earth, and Mars), not just its mirror partner Mars. Saturn’s mirror partner Earth is predominantly driven by Jupiter, not Saturn.
• Orbital distance: The geometric mean distance of mirror pairs does not cluster at the asteroid belt. The pairs are not symmetric about the belt in either linear or logarithmic distance space.
• Orbital resonance: Mirror pairs (Mars/Jupiter, Earth/Saturn, Venus/Uranus) are not in known mean-motion resonances with each other.

The mirror symmetry instead reflects the mathematical structure of the Fibonacci coupling assignment: the same selection rule $\text{idx}(F) = 2k - 4$ operates independently on both sides of the asteroid belt, producing matching quantum numbers at each ordinal level.

Period-sharing partner constraints

Planets that share the same oscillation period form constrained pairs. For Venus and Neptune (both $T = 2H$, both in $\psi_1$):

This means $2 \times i_V \times \sqrt{m_V} = 5 \times i_N \times \sqrt{m_N}$ — the most precise Fibonacci identity in the dataset. Since both planets share the same $b = 1$, the constraint reduces to: the ratio of their $F$ values exactly compensates the ratio of their mass-weighted amplitudes.

For Earth and Uranus (both $T = H/3$, different $\psi$-groups):

The ratio of their $F \times \eta$ products equals the $\psi$-group ratio $3/2$ — linking the mirror symmetry structure to the $\psi$-level hierarchy.

The Fibonacci Quantization Principle

Combining the two quantum numbers with the $\psi$-constant structure gives the complete master equation:

where:

• $\psi_g$ is the group constant ($\psi_1$, $\psi_2 = \tfrac{3}{2}\psi_1$, or $\psi_3$)
• $b$ is the period denominator (a Fibonacci number, from $T = H \times a/b$)
• $F$ is the coupling quantum number (a Fibonacci ratio, from the mirror symmetry)
• $i \times \sqrt{m}$ is the mass-weighted inclination amplitude

This parallels quantum mechanics: just as atomic energy levels are characterized by quantum numbers ($n$, $l$, $m_l$) that determine discrete allowed states, each planet’s inclination dynamics is characterized by Fibonacci quantum numbers ($b$, $F$, $g$) that determine its allowed coupling to the solar system’s collective oscillation structure.

The analogy to Balmer’s spectral formula is apt: Balmer discovered that hydrogen wavelengths follow $\lambda = B \cdot n^2/(n^2 - 4)$ before the underlying quantum theory was known. Similarly, the Fibonacci quantization principle describes an observed pattern — the selection rule that determines which $F$ each planet receives remains an open question for future theoretical work.

Relation to Prior Work

Fibonacci numbers and the golden ratio have been studied in natural systems for centuries. The following comparison shows how the three laws documented here relate to — and differ from — the existing literature on Fibonacci patterns in planetary science and beyond.

Existing Fibonacci research in planetary science

Molchanov (1968) — Integer resonances in orbital frequencies

Molchanov proposed that all nine planetary orbital frequencies satisfy eight simultaneous linear equations with small integer coefficients (1, 2, 3, 5, 6, 7). His framework used general small integers — not specifically Fibonacci numbers — and dealt exclusively with orbital frequencies (mean motions). He did not incorporate planetary masses, eccentricities, or inclinations. His work was criticized by Backus (1969) for lacking statistical significance, though Molchanov defended the results with probability estimates of ~$10^{-10}$ for chance occurrence. The debate was never fully resolved, but the concept of dissipative evolution toward resonance remains influential.

Key difference: Molchanov found approximate integer relations among frequencies. Our laws use specifically Fibonacci numbers/ratios, applied to mass-weighted eccentricities and inclinations — entirely different parameters and a different number-theoretic structure.

Aschwanden (2018) — Harmonic resonances in orbital spacing

Aschwanden identified five dominant harmonic ratios — (3:2), (5:3), (2:1), (5:2), (3:1) — governing the spacing between consecutive planet and moon orbits. His model achieves ~2.5% accuracy on semi-major axis predictions, significantly better than the Titius-Bode law. The analysis is purely kinematic: only orbital periods and semi-major axes are used. Planetary mass, eccentricity, and inclination do not appear in his model. Aschwanden himself did not frame his results in terms of Fibonacci numbers, though subsequent work (Aschwanden & Scholkmann, 2017) noted that the dominant ratios involve numbers that happen to be Fibonacci.

Key difference: Aschwanden analyzed distance/period ratios between consecutive pairs. Our laws analyze absolute values of a mass-weighted quantity for individual planets. His model says nothing about what determines a planet’s eccentricity or how eccentricity relates to mass.

Pletser (2019) — Fibonacci prevalence in period ratios

Pletser tested whether orbital period ratios between successive bodies preferentially align with irreducible fractions formed from Fibonacci numbers (1 through 8). He found ~60% alignment vs ~40% expected by chance, and showed this tendency increases for minor planets with smaller eccentricities and inclinations. His analysis covers the Solar System, satellite systems of giant planets, and exoplanetary systems.

Key difference: Pletser used eccentricity and inclination only as selection filters (choosing subsets of asteroids with more circular/coplanar orbits). He did not analyze whether eccentricities or inclinations themselves form Fibonacci ratios. Mass-weighting is entirely absent. His work represents the state of the art as of 2019 — and none of our three laws appear in his analysis.

Fibonacci in other natural systems

Yamagishi & Shimabukuro (2008) — Fibonacci in DNA nucleotide frequencies

This paper showed that nucleotide frequencies in the human genome can be approximately described using Fibonacci/golden-ratio relationships within an optimization framework. The connection to our work is conceptual: both cases invoke optimization principles where nature appears to select solutions related to the golden ratio. However, the physical mechanisms are entirely different (mutational biology vs gravitational dynamics), and the paper’s claims about universality remain contested (Idriss & El Kossifi, 2018).

Prusinkiewicz & Lindenmayer (1990) — Fibonacci in phyllotaxis

This foundational text documents how plant organ arrangements (leaves, seeds, florets) follow Fibonacci parastichy numbers, driven by the golden divergence angle of 137.5°. The underlying mechanism — optimal packing through inhibitory field models — produces Fibonacci patterns through self-organization. The conceptual parallel to our work is that in both plants and planetary systems, an optimization/self-organization process produces Fibonacci-governed structures, though the physical mechanisms differ completely (chemical inhibition fields vs gravitational resonance).

Comparison table

What’s Novel

The three Fibonacci laws introduce concepts absent from the published literature:

1. Mass-weighted orbital parameters as Fibonacci variables

The quantity $\xi = X \times \sqrt{m}$ — combining a planet’s eccentricity (or inclination amplitude) with the square root of its mass — has not been used as a Fibonacci variable in any prior work. All existing studies (Molchanov, Aschwanden, Pletser) work with purely kinematic quantities (periods, distances, frequencies). Introducing mass connects Fibonacci structure to a planet’s physical properties, not just its orbital timing.

2. Fibonacci structure in eccentricities and inclinations

Prior work has found Fibonacci patterns only in orbital period ratios and distance spacings. Nobody has previously shown that the eccentricities or inclination amplitudes of planets — the shapes and tilts of their orbits — are connected through Fibonacci ratios. This extends Fibonacci’s reach from one orbital element (semi-major axis / period) to three (eccentricity, inclination, and period).

3. Fibonacci ladders: ordered sequences across multiple planets

Law 1 shows that the four inner planets form an ordered Fibonacci ladder 8 in mass-weighted eccentricity. This is not a pairwise ratio between two bodies (as in Pletser) but a systematic sequence across four planets, where consecutive multiplier ratios (5/2, 2, 8/5) themselves converge toward $\varphi$.

4. Fibonacci additive identities in orbital mechanics

Law 3 uses the Fibonacci additive property ($3 + 5 = 8$) applied to mass-weighted parameters — the LHS is a sum of two planet contributions equaling a third. All prior work uses only multiplicative ratios ($p/q$). The additive form connects three planets simultaneously rather than in pairs.

5. Cross-element consistency

The same Fibonacci structure ($X \times \sqrt{m}$ with Fibonacci weights) governs both eccentricity and inclination in the Earth–Jupiter–Saturn triad (Law 3). This dual consistency — the same mathematical form working for two independent orbital elements — has no precedent in the literature.

6. Predictive power for base orbital elements

The laws predict base eccentricities and inclination amplitudes for 7 of 8 planets to within 1%. No prior Fibonacci analysis of planetary systems has produced quantitative predictions for eccentricities or inclinations.

7. Two Fibonacci quantum numbers and mirror symmetry

The decomposition $d = b \times F$ reveals that each planet carries two independent Fibonacci quantum numbers — a period number $b$ and a coupling number $F$. The Fibonacci index of $F$ forms a mirror-symmetric sequence across the asteroid belt, connecting inner and outer planets through an unexpected structural symmetry. No prior work has identified such a quantum-number-like classification of planets using Fibonacci numbers.

Directions for Future Research

This section collects open questions and promising research directions. Items are organized by topic to allow easy updating as follow-up work is completed.

Open questions from current laws

• Mercury’s inclination: No clean Fibonacci identity found for Mercury’s very small inclination amplitude (±0.012°). Its low mass and tiny amplitude make $\xi$ very small — a higher-order Fibonacci ratio (involving 13 or 21) or a composite approach similar to Mercury’s period denominator ($11 = 3 + 8$) may be required.
• Mercury’s period and the Fibonacci–Lucas boundary: Mercury’s period denominator $b = 11$ is not a Fibonacci number but is a Lucas number ($L_5 = 11$). Since $L_n = F_{n-1} + F_{n+1}$, this connects to $11 = 3 + 8 = F_4 + F_6$. Understanding whether Mercury belongs to a Lucas-number extension of the Fibonacci quantum structure could explain why it is excluded from the two-quantum-number mirror symmetry.
• Jupiter–Saturn absolute base eccentricities: The ratio $e_J/e_S = 0.876$ is fixed by the masses; absolute base eccentricities require one additional constraint. Secular perturbation theory proper eccentricities or Laskar’s La2004/La2010 long-term numerical solutions could provide this.

Open questions from the two-quantum-number structure

• Selection rule for $F$ — partially resolved: The empirical formula $\text{idx}(F) = 2k - 4$ (for $k \geq 2$, with $-4$ for $k = 1$) correctly predicts the Fibonacci index of $F$ for all 7 planets from their ordinal distance $k$ from the asteroid belt. The remaining open question is why this formula holds — what physical mechanism enforces the even-index constraint and the $2k - 4$ dependence? A derivation from KAM theory, secular perturbation theory, or angular momentum conservation would complete the analogy to quantum mechanical selection rules.
• Why Mars and Jupiter share $F = 1/5$ — explained by selection rule: Both are at $k = 1$ (belt-adjacent), so the formula assigns $\text{idx}(F) = -4$ to both. The deeper question remains: why does the belt-adjacent level have a $+4$ gap to the next level, rather than the $+2$ steps seen elsewhere? The two-phase structure (denominator collapse then numerator ascent) suggests the belt acts as a structural boundary where the coupling mechanism changes qualitatively.
• Neptune’s missing inner partner: The mirror symmetry at $\text{idx}(F) = +4$ has Neptune but no inner planet. Mercury ($a = 0.39$ AU) occupies the corresponding orbital position, but its $b = 11$ (Lucas, not Fibonacci) excludes it from the structure. Is this connected to Mercury’s anomalous dynamical properties (high eccentricity, near-resonant period ratios with Venus)?
• Eccentricity quantum numbers: The $d = b \times F$ decomposition works cleanly for inclination (all 7 planets) but only partially for eccentricity (outer planets J, S, U have exact Fibonacci $F_\text{ecc}$; inner planets do not). Understanding this asymmetry could reveal why inclination and eccentricity are organized differently.

Physical origin of $\sqrt{m}$

The quantity $e \times \sqrt{m}$ (or $i \times \sqrt{m}$) may relate to established dynamical quantities:

• Angular momentum deficit (AMD): AMD per planet is $\text{AMD}_i \propto m_i \sqrt{a_i} (1 - \sqrt{1-e_i^2}) \approx m_i \sqrt{a_i} \cdot e_i^2/2$ for small $e$. Our $\xi = e \sqrt{m}$ is not exactly AMD but shares the mass-eccentricity coupling. The relationship between $\xi$ and AMD deserves investigation.
• Delaunay action variables: In Hamiltonian celestial mechanics, the Delaunay variables involve combinations of mass, semi-major axis, and eccentricity. Our quantity might correspond to a simplified action variable.
• Why $\sqrt{m}$ specifically? The square root of mass (rather than $m$, $m^{1/3}$, or $m^{2/3}$) may connect to energy equipartition or virial theorem arguments. Testing other mass exponents systematically could clarify whether $\sqrt{m}$ is uniquely optimal or part of a family of viable exponents.

Connection to KAM theory

Pletser (2019) connected Fibonacci period ratios to KAM (Kolmogorov-Arnold-Moser) theory: the golden ratio is the “most irrational” number, meaning orbits near golden-ratio resonances are the most stable against perturbations. This theoretical framework could explain why Fibonacci numbers specifically — not just any small integers — appear in the mass-weighted parameters. Investigating whether our laws can be derived from KAM stability conditions would provide a rigorous physical foundation.

Statistical significance

Molchanov’s 1968 work was criticized by Backus (1969) for not proving that the observed resonances were statistically significant compared to random numbers. The same critique applies here: with 20 Fibonacci ratios and 28 planet pairs, some matches will occur by chance. To address this, we performed a comprehensive significance analysis using three independent null models and four test statistics, explicitly accounting for the look-elsewhere effect.

Methodology. Four 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 (28 eccentricity + 28 inclination) fall within 5% of any Fibonacci ratio?
2. Eccentricity ladder — For the best 4-planet subset, how many $\xi$-ratios match Fibonacci ratios (within 3%)?
3. $\psi$-constant spread — For the best 3-planet subset with Fibonacci weights $d \in \{1,2,3,5,8\}$, how small is the relative spread of $d \times i \times \sqrt{m}$?
4. Additive triad error — For the best 3-planet subset and Fibonacci triple ($a + b = c$), how small is the relative error of $a \cdot X \cdot \sqrt{m_1} + b \cdot X \cdot \sqrt{m_2} = c \cdot X \cdot \sqrt{m_3}$?

Each statistic is optimized over all possible planet combinations and weight assignments, so the look-elsewhere effect (1,120 to 16,800 implicit comparisons per test) is automatically accounted for.

Three null distributions were tested:

• Permutation (exhaustive, $8! = 40{,}320$ trials): same observed values, randomly reassigned to planets. Tests whether the assignment to specific masses matters.
• Log-uniform Monte Carlo (100,000 trials): eccentricities drawn from $[0.005, 0.25]$ and inclination amplitudes from $[0.01°, 3.0°]$, both log-uniform. Tests against random planetary systems with realistic value ranges.
• Uniform Monte Carlo (100,000 trials): same ranges, flat distribution.

In all cases, planetary masses are held fixed at their observed values — the most conservative choice.

Results:

Interpretation:

• Law 2 is individually highly significant ($p \leq 0.003$) across all three null models. The inclination $\psi$-constant linking Venus, Earth, and Neptune through Fibonacci weights $\{2, 3, 5\}$ is very unlikely to arise by chance — even after accounting for all 3,360 possible triplet-weight combinations.

• The overall Fibonacci structure is significant (Fisher’s combined $p \leq 0.006$), robust against the choice of null distribution.

• Laws 1 and 3 are not individually significant after look-elsewhere correction ($p \approx 0.35{-}0.48$). The eccentricity ladder and additive triad produce small errors (0.1–0.5%) for the observed planets, but the test statistics — which optimize over all possible planet subsets — find similarly good matches in random systems often enough that these laws alone would not pass a significance threshold. This does not mean they are wrong; it means they cannot be established from significance testing alone and require physical justification (e.g., from KAM theory or angular momentum conservation).

• The pairwise count (18 of 56 ratios matching Fibonacci) is marginally significant ($p \approx 0.04{-}0.08$), depending on the null model.

This analysis directly addresses the Backus (1969) critique: the Fibonacci structure as a whole is statistically significant ($p < 0.01$), with Law 2 providing the strongest individual evidence. The script fibonacci_significance.py is available for independent verification — see Computational verification below.

Exoplanet systems

If the Fibonacci laws reflect a universal self-organizing principle, they should appear in other planetary systems where masses and eccentricities are known:

• TRAPPIST-1: 7 planets in a known resonance chain, with mass estimates available
• Kepler-90: 8 planets — same count as the Solar System
• Mass and eccentricity estimates for exoplanets are less precise than for Solar System planets, but even approximate tests would be significant

Fibonacci in the angular momentum budget

Since $e \times \sqrt{m}$ is dimensionally related to angular momentum components, testing whether the total angular momentum of the Solar System can be decomposed into Fibonacci-weighted contributions from individual planets could provide a powerful physical interpretation.

Time evolution of Fibonacci relationships

The laws use “base” (oscillation midpoint) values for eccentricity and inclination. Key questions:

• Do the Fibonacci relationships hold only at the base values, or approximately throughout the oscillation cycle?
• If they are exact only at the midpoint, this would suggest the Fibonacci structure is a long-term attractor state toward which the Solar System has been driven by dissipative evolution — consistent with Molchanov’s (1968) hypothesis of resonance capture.

Additional references to investigate

The following papers may contain relevant connections not yet explored:

• Bank & Scafetta (2021): “Scaling, Mirror Symmetries and Musical Harmonies among the Distances of the Planets of the Solar System” — compares harmonic models including 12-tone equal temperament
• Aschwanden & Scholkmann (2017): “Exoplanet Predictions Based on Harmonic Orbit Resonances”, Galaxies 5(4), 56 — found 73% of exoplanet period ratios involve Fibonacci numbers
• Idriss & El Kossifi (2018): “Is the golden ratio a universal constant for self-replication?” — cautionary analysis showing the golden ratio is not uniquely special in all self-organizing systems
• Read (1970): “Fibonacci Series in the Solar System”, The Fibonacci Quarterly — early attempt to fit Fibonacci sequences to moon distances

References

Fibonacci and orbital resonance

1. Molchanov, A.M. (1968). “The resonant structure of the Solar System.” Icarus, 8(1-3), 203-215. ScienceDirect 
2. Backus, G.E. (1969). “Critique of ‘The Resonant Structure of the Solar System’ by A.M. Molchanov.” Icarus, 11, 88-92.
3. Molchanov, A.M. (1969). “Resonances in complex systems: A reply to critiques.” Icarus, 11(1), 95-103.
4. Aschwanden, M.J. (2018). “Self-organizing systems in planetary physics: Harmonic resonances of planet and moon orbits.” New Astronomy, 58, 107-123. arXiv 
5. Aschwanden, M.J., & Scholkmann, F. (2017). “Exoplanet Predictions Based on Harmonic Orbit Resonances.” Galaxies, 5(4), 56. MDPI 
6. Pletser, V. (2019). “Prevalence of Fibonacci numbers in orbital period ratios in solar planetary and satellite systems and in exoplanetary systems.” Astrophysics and Space Science, 364, 158. arXiv 
7. Read, R.C. (1970). “Fibonacci Series in the Solar System.” The Fibonacci Quarterly, 8(4).
8. Bank, M.J., & Scafetta, N. (2021). “Scaling, Mirror Symmetries and Musical Harmonies among the Distances of the Planets of the Solar System.” Frontiers in Astronomy and Space Sciences, 8, 758184.

Fibonacci in other natural systems

9. Yamagishi, M.E.B., & Shimabukuro, A.I. (2008). “Nucleotide frequencies in human genome and Fibonacci numbers.” Bull. Math. Biol., 70(3), 643-653. PubMed 
10. Prusinkiewicz, P., & Lindenmayer, A. (1990). The Algorithmic Beauty of Plants. Springer. PDF 
11. Idriss, S.A., & El Kossifi, Y. (2018). “Is the golden ratio a universal constant for self-replication?” Journal of Theoretical Biology, 445, 33-40. PMC 

Computational verification

The optimization scripts that derived and verified these laws are available on request:

• fibonacci_all_planets.py — All three laws, predictions, and verification
• fibonacci_significance.py — Monte Carlo and permutation significance tests (produces the p-values in Statistical significance)
• fibonacci_psi_search.py — Extended $\psi$-constant search: pairwise scans, multi-planet groups, multiple $\psi$-levels, and Fibonacci connection network analysis
• fibonacci_d_search.py — Systematic search for formulas predicting $d$ from known planetary quantities (40+ quantities, power-law fits, period-fraction analysis, combinatorial Fibonacci formulas)
• fibonacci_two_quantum.py — Two-quantum-number deep exploration: $d = b \times F$ decomposition, partner constraints, eccentricity extension, algebraic structure
• fibonacci_selection_rule.py — Mirror symmetry analysis: Fibonacci index ladder, selection rule search, cross-parameter consistency
• fibonacci_mirror_deep.py — Deep mirror symmetry investigation: even-index constraint, golden ratio powers, selection rule derivation, gravitational coupling test, orbital distance analysis
• fibonacci_identity_simplified.py — Jupiter–Saturn simplified identity analysis
• fibonacci_identity_optimization.py — Base eccentricity and mass correction optimization

Continue to Scientific Background for the physical context, or see Formulas for the complete calculation formulas.