Fibonacci Laws — Derivation

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.

At first glance, these two properties seem unrelated — eccentricity is about orbital shape, inclination is about orbital tilt. They are governed by different physical mechanisms and measured in different units. There is no obvious reason they should follow the same mathematical pattern.

Yet 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: $1/5$, $1/3$, $2/3$, $3/2$, etc.

The three laws documented below constrain:

1. Law 1 — Eccentricity ratios form Fibonacci ladders: $\xi_A / \xi_B = F_i / F_j$
2. Law 2 — Inclination amplitudes satisfy $d \times \eta = \psi$ with Fibonacci quantum numbers $d$ and a universal constant $\psi$
3. Law 3 — A weighted Fibonacci triad: $3\eta_\text{Earth} + 5\eta_\text{Jupiter} = 8\eta_\text{Saturn}$

The most far-reaching consequence is a zero-parameter prediction: the universal inclination constant $\psi_1$ 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 no 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 empirical verification and error analysis are presented in the sections below, with a combined statistical significance of $p < 10^{-4}$.

Prediction summary

The table below shows what the three laws concretely predict for each planet. Inclination amplitudes follow from $\psi_1$ alone (zero free parameters). Base eccentricities follow from a single free parameter: Earth’s base eccentricity $e_E = 0.015321$ (determined independently from the 3D simulation).

Inclination amplitudes — predicted from $\psi_1 = F_5 \times F_8^2 / (F_3 \times H)$. The “Predicted” column is the oscillation amplitude (how far the orbit tilts from its mean). The J2000 and Mean columns show the total inclination at the current epoch and the time-averaged midpoint respectively — these are context, not direct comparisons. For error analysis, see Predicted inclination amplitudes.

Mercury’s weight $d = 21/2 = F_8/F_3$ is a ratio of Fibonacci numbers. Saturn’s weight $d = 13/11 = F_7/L_5$ mixes a Fibonacci and a Lucas number. Both planets’ period denominators are Lucas numbers ($b = 11$), so the $d = b \times F$ decomposition does not apply directly (see quantum number table).

Base eccentricities — Inner planets form a Fibonacci ladder $\xi_V : \xi_E : \xi_{Ma} : \xi_{Me} = 1 : 5/2 : 5 : 8$, where $k$ is each planet’s ladder position relative to Venus ($k = 1$). Earth’s independently known $e_E = 0.015321$ (from the 3D simulation) sits at $k = 5/2$ and anchors the predictions via $\xi_{\text{planet}} = (k / k_E) \times \xi_E$. Jupiter and Saturn are linked by their Law 3 ratio. Uranus and Neptune form an outer pair ($\xi_U = 5\xi_N$).

The inner ladder positions $1 : 5/2 : 5 : 8$ are all Fibonacci numbers, with consecutive ratios ($5/2$, $2$, $8/5$) converging toward $\varphi$. Jupiter and Saturn are linked by the Law 3 ratio ($\xi_J/\xi_S = 13/8$). Uranus and Neptune form an outer pair constrained by $\xi_U = 5\xi_N$ (ratio $= F_5$).

The structure is interlocking: changing any single planet’s value breaks the Fibonacci ratios with every other planet in the same law.

Technical summary

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). When distinguishing the two parameters, we write $\xi = e \times \sqrt{m}$ for eccentricity and $\eta = i \times \sqrt{m}$ for inclination. The Fibonacci laws constrain how these 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

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

The mass-weighted eccentricity $\xi = e_{\text{base}} \times \sqrt{m}$ satisfies:

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 ψ-Constant

Each planet’s mass-weighted inclination amplitude, multiplied by a Fibonacci quantum number, equals the same universal constant.

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%.

Predictions (calibrated from Earth)

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

These are the one-free-parameter predictions (using Earth’s observed inclination). The zero-free-parameter predictions below use $\psi_1$ derived from $H$ and give slightly different values.

Pairwise Fibonacci ratios

The equivalent pairwise statement — the ratio $i \times \sqrt{m}$ between any two of the Law 2 planets is a Fibonacci ratio. The table also includes the cross-group Jupiter/Mars pair ($\psi_3$):

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.

Multiple ψ-levels

The $\psi$-constant is not unique — there are three levels, and their ratios are themselves Fibonacci ratios. The full derivation from first principles is given in Deriving ψ₁ from the Holistic Year; here is the summary:

Venus bridges $\psi_1$ and $\psi_2$ (participating with weights $d=2$ and $d=3$), and Mars bridges $\psi_1$ and $\psi_3$ (participating with weights $d=13/5$ and $d=3$). The ratios $3/2$ (error: 0.31%) and $15/13$ are algebraic identities forced by these dual memberships, not empirical fits.

Uranus ($\psi_2$ only), Jupiter ($\psi_3$ only), and Mercury ($\psi_3$, $d = 21/2$) provide independent confirmations — they were not used to define the ratios.

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

The mass-weighted inclination amplitudes of Earth, Jupiter, and Saturn satisfy the Fibonacci addition rule.

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. Since $T = H/d$ where $d$ is the weight, Law 3 can be rewritten as:

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.

Predictions

Predicted base eccentricities

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

Predicted inclination amplitudes

With $\psi_1$ derived from $H$ (see Deriving ψ₁ from the Holistic Year below), inclination amplitudes for all 8 planets require zero free parameters — only $H$, planetary masses, and Fibonacci quantum numbers:

Saturn’s weight $d = 13/11$ mixes Fibonacci and Lucas numbers (see quantum number table). The Law 3 triad $3\eta_E + 5\eta_J = 8\eta_S$ provides an independent check at $-0.69\%$. All 8 predictions are within 0.75%.

Predictive system status

Inclination predictions cover all 8 planets directly via $\psi_{1,2,3}$ (Saturn via $\psi_1$ with $d = 13/11 = F_7/L_5$, independently confirmed by the Law 3 triad). Eccentricity predictions: 3 inner planets from Earth’s base eccentricity (Law 1), Neptune from the outer pair identity ($\xi_U = 5\xi_N$), plus the Jupiter–Saturn ratio (Law 3). Total: one free parameter ($e_E = 0.015321$) plus $H$, planetary masses, and Fibonacci quantum numbers predicts both inclination amplitudes and base eccentricities across all 8 planets.

Additional Fibonacci Relationships

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

Inclination identities

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 connects to Jupiter and Mars via $\psi_3$ ($d = 21/2$)

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 both eccentricity and inclination.

Planet team assignments

Each planet participates in one or more structural groups. Venus and Mars have dual membership — they belong to two $\psi$-groups simultaneously, which forces the ratios $\psi_2/\psi_1 = 3/2$ and $\psi_3/\psi_1 = 15/13$ to be algebraic identities (not fitted parameters). Saturn participates in $\psi_1$ with $d = 13/11$, independently confirmed by the Law 3 triad.

Saturn’s $d = 13/11 = F_7/L_5$ is the only weight requiring a Lucas number. The Law 3 triad $3\eta_E + 5\eta_J = 8\eta_S$ provides an independent cross-check ($-0.69\%$). In the quantum number decomposition, Saturn has $b = 8$, $F = 1$, giving a structural $d = 8$ — the triad weight.

Summary of the Three Laws

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

Together with the additional pairwise identities (Mars–Jupiter, Venus–Uranus, Uranus–Neptune) and the three $\psi$-levels ($\psi_1$, $\psi_2 = \frac{3}{2}\psi_1$, $\psi_3 = \frac{15}{13}\psi_1$), 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.

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. The group constant $\psi_1$ itself can be derived from $H$: $\psi_1 = F_5 \times F_8^2 / (2H) = 2205/667{,}776$, matching the empirical value to 0.07%. The ψ-level ratios $\psi_2/\psi_1 = 3/2$ and $\psi_3/\psi_1 = 15/13$ are algebraic identities forced by dual-membership planets (Venus and Mars). Together, these predict inclination amplitudes for all 8 planets from $H$ alone with zero free parameters. For eccentricity, one free parameter ($e_E = 0.015321$) remains.

The sections below develop this structure in full: the two-quantum-number decomposition, the derivation of ψ₁ from H, the eccentricity ladder structure, and the physical foundations (AMD, KAM theory) that explain why Fibonacci numbers appear.

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 eight 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

†Mercury’s period denominator $b = 11$ is a Lucas number, not Fibonacci, so the standard $d = b \times F$ decomposition does not apply. Mercury participates in $\psi_3$ through its Fibonacci weight $d = 21/2 = F_8/F_3$ directly, and in the mirror symmetry at $k = 4$ via the alternative factoring $d = (21/13) \times (13/2)$ with $\text{idx}(13/2) = +4$.

‡Saturn’s structural $d = b \times F = 8$ is the Law 3 triad weight, but its effective $\psi$-constant weight is $d = 13/11 = F_7/L_5$ (Fibonacci/Lucas). The triad provides an independent cross-check.

For the seven planets with Fibonacci period denominators, 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 $F = 1$. Venus at $\text{idx}(F) = +2$ is mirrored by Uranus, and Mercury at $\text{idx}(F) = +4$ completes the pattern as Neptune’s inner mirror partner.

†Mercury’s period denominator $b = 11$ is a Lucas number ($L_5 = 11$), not Fibonacci, so the standard $d = b \times F$ decomposition does not apply. However, Mercury’s $d = 21/2 = F_8/F_3$ can be factored as $(21/13) \times (13/2)$, where $13/2 = F_7/F_3$ has $\text{idx} = 7 - 3 = +4$ — matching Neptune’s $\text{idx}(F) = +4$. The factor $21/13 = F_8/F_7$ approximates the golden ratio $\varphi$, replacing the integer period denominator $b$ of the other planets.

Why Lucas, not Fibonacci? The period denominators of the Law 3 planets form an additive chain: $3 + 5 = 8$ (Earth + Jupiter = Saturn) and $5 + 8 = 13$ (Jupiter + Saturn = Mars) are adjacent Fibonacci sums, producing the next Fibonacci number. Mercury’s $b = 11 = 3 + 8 = F_4 + F_6$ is a non-adjacent Fibonacci sum — it skips $F_5 = 5$ (Jupiter) because Jupiter is on the other side of the asteroid belt. The general identity $L_n = F_{n-1} + F_{n+1}$ shows that non-adjacent Fibonacci sums always produce Lucas numbers. The Fibonacci-to-Lucas transition is thus the algebraic signature of the belt barrier: Mercury, at the outermost mirror position ($k = 4$), cannot access the adjacent Fibonacci term without crossing the belt, so its period denominator is forced from Fibonacci to Lucas.

This explains a further symmetry in the period fractions $(a, b)$: Mercury has $(8, 11) = (b_\text{Saturn}, b_\text{Earth} + b_\text{Saturn})$, while Mars has $(3, 13) = (b_\text{Earth}, b_\text{Jupiter} + b_\text{Saturn})$. Both belt-boundary planets have period fractions whose numerator is one Law 3 denominator and whose denominator is the sum of two others.

The golden-ratio factor $21/13 \approx \varphi$ in Mercury’s decomposition $d = (21/13) \times (13/2)$ has a KAM interpretation: at the outermost mirror position, the discrete Fibonacci structure asymptotes to its continuum limit — the golden ratio. For all other planets, the “period factor” in $d = b \times F$ is an integer or simple fraction; for Mercury, it is $\varphi$ itself. This is consistent with KAM theory, where $\varphi$ marks the maximally stable frequency ratio.

Secular coupling analysis confirms this is an algebraic identity, not a physical coupling effect: Mercury’s eigenmode is $99.97\%$ pure (the most isolated in the system, with the lowest participation entropy), and its coupling is dominated by Jupiter ($68.5\%$), not Earth and Saturn ($13.9\%$). The additive structure $b_\text{Earth} + b_\text{Saturn} = b_\text{Mercury}$ emerges from the Fibonacci chain, not from direct gravitational interaction.

Mercury’s Lucas number also appears in the $\varphi^4$ belt barrier identity: $\eta_E / \eta_S = 13/33 = F_7 / (F_4 \times L_5)$, where $L_5 = 11$ is Mercury’s period denominator and $F_4 = 3$ is Earth’s. The number $88 = 8 \times 11 = F_6 \times L_5$ (Saturn $\times$ Mercury) emerges algebraically from the Law 3 combination $3\eta_E + 5\eta_J = 8\eta_S$ when the $\psi$-level ratios are substituted. The same Lucas number that determines Mercury’s precession period also sets the coupling suppression across the belt. (See fibonacci_mercury_lucas.py for the full analysis.)

The even-index constraint

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 eight planets. Mercury’s $\text{idx}(F) = +4$ is verified through the alternative decomposition $d = 21/2 = (21/13) \times (13/2)$, where $\text{idx}(13/2) = +4$ (see mirror symmetry table). The selection rule 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.

Why $k = 1$ gives $F = 1/5$, not $1/3$: The Phase 1 step of $+4$ (instead of $+2$) is explained by the KAM interpretation: belt-adjacent planets experience maximal cross-belt coupling, so stability requires the most irrational frequency ratio — the deepest Fibonacci ratio $1/5 \approx \varphi^{-4}$. Eigenmode analysis (fibonacci_k1_anomaly.py) confirms that Mars and Jupiter have the two lowest participation entropies among the inner and outer planet groups respectively, meaning each is dominated by a single secular eigenmode more completely than any of its neighbors. The belt creates a coupling cliff that forces both adjacent planets into maximally isolated eigenmodes.

Theoretical basis of the selection rule

The empirical formula $\text{idx}(F) = 2k - 4$ has four key properties, three of which now have theoretical explanations from secular perturbation theory:

1. The even-index constraint follows from the D’Alembert rules of secular perturbation theory.
2. The mirror symmetry across the belt follows from the approximately block-diagonal structure of the secular coupling matrix.
3. The step size of 2 corresponds to one factor of $\varphi^2 \approx 2.618$ per planet separation, matching the exponential decay of Laplace coefficients.

The fourth property — the belt-adjacent anomaly ($k = 1$ gives $\text{idx} = -4$ rather than $-2$) — is qualitatively explained by three reinforcing mechanisms:

1. Coupling discontinuity. The belt semi-major axis ratio $\alpha_\text{belt} = a_\text{Mars}/a_\text{Jupiter} = 0.293$ is roughly half that of neighboring pairs ($\alpha_\text{Earth-Mars} = 0.656$, $\alpha_\text{Jup-Sat} = 0.545$). Since secular coupling scales as $\alpha \times b^{(1)}_{3/2}(\alpha)$, the belt coupling product ($0.305$) is the lowest of all adjacent pairs. The geometric mean of neighboring coupling products divided by the belt coupling is $8.4 \approx \varphi^{4.4}$ — close to the $\varphi^4$ expected for a double index step.