Physical Origin — Technical Reference

This is the full technical reference for the physical origin of the Fibonacci Laws, synthesized from 30+ investigation scripts (docs/formulas/fibonacci_*.py). It provides the complete evidence chain, derivations, and quantitative results behind the accessible treatment in Physical Origin.

1. What Needs Explaining

1.1 The primary structure (Laws 1-6)

1.2 The secondary structure (ω pattern)

All 8 planets have ω = 360° × (Fibonacci fraction), with errors 0.02-0.43%, all using the Law 6 triad {3, 5, 8, 13}. Statistically significant at p = 0.037 (all 7 non-Earth planets) and p = 0.006 (best 4). No unified formula exists — each planet uses other planets’ quantum numbers. See fibonacci_omega_montecarlo.py for the Monte Carlo analysis.

1.3 The eccentricity ladder

d × ξ × √m = constant for all 8 planets (J2000 eccentricities), with spread 0.04%. Monte Carlo testing: p < 10⁻⁵ (0/100,000 random AMD-matched distributions). This is a formation constraint, not dynamically maintained — no secular eigenmode preserves the ladder. See fibonacci_j2000_eccentricity.py.

1.4 Specific features requiring explanation

• Why Fibonacci numbers specifically (not, say, powers of 2)?
• Why √m mass weighting (not m, or m^(1/3))?
• Why the specific quantum numbers {1, 2, 3, 5, 8, 11, 13, 21, 34}?
• Why mirror symmetry across the asteroid belt?
• Why the even-index selection rule (idx = 2k − 4)?
• Why R ≈ 311 (a prime, not a Fibonacci number)?
• Why ω uses other planets’ m-values (inter-planet coupling)?

2. Layer 1: KAM Theory — Why Fibonacci

2.1 The mathematical foundation

The Kolmogorov-Arnold-Moser (KAM) theorem (1954-1963) proves that in a nearly integrable Hamiltonian system (like a planetary system with small mutual perturbations), most quasi-periodic orbits survive if their frequency ratios satisfy a Diophantine condition — they must be sufficiently irrational.

The key insight: the golden ratio φ = (1+√5)/2 is the most irrational number. Its continued fraction [1; 1, 1, 1, …] has the slowest possible convergence, meaning φ is hardest to approximate by rationals. In KAM theory, this translates to maximum stability.

2.2 Greene-Mackay result

Greene (1979) and Mackay (1983) showed that in the standard map (a paradigm for Hamiltonian chaos), the golden-ratio torus is the last KAM torus to break as perturbation strength increases. Noble numbers (those with continued fractions ending in all 1’s — i.e., converging to φ) are the most stable frequency ratios.

2.3 From golden ratio to Fibonacci

The Fibonacci sequence is the algebraic skeleton of the golden ratio:

• F(n+1)/F(n) → φ as n → ∞
• Every noble number can be expressed as (aφ + b)/(cφ + d) with a,b,c,d from Fibonacci numbers
• The convergents of φ are exactly F(n)/F(n+1)

Therefore: if KAM theory selects golden-ratio-compatible frequency ratios for maximum stability, it automatically selects Fibonacci ratios as the discrete approximants.

2.4 Evidence in the Solar System

The secular eigenfrequencies of the Solar System are near (not exactly at) Fibonacci ratios, as KAM theory predicts:

• The frequencies cluster near noble number ratios but are perturbed away from exact resonance
• Exact Fibonacci ratios would be commensurabilities (resonances) — KAM theory says stable orbits are near but not at rational ratios
• The Holistic model’s Fibonacci structure represents the organizing framework around which secular dynamics oscillates

3. Layer 2: Formation-Epoch Freezing

3.1 The mechanism

KAM theory explains why Fibonacci ratios are preferred — but why are they so precisely realized? The answer lies in the formation epoch:

1. Protoplanetary disk phase: planets form and migrate within a gas disk. Dissipative forces (gas drag, disk torques) push orbits toward configurations that minimize AMD (Angular Momentum Deficit)
2. KAM selection: among AMD-minimizing configurations, Fibonacci-organized ones are dynamically preferred because they maximize stability margins (KAM theory)
3. Disk dissipation: when the gas disk dissipates (~3-10 Myr after formation), the dissipative mechanism shuts off, and the Fibonacci configuration is frozen
4. Long-term preservation: without dissipation, secular dynamics oscillates around the frozen configuration but cannot destroy it. KAM tori protect the overall architecture

3.2 Evidence: eccentricity ladder as formation constraint

The eccentricity Fibonacci ladder (d × ξ × √m = constant) is the clearest evidence for formation-epoch freezing:

• It is not dynamically maintained: no secular eigenmode preserves the ladder. All 8 BvW modes have >200% spread in d × E × √m
• N-body simulations confirm: minimum spread over 10 Myr is 3.75%, never approaching the observed 0.04% (see fibonacci_nbody_proper.py)
• J2000 values are special: 5/8 planets have base eccentricity ≈ J2000 to less than 0.1%. The ladder is satisfied NOW (at J2000), not at some past or future epoch
• Monte Carlo significance: 0/100,000 random AMD-matched distributions achieve comparable spread (p < 10⁻⁵)

Interpretation: the protoplanetary disk dissipated when the eccentricity distribution happened to satisfy the Fibonacci ladder. This was not coincidence — dissipative evolution drove the system toward this configuration, and KAM stability locked it in.

3.3 Evidence: ω pattern as formation memory

The argument of perihelion investigation confirms the same picture:

Every expression is built from the Law 6 triad (3, 5, 8, 13) — the same period denominators that govern the Earth-Jupiter-Saturn resonance (Law 6: 3 + 5 = 8) and axial precession (H/13). Each planet’s ω encodes other planets’ quantum numbers rather than its own — Mercury uses Saturn’s 8, Venus uses Saturn’s 8 · Earth’s 3 · Mars’s 13, Neptune uses Jupiter’s 5 — reflecting the inter-planet gravitational coupling at formation.

The oscillation is a coordinate artifact from a frame mismatch, not physics — the true ω is constant in a consistent reference frame. Earth is special: its perihelion and ascending node precess at different rates (H/16 vs H/3), so ω cycles through 360° over the Holistic Year. In the meeting-frequency frame, ω = 180° exactly.

• ω is constant because perihelion and ascending node precess at the same ICRF rate — a frozen angular relationship
• Each planet’s ω uses OTHER planets’ m-values → reflects inter-planet gravitational coupling during formation (see fibonacci_omega_formula_search.py)
• No secular eigenmode explains ω → the pattern predates the current secular dynamics (see fibonacci_omega_eigenmodes.py)
• Statistically significant (p = 0.037) but weaker than eccentricity (p < 10⁻⁵) → a secondary formation constraint, not a primary structural law

3.4 The Earth eccentricity lever

Earth’s base eccentricity (e_E = 0.015321) is the single most important formation parameter:

• It is an oscillation midpoint, not J2000 (−8.3% from J2000)
• Adjusting e_E from J2000 reduces eccentricity ladder spread from 10.3% → 1.4% (89% of improvement)
• The ladder constraint itself predicts e_E = 0.015323 (0.016% match with the model value)
• This is the one free eccentricity parameter — all others are J2000 values or derived

See fibonacci_base_identity.py and fibonacci_j2000_eccentricity.py for the full analysis.

4. Layer 3: AMD-Natural Variables and √m Weighting

4.1 Why √m?

The Angular Momentum Deficit (AMD) provides the answer:

AMD_i ≈ (√a_i / 2) × (ξ_i² + η_i²)

where ξ_i = e_i × √m_i and η_i = i_i × √m_i.

AMD is the fundamental conserved quantity in secular planetary dynamics (Laskar 1997). It measures how far each orbit deviates from circular and coplanar. The natural variables for AMD decomposition are exactly ξ = e√m and η = i√m — the same mass-weighted quantities that appear in Laws 2 and 4.

4.2 Uniqueness of √m

Empirical testing confirms √m is unique:

• α = 0.50 (i.e., m^0.50 = √m) gives inclination ladder spread of 0.11%
• The next best exponent gives >28% spread
• No other mass power produces comparable Fibonacci structure

See fibonacci_amd_structure.py for the AMD decomposition analysis.

Interpretation: the Fibonacci structure acts on AMD-natural variables because AMD is what dissipative processes minimize during formation. The golden ratio optimality (KAM) operates in the same variable space as AMD conservation.

5. The Quantum Number Structure

5.1 Selection rules

The Fibonacci coupling numbers F follow strict selection rules:

• Even-index constraint: idx(F) = 2k − 4, where k is the planet’s position index
• Mirror symmetry: 4 inner + 4 outer planets mirror across the asteroid belt
• Step size = 2: one φ² factor per planet separation (coupling decay rate)

5.2 Physical origin of selection rules

5.3 The φ⁴ belt barrier — resolved

The cross-belt coupling ratio is B_Mars→Earth / B_Jup→Saturn = 0.144 ≈ φ⁻⁴. A four-factor decomposition sums to exactly −4.00 in log_φ:

The dominant term (b1) is algebraically exact from quantum numbers: η_E/η_S = (ψ₁/3)/(ψ₁×11/13) = 13/33. This means 97% of the φ⁻⁴ barrier is structurally forced by the Fibonacci quantum numbers themselves.

The remaining 3% is KAM-selected: perturbation analysis (10k trials, ±20% SMA) shows only 17% of configurations land within 0.5 of −4.00 in log_φ. Jupiter’s actual semi-major axis (5.20 AU) sits precisely where the belt coupling ratio crosses φ⁻⁴. The pure-geometry metric gives φ^4.4 (overshoots because it omits mass factors).

See fibonacci_phi4_derivation.py and fibonacci_cancellation_investigation.py for the full derivation and perturbation analysis.

5.4 Mercury completes the mirror

Mercury at k=4 fills Neptune’s mirror partner, giving a complete 4+4 structure:

See fibonacci_k1_anomaly.py for the belt-adjacent anomaly investigation.

6. The R ≈ 311 Mystery

6.1 What R is

R = ψ₁/ξ_V ≈ 310.83 — the ratio of the inclination scale (ψ₁) to the eccentricity scale (ξ_V). It connects Laws 2-3 to Laws 4-5: how much larger are inclination quantum numbers than eccentricity quantum numbers?

6.2 311 is a Fibonacci primitive root prime

311 is prime, and it is a Fibonacci primitive root (FPR) — meaning φ mod 311 generates the entire multiplicative group F*₃₁₁. Equivalently:

• The Pisano period π(311) = 310 = 311 − 1 (maximal possible)
• φ mod 311 has order 310 (generates all non-zero residues)
• No Fibonacci “dead zones” exist mod 311 — maximally compatible with Fibonacci chains

Only ~30% of primes have this property. There are 33 FPR primes ≤ 600 (OEIS A003147).

6.3 Two-layer explanation

Layer 1 — Why an FPR prime? An FPR prime ensures that Fibonacci coupling chains can connect ALL quantum number combinations without dead zones. A non-FPR prime would have forbidden couplings, constraining the system.

Layer 2 — Why 311 specifically? The formation-epoch eccentricity scale places R = 310.83. The closest FPR prime is 311 (distance 0.17). The next FPR primes are 271 (distance 40) and 359 (distance 48). The system “snaps” to the nearest FPR prime.

Additional properties:

• α(311) = 310 = π(311) → Type 1 (maximally efficient entry point)
• R ≈ 311 − 2/13 = 311 − F₃/F₇ at 0.007% — clean Fibonacci expression
• Continued fraction: R = [310; 1, 4, 1, 2, 1, 2, 128…], 311/1 is best integer convergent

6.4 TRAPPIST-1 confirmation

TRAPPIST-1’s optimal super-period is N = 311 × P_b (maximum deviation 0.12%). Two independent planetary systems both selecting 311 has probability P ≈ 2 × 10⁻⁶ by Monte Carlo. See fibonacci_trappist1_deep.py and fibonacci_311_deep.py.

6.5 311 is external to the quantum numbers

311 cannot be built from the quantum number set {1, 2, 3, 5, 8, 11, 13, 21, 34}. It is the one truly external parameter — set by the overall scale of eccentricity at the formation epoch.

6.6 Open question

Why does dissipative formation converge to FPR primes? The KAM attractor hypothesis: FPR primes maximize the density of Fibonacci-compatible frequency ratios, so dissipative evolution preferentially finds FPR-compatible configurations. This remains unproven.

7. The Complete Origin Story

7.1 The three-phase narrative

Phase 1: Mathematical necessity (KAM theory) The golden ratio φ is the most irrational number. KAM theory proves that frequency ratios near φ (and its convergents F(n)/F(n+1)) are maximally stable against perturbation. This is pure mathematics — it does not depend on the specific masses, distances, or history of our Solar System.

Phase 2: Dissipative selection (formation epoch) During the protoplanetary disk phase (~3-10 Myr), dissipative forces (gas drag, disk torques, planetary migration) minimize AMD while maintaining dynamical stability. Among all AMD-minimizing configurations, those organized by Fibonacci ratios have the widest KAM stability margins. The dissipative evolution therefore converges toward Fibonacci-organized configurations.

This phase sets:

• The eccentricity ladder (d × ξ × √m = constant) → frozen at disk dissipation
• The ω pattern (perihelion-to-node angles) → frozen inter-planet coupling geometry
• The inclination constants (ψ₁, ψ₂, ψ₃) → frozen AMD partition
• The eccentricity scale R ≈ 311 → snapped to nearest FPR prime
• The quantum number assignments (d, b, F for each planet) → set by position and coupling

Phase 3: Long-term preservation (4.5 Gyr to present) After disk dissipation, the system evolves under conservative (non-dissipative) Hamiltonian dynamics. KAM tori protect the overall Fibonacci architecture. Secular oscillations modulate individual orbital elements (e.g., Earth’s eccentricity oscillates around 0.015321 with amplitude ~0.06), but the organizing Fibonacci structure persists because:

• The AMD-natural variables (ξ, η) oscillate within KAM-bounded regions
• The global AMD budget is approximately conserved
• No secular eigenmode can destroy the Fibonacci ladder (tested: all modes give >200% spread)
• The quantum number assignments are topological — they encode the orbit’s position in the coupling hierarchy, not its instantaneous state

7.2 What each layer explains

8. Hierarchy of Certainty

9. Relationship to the Published Paper

The published paper “Fibonacci Laws of Planetary Motion” documents the observational structure (Laws 1-6) and its statistical significance. This document addresses the physical origin question that the paper raises but does not fully answer.

The paper’s §14 (Discussion) touches on KAM theory and formation constraints briefly. This reference provides the comprehensive treatment, incorporating all post-publication investigations:

10. References

• KAM theory: Kolmogorov 1954, Arnold 1963, Moser 1962
• Golden ratio stability: Greene 1979 (standard map), Mackay 1983 (renormalization)
• Planetary KAM: Morbidelli & Giorgilli 1995 (superexponential stability)
• AMD: Laskar 1997 (AMD conservation), Laskar & Petit 2017 (AMD-stability)
• Secular theory: Brouwer & van Woerkom 1950 (eigenmodes)
• TRAPPIST-1: Grimm et al. 2018 (TTV masses and eccentricities)
• FPR primes: OEIS A003147

Investigation scripts

All scripts are in docs/formulas/:

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