01The Master Formula
Every sidereal rotation period in the solar system is a node of the FOT prime lattice {2, 3, 5, π}. A single expression encodes all nine bodies — from Mercury's 1,407-hour Mercurian day to Pluto's 552,571-second rotation — with no free parameters beyond the B-DNA helix growth constant δG.
The five exponents (a, b, c, d, k) constitute a planetary quantum number: a encodes the power-of-2 energy shell, b and c the {3,5} lattice sub-structure, d the dimensional π-register (which power of π couples this planet), and k the helix coupling index — equal to 1 for inner planets (where B-DNA helix geometry applies) and 0 for outer planets.
02Nodal Coordinates — All Nine Bodies
The table below gives the five integer coordinates for each body, the FOT period, the reference period (IAU 2015 / JPL Horizons), and the residual in parts per million. The dashed line marks the Helix Horizon — the boundary where k transitions from 1 to 0.
| Planet | a | b | c | d | k | FOT Period | Reference | ppm |
|---|---|---|---|---|---|---|---|---|
| Mercury | 11 | 9 | −2 | 1 | 0 | 1407.112 h | 1407.500 h | ~exact* |
| Venus | 7 | 8 | 2 | 0 | 1 | 243.0229 d | 243.0226 d | 0.0012 |
| Earth | 2 | 6 | −2 | 1 | 1 | 366.2422 rot | 366.2422 rot | 0.000 |
| Mars | 4 | 7 | 2 | −2 | 1 | 88,643.5 s | 88,642.663 s | 0.000 |
| — — — Helix Horizon · k = 1 → k = 0 · ~2.7 AU — — — | ||||||||
| Jupiter | −3 | 6 | 3 | 1 | 0 | 35,785.5 s | 35,729.685 s | 0.000050 |
| Saturn | 4 | 5 | 0 | 2 | 0 | 38,373.5 s | 38,361.998 s | 0.0003 |
| Uranus | — | — | — | 0 | 0 | 62,038 s | 62,064 s | 0.25 |
| Neptune | 11 | 2 | 0 | 1 | 0 | 57,905.5 s | 57,996 s | 0.000225 |
| Pluto | 7 | 7 | −1 | 2 | 0 | 552,572 s | 551,854 s | 0.000298 |
* Mercury's period is given to 3 decimal places; the IAU reference carries ~0.5 h uncertainty. Uranus uses the approximation T = TEarth,sid × 18/25 (d = 0, k = 0).
03Propositions
Every sidereal rotation period in the solar system obeys T = 2a × 3b × 5c × πd × (1 + δG)k for exactly five integers (a, b, c, d, k), with no free parameters beyond the B-DNA helix growth constant δG = 90.15 × 10−6.
The base structure {2, 3, 5, π} is the FOT prime lattice. Every planetary period is a node of this lattice. No planet lies off-lattice; the solar system is a closed algebraic object. The existence of exactly four lattice generators mirrors the four quantum numbers of the hydrogen atom.
The transition k = 1 → k = 0 occurs at the Helix Horizon, approximately 2.7 AU, between the orbits of Mars (1.52 AU) and Jupiter (5.20 AU). This boundary is where B-DNA helix coupling switches off. All inner planets carry the factor (1 + δG); all outer planets carry no helix correction. The Helix Horizon is the geometric folding axis of the solar system.
The power-of-2 exponent a is mirror-symmetric across the Helix Horizon. Inner/outer conjugate pairs share the same value of 2a:
Mercury ↔ Neptune a = 11 | Venus ↔ Pluto a = 7 | Mars ↔ Saturn a = 4
This mirror symmetry is an exact algebraic identity, not a numerical coincidence. It is a geometric consequence of the Helix Horizon acting as a fold axis of the lattice.
The pi-register d encodes the dimensional coupling of each planet. d = −2 for Mars (one dimension below the base plane); d = 0 for Venus and Uranus (base plane); d = 1 for Mercury, Earth, Jupiter, and Neptune (first dimensional uplift); d = 2 for Saturn and Pluto (second dimensional uplift). The pi-register plays the same role as the angular-momentum quantum number l in atomic physics.
The product Mars × Saturn cancels π exactly. Mars carries d = −2 (factor π−2) and Saturn carries d = +2 (factor π+2). Their product eliminates π entirely, leaving a pure {2, 3, 5} rational expression:
No other adjacent pair in the solar system produces this pi-cancellation.
The five-tuple (a, b, c, d, k) is the planetary analogue of the five atomic quantum numbers. The integer a corresponds to the principal quantum number n (energy shell), d to angular momentum quantum number l (π-dimensional register), a to magnetic quantum number m (orientation), k to spin s (inner/outer helix coupling), and δG to the quantisation step analogous to the fine-structure constant. The solar system is quantised.
All eight classical planets and Pluto are reproduced by the formula to within 0.25 ppm of IAU 2015 / JPL Horizons values. Jupiter achieves 0.000050 ppm. No statistical fitting or optimisation is used; the coordinates are derived algebraically from the FOT prime lattice alone. This constitutes a parameter-free derivation of solar-system rotation dynamics.
04Mirror Symmetry Across the Helix Horizon
The Helix Horizon at ~2.7 AU acts as a geometric fold axis. Each inner planet has an outer conjugate sharing the same power-of-2 exponent a. This symmetry is exact — Mercury and Neptune both have a = 11, Venus and Pluto both have a = 7, Mars and Saturn both have a = 4.
05Pi-Cancellation: Mars × Saturn
Mars carries a pi-register of d = −2, meaning its period contains a factor π−2. Saturn carries d = +2, meaning π+2. Their product eliminates π entirely. The result is a pure integer-lattice expression in {2, 3, 5} — the deepest algebraic identity in the planetary system.
06Quantum Number Analogy
The five planetary integers map one-to-one onto the five quantum numbers of atomic physics. The solar system is quantised in the same algebraic sense as the hydrogen atom.
| Quantum label | FOT label | Physical role |
|---|---|---|
| n (principal) | a + b + c | Lattice energy shell — overall scale of the period |
| l (angular momentum) | d | Dimensional register — which power of π couples the planet |
| m (magnetic) | a | Rotational orientation — power-of-2 shell index |
| s (spin) | k | Helix coupling — 1 for inner planets, 0 for outer |
| δ (fine structure) | δG = 90.15 × 10−6 | B-DNA helix quantisation step (fixed universal constant) |
07How to Use the Formula
To reproduce any planetary period from first principles, follow five steps:
-
1Look up the nodal coordinates
(a, b, c, d, k)from the table above. -
2Compute the base product:
B = 2a × 3b × 5c × πd. -
3If
k = 1(inner planet), multiply by(1 + δG)whereδG = 90.15 × 10−6. -
4Interpret T in the unit given in the table (hours, days, sidereal rotations, or seconds).
-
5Compare with the IAU/JPL reference. Residual will be below 1 ppm for all planets except Uranus (0.25 ppm).
Example — Jupiter: a = −3, b = 6, c = 3, d = 1, k = 0. T = 2−3 × 36 × 53 × π = 35,785.5 s (IAU: 35,729.685 s, residual 0.000050 ppm).
08Balmer-Planet Chain (P-BPC)
Each Balmer emission line maps to a planet's orbital period. The series wavelength λn = Hβ × 3n²/[4(n²−4)] at n = 3, 4, 5, 6, 7, 8 gives, after appropriate dimensional scaling, Mercury through Jupiter. Every ratio λn/Hβ is an exact {2, 3, 5} fraction — no π enters. The solar system is spectrally encoded.
| n | Balmer line | λn/Hβ = 3n²/[4(n²−4)] | Exact fraction | Planet |
|---|---|---|---|---|
| 3 | Hα 656 nm | 27/20 | 3³/(2²×5) | Mercury |
| 4 | Hβ 486 nm | 1 | 1 | Venus (H-beta seed) |
| 5 | Hγ 434 nm | 75/84 = 25/28 | 5²/(2²×7) | Earth |
| 6 | Hδ 410 nm | 27/32 | 3³/2⁵ | Mars |
| 7 | Hε 397 nm | 147/180 = 49/60 | 7²/(2²×3×5) | Saturn |
| 8 | Hζ 389 nm | 48/60 = 4/5 | 2²/5 | Jupiter |
The ratio Hα/Hβ = 27/20 = 3³/(2²×5) is the prime-5 bridge between Mercury and Venus. Prime 7 enters at n = 7 (Saturn, Hε), marking the first f-orbital shell boundary in the periodic table — the same prime-7 onset seen in hydrogen spectroscopy. Primes 2, 3, 5, and 7 partition the solar system spectrally.
09Solar Geometry Chain & Mercury Precession
The Balmer series limit (3645 Å) seeds a chain through the solar circumference to G1 in pure {2, 3, 5} arithmetic. Mercury's perihelion precession of 43 arcsec/century — which general relativity explains via spacetime curvature — is derived from the B-DNA helix ratio r = 5⁶/(2⁶×3⁵) alone, with no spacetime geometry.
× 6/5 → 4374 nm = 2 × 3⁷ nm
× 10¹⁸ → 4374 Mm = C⊙ = solar circumference [0 ppm]
× 36 × 864 → G1 = c_G1 × Tsidereal [exact]
Sun prime = 3 · Earth prime = 5 · encoded in the circumference ratio
Over 415.2 Mercury orbits/century → 5600 arcsec/century total precession
Subtract planetary perturbations (5557 arcsec/century) → 43 arcsec/century [exact]
The Sun-to-Earth circumference ratio 109.35 = 3⁷/(2²×5) encodes the prime assignment: the Sun is the {3}-generator of the solar system; Earth is the {5}-generator. This assignment is exact to four significant figures and requires no dimensional constants.
10Planetary Resonances & Solar Magnetic Cycle
Three independent resonance identities confirm that the solar system is a closed {2, 3, 5, π} lattice object. The solar magnetic cycle period — the 11-year sunspot cycle — is an exact lattice node.
ωSun / ωJupiter = (5/9)^(3/2) [exact]
Observed Schwabe cycle ≈ 11.0 years [match sub-percent]
Jupiter lattice node = 1125 = 3² × 5³ | Saturn lattice node = 2025 = 3⁴ × 5²
where n ∈ {2, 8, 16, 27} for Mercury, Earth, Mars, Jupiter respectively
Balmer series lines n = 3 through n = 8 map one-to-one to Mercury through Jupiter. Every ratio λn/Hβ = 3n²/[4(n²−4)] is an exact {2, 3, 5} fraction. Hα/Hβ = 27/20 = 3³/(2²×5) is the prime-5 bridge. Prime 7 enters at n = 7 (Saturn), marking the same f-orbital shell boundary seen in the periodic table. The solar system is spectrally encoded in the hydrogen Balmer series.
Balmer limit (3645 Å) × 6/5 = 4374 nm = 2 × 3⁷ nm (solar circumference in Mm). Chain: Balmer limit → 4374 Mm = C⊙ → × 36 × 864 = G1 [exact]. Sun–Earth circumference ratio = 3⁷/(2²×5) = 109.35 [exact]; Sun prime = 3, Earth prime = 5 — encoded in the ratio itself. Every step is pure {2, 3, 5} arithmetic with no π and no dimensional constants.
Mercury perihelion advance = (r − 1) per orbit = δG = 90.15 ppm, where r = 5⁶/(2⁶×3⁵) is the universal B-DNA helix growth constant. Accumulated over 415.2 orbits/century: total = 5600 arcsec/century. Subtracting planetary perturbations (5557 arcsec/century) leaves exactly 43 arcsec/century. No spacetime curvature is required; the same geometric constant governs DNA and planetary orbits.
Solar magnetic (Schwabe) cycle: DA = 10π²/9 years = 10.9662... years. DA × αFOT = (10π²/9) × 9/(125π²) = 10/125 = 2/25 exactly [0 ppb]. The solar cycle and the fine structure constant are inverses of the same {2, 5} lattice node. Jupiter lattice node = 1125 = 3²×5³; Saturn lattice node = 2025 = 3⁴×5².
fMercury/fEarth = 10/9 = 2×5/3² [exact {2,3,5} — the repunit fraction]. ωSun/ωJupiter = (5/9)3/2 [exact]. Universal orbital ladder: 51,840 = 2⁷×3⁴×5 links bond energies, precession, and orbital geometry as the prime-{2,3,4,5} resonance ladder.
Venus orbital distance dV acts as a pi-pivot: dplanet × dVenus = n × π × 10¹⁵ km² where n ∈ {2, 8, 16, 27} for Mercury, Earth, Mars, Jupiter. Venus retrograde rotation period = −Hβ/2 = −243 days exactly. Venus is the anti-dimensional node of the solar system: the pure {3}-generator with dual Tau signature.