The N-body Exploration

To test its own planetary claims, the model built a dynamical laboratory: a nine-body integrator seeded from the observed J2000 state, a Wisdom–Holman engine able to run millions of years, a frequency analysis (NAFF), and a series of bound experiments in which the model’s laws were imposed on the dynamics and the price was measured. Everything below is public in the repository — the instruments live in tools/explore/ and the full research record is doc 109 .

Three kinds of “perihelion rate”

Much confusion about planetary precession comes from one word covering three different quantities:

Saturn genuinely moves retrograde in today’s window — and its long-term mean is genuinely prograde. Neither number is wrong; they are different quantities. The outer planets’ window “rates” swing by thousands of arcseconds per century between adjacent windows (the 883-year great inequality, the 59-year conjunction cycle), so no single rate exists for them on centuries-long spans. Any claim about a planetary period must state which type it means.

What the audit found

The audit put the model’s own integration beside the JPL ephemeris, planet by planet, window by window:

• Newton plus the measured masses reproduce every planet’s window rate and every node rate — except Mercury’s perihelion, which is short by 43.0 ″/cy in both test windows. Switching on the first post-Newtonian (relativistic) term closes exactly that row (−43.0 → −0.0) and moves nothing else.
• Masses cannot carry the 43. A joint mass adjustment exists that fits every perihelion, but it breaks the node rates by 10–65 ″/cy where the measured masses reproduce every node to 0.1. Perihelia and nodes are set by the same masses through different geometry — only Mercury’s perihelion is off.
• A short-range fifth force can mimic relativity on the perihelia (a Yukawa term tuned to Mercury reproduces all four inner-planet residuals) — but it shifts the effective gravitational parameter Mercury feels by five parts in ten million, where ranging fits every planet with one value to one part in ten billion. Excluded by three orders of magnitude, theory-vs-observation.
• The engine reproduces the standard secular frequencies from scratch: g₅ = 4.2562 ″/yr (Laskar: 4.2575), g₆ = 28.2453 (28.2455), s₆ = −26.3477 (−26.3475). With the relativistic term on, Mercury’s own frequency g₁ moves from 5.103 to 5.576 ″/yr (Laskar: 5.5965) — the “43” seen at the long-term level.

The eccentricity shapes and the dominance rule

Each planet’s eccentricity and perihelion are one object: the vector z = e·e^(iϖ), which traces a shape over time — a large circle for the planet’s own mode with smaller epicycles borrowed from its neighbours (the same two-arrow construction as the Earth wobble-centre diagram on this site). The perihelion “rate” at any moment is just how fast the trace happens to be circling the origin:

The free universe: every planet’s eccentricity vector as a shape, measured from the model’s own engine. Mercury is a clean ring far from the origin; Venus and Earth are flowers passing through it — which is why their instantaneous rates carry no structural information. This explains, quantitatively, which planets ride near their long-term mean and which wander:

“Moves at its mean” is a dominance property: planets that own their shapes ride their means; borrowers wander. Earth today runs about 31 % above its long-term mean — its present apsidal period near H/3 is a reading of where the shape currently is, not a permanent rate.

The bound experiments — three universes

Can the model’s period laws be imposed on the dynamics? A class of work-free steering forces (they change no orbital energy, and are gated to exactly zero force throughout the observed era, so nothing measurable today is touched) was built to find out. The first experiment held Earth’s eccentricity on the model’s H/3 law while the free dynamics wanted to dive toward zero:

The first bound, 160 kyr: the free trajectory (blue) passes near zero eccentricity at +26 kyr; the bounded one (rust) stays on the annulus the H/3 law permits. The angular momentum this requires is delivered through the Law-5 balance weights — 99.9 % lands on the four giant planets at a cost (Δe ≤ 3·10⁻⁵) invisible to any observation.

Three complete universes were then integrated against a free control:

The comb universe over one complete 8H cycle (2,682.5 kyr). Because every frequency is a multiple of 2π/8H, the whole secular system repeats exactly every 8H, and each planet’s perihelion winds an exact integer number of turns per cycle — every mean perihelion period is exactly 8H/N:

Two structural results came with these universes. The stable ones only exist in a narrow corridor beside the Newtonian eigenfrequencies — the bound inherits its stability from the free system rather than creating it. And Earth’s eccentricity law and perihelion law need each other: with the eccentricity free, the perihelion cannot be held at all (a near-circular orbit has no controllable apse).

The hybrid universe answered the direct question can Earth run at H/3? — dynamically, a full yes:

Earth over 800 kyr: the free flower (blue) against the closed curve the H/3 law draws (rust), retraced every 111.8 kyr with eccentricity and perihelion locked to the same clock. The integration reproduces the law’s own constants — the central line reads base′ to 0.2 % and the modulation reads base′/2 exactly. The seven neighbours barely notice: Earth is too light to disturb the giants.

The verdict — and the one measured referee

Every comparison above is theory-vs-theory except one. The deep-time eccentricity metronome is measured: 405.6 ± 2.4 kyr, phase-stable in the geological record for over 200 million years (Triassic lake beds, Mesozoic carbonates, Cenozoic carbon isotopes). Against it:

• the free dynamics rings at 405 kyr (the model’s own engine reproduces the metronome from gravity alone);
• the comb universe rings at 447 kyr — beats of comb lines are comb lines, and 405 is not one (8H/6 = 447, 8H/7 = 383);
• the hybrid rings at 111.8 kyr only — the metronome vanishes.

The rock selects the free dynamics for the planets’ deep-time motion. (The 405-kyr component is weak in the most recent ~1 Myr of climate records — the known “400-kyr problem” — but the deep record is where the test lives, and there the line is strong and phase-coherent.)

What this settles

The exploration ends in a clean division of domains:

• The planets’ orbits belong to the dynamics. The model’s own engine reproduces the standard secular system, and every lattice-native deep-time law for the orbits fails the same measured referee.
• Earth’s rotation family belongs to the lattice. The axial precession (H/13, ~25,794 years), the length-of-day stack, the eclipse and paleo results — none of them rest on the planetary divisors, and none were touched by any experiment here.
• The H/3 apsidal law (~111,772 years) is the epoch-local tangent of Earth’s free eccentricity vector: throughout the observed era the law and the dynamics are the same motion — which is exactly why the model matches observation everywhere it is tested — while at deep time Earth rides its epicycles, 405-kyr beat included.
• Open questions, honestly stated: the steering forces of the bound experiments are a mechanism hypothesis with no physical carrier identified; and whether Earth’s H/3 stability plays a role in why Earth is special (a stable spin and a stable apsidal environment as a precondition for climate stability) is a hypothesis under exploration, not a result.