Mathematical Foundations

The model is two engines and three ledgers: a planetary N-body engine integrated from one cited J2000 state with no fitted constants, a lunisolar precession channel for Earth’s spin and time whose constants are derived or cited, and a fitted Earth-rotation stack behind verification gates. This page is the technical reference for that construction — what the clock is built from, which inputs the model takes, which values it predicts without having been tuned to them, and what would falsify it. The comparison with standard theory, model against IAU 2006, La2004 and DE440 epoch by epoch, lives on Scientific Background. Readers who want the concepts first should start at How It Works.

This page establishes:

• The clock’s J2000 rate is the cited IAU 2006 general precession, its history is composed from measured physics, and the year lengths and movements follow from it and the N-body series — The clock.
• Earth’s five long periods are dynamical readings against the clock, not fractions of a fixed grid — The five periods against the clock.
• The inputs sit in three ledgers and the planetary ledger has no free parameters — Parameter accounting.
• Which values were inputs and which are genuine predictions, and why matching J2000 proves nothing — Calibration vs prediction.
• The named observations that would falsify the model, and the limits it states for itself — Testable predictions, Uncertainties and limitations.

The clock

Earth’s side of the model is read against one clock, the mean lunisolar precession period T_p(t), and against the orbit the N-body engine supplies.

The clock’s J2000 rate is cited. The J2000 rate p₀ is the IAU 2006 general precession p_A, 50.288″/yr, taken as an input; one full circle at that rate is 25,771.58 years. Nothing is tuned to reach it.

Its history is composed. Through time the rate is composed from the Sun’s and the Moon’s torques on the spin, ψ̇(t) = [ω(t)/ω₀]·p₀·[f_S + (1 − f_S)(a₀/a_M(t))³]: the spin from the tidal recession history, the lunar torque on the Moon’s distance. Both are measured histories behind fail-proven gates, and the clock lengthens at geological time with them and with solar mass loss (Earth’s Clock, Expanding Resonance).

The year lengths follow. The difference between the sidereal and the tropical year, accumulated over one precession period, is exactly one tropical year, so the IAU sidereal year and p₀ give the J2000 tropical year, 365.242190469 days — derived, not entered. The J2000 day is the SI day of 86,400 s, and the year and day lengths of date are one family on the clock and the engine’s orbit (Days & Years).

The movements are the N-body series. Earth’s eccentricity, orbit plane and perihelion of date are the series integrated from the J2000 state; the obliquity is the spin axis integrated about that plane from the IAU 2006 J2000 obliquity. No correction term on a fixed period grid sits on any of them (Earth’s Clock).

The 1246 AD alignment

Meeus’s formula (Astronomical Algorithms, 1998, Chapter 26) calculates longitude of perihelion as a polynomial in time:

ω = 102.93735° + 1.71953°T + 0.00046°T² + ...

where T is centuries from J2000. Solving for ω = 90° gives T ≈ −7.54 (~1246 AD). The Meeus formula has stated precision of ~0.01° over ±2000 years; the “alignment” is not exact to a specific day — perihelion and solstice were within ~1 week of each other around 1246 AD. The alignment is a dated event of the IAU mean elements, not an input of the model: its own orbital series, whose secular element at J2000 sits slightly below the mean element, crosses 90° at 1247.2 AD — the mean-element and secular-element conventions, labelled. The perihelion reaches the June solstice half a perihelion-of-date cycle later.

The five periods against the clock

The five Milankovitch-type periods are not defined as fractions of any fixed grid. Each is the model’s own dynamical value, read against the clock:

The five are not independent. The perihelion-of-date period is the beat of the clock with the apsidal motion: the two motions counter-rotate, so the frequencies add, 1/T_peri = 1/T_p + 1/T_aps. The obliquity cycle is the beat of the spin axis’s own precession with the nodal motion, 1/T_obl = 1/T_ψ̇ − 1/T_s₃, where T_ψ̇ = 25,722 yr is the pole’s inertial period — 0.097 ″/yr faster than the equinox clock T_p, which is the planetary precession. All five vary around their means (the of-date axial period wanders between 24,802 and 26,560 years across ±26 kyr; the apsidal ratio T_aps/T_p between 1.08 and 9.81). The concept-level account is on Earth’s Clock; full analysis: Supporting Evidence §8.

Parameter accounting — two engines, three ledgers

The model is two engines, and its numbers fall into three ledgers. No single “N free parameters” headline is published — the ledgers are the accounting.

Ledger 1 — the planetary N-body engine’s inputs (the planets: dynamics — cited, not fitted). Free parameters: none. The planetary side is derived from published measurements taken as-is: one J2000 heliocentric state (JPL Horizons vectors, one epoch), the DE440 mass ratios, and the standard 1PN relativistic correction. Everything downstream — the element chains, the secular mode tables, the 405.6-kyr eccentricity metronome — derives from that seed. Integrator order and step are reproducibility conventions, not parameters. Earth’s published orbit rides this ledger too: e(t), the inclination and node, ϖ of date and the obliquity hybrid derive from the same N-body seed, the IAU 2006 J2000 obliquity and the IAU 2006 general precession.

The descriptors ride it too. The per-planet long-period descriptors published alongside the orbits — the perihelion and node periods, and the inclination and eccentricity — are engine output as well: the periods are the chain’s leading secular eigenfrequencies (1,296,000″ ÷ g and ÷ s), and the shape quantities are read at the epoch asked for rather than as a mean plus an amplitude.

Ledger 2 — the lunisolar precession channel (Earth’s spin & time). The channel’s constants are derived or cited; two measured histories sit behind fail-proven gates. None is a free parameter:

Earth’s eccentricity, perihelion and obliquity are not parameters of this ledger: they are the N-body series and the spin-axis integration on it (Ledger 1). The simulator’s 3D scaffold is posed by a few recorded J2000 constants; no published quantity rides them — the engine places the Earth frame, the Sun and the Moon.

Ledger 3 — the fitted Earth-rotation stack (named, gated, hashed). The ΔT/LOD stack’s fitted constants — four millennial cycle terms stated as periods in years (the Bond, Hallstatt and two Jose cycles) and the core-mantle swing — with the recession history’s regime knots and the GIA τ, all under the constants hash. Each is a bounded harmonic or measured term, never a polynomial (polynomials compound at deep time). Every array is fitter-owned, provenance-tracked, and behind fail-proven verification gates; the paleo-anchors gate fails on an unexplained improvement too. These fit the Earth-rotation record; none of them touch the planetary dynamics of Ledger 1 or the movements of Earth’s orbit and spin. (The climate formula’s per-line amplitudes are fitted to the climate records and sit outside the motion model — Climate Formula.)

(No single free-parameter count is published. For comparison, Laskar’s (1993) obliquity formula has ~6 free parameters and standard precession theory uses multiple fitted constants; the model’s inputs sit in the three ledgers above, and the planetary ledger has none at all.)

Calibration vs prediction

Any model with adjustable parameters can be made to fit data. A common criticism of phenomenological models is circular reasoning: if you tune parameters to match data, then cite that match as evidence, you have proven nothing — “if the model’s constants were found by fitting to observations, then claiming the model matches observations is meaningless.” The concern is legitimate, and the scientific question is whether the model predicts values that were not used in its construction. This section explicitly separates inputs from predictions.

What was calibrated (inputs)

These values were directly used to build the model:

What is predicted (NOT used in calibration)

These values are genuine predictions — they were NOT used to construct the model:

*The J2000 eccentricity, perihelion and inclination are not entered as values: they come out of the integrated J2000 state. Because that state is itself an input, these rows are consistency and convention checks rather than independent tests; the predictions proper are the rows at other epochs.

What is constrained (validation, not proof)

The climate-cycle question, answered honestly. The post-MPT ~100-kyr band was known to the author while the model was built, and it was part of the motivation for building it — so it cannot validate the model. The model does not attribute the band to Earth’s apsidal period: the climate formula carries it as the planets’ eccentricity beats read from the model’s own modes, and Earth’s own apsidal period is not a line of the formula (Climate Formula). The band is therefore neither an input to the formula’s line periods nor a prediction of them; the formula’s own status as description versus prediction is measured separately (T7, on the Climate Formula page).

The circularity concern

Matching an input proves nothing — it only confirms the construction worked. Not all comparisons are equal:

1. True inputs (the J2000 state and masses, the IAU general precession, sidereal year and J2000 obliquity, the LLR recession rate, the historical ΔT record): the model reproduces these by construction.
2. Structural choices (the composed form of the precession rate; bounded harmonic terms only in the ΔT stack, never a polynomial): these constrain what the model can do but fix no value.
3. Post-hoc validations (Mercury, planetary inclinations, the 1246 AD alignment, the eclipse record): these were checked against the built model.

What the model cannot claim: that matching J2000 values validates it (they were inputs); that matching the ~100k climate cycle validates it (it was known during construction); that any parameter fitted to data constitutes evidence. What it can claim: obliquity predictions at dates other than J2000 agree with Laskar; perihelion longitude predictions at dates other than J2000 agree with Meeus, and the series reaches the 1246 AD alignment without having been placed on it; eccentricity and inclination emerge from the integration without being entered as values. The model’s predictive power should be judged by how well it predicts values at dates other than J2000, by its testable predictions that differ from standard theory, and by whether its long-term predictions (eccentricity cycle, precession reversal) prove correct — NOT by how well it matches the data used to construct it.

Is “we are not at the mean” an escape hatch? The model’s J2000 periods are readings of quantities that wander over tens of thousands of years, and currently observed values differ because we are at a specific phase of that wander:

A valid concern is that any discrepancy could be attributed to “not being at mean.” It cannot: the model predicts specific values at specific dates (not just means) and predicts how those values change over time (specific rates), and those predictions can be compared to observations over decades.

What would constitute independent validation?

The model can be tested by observations that were not used in calibration, cannot be adjusted after the fact, and differ meaningfully from standard theory.

Strong tests:

Weak tests (similar predictions):

Independent historical verification. The 3D simulation is checked against over 8,000 verification data points from authoritative astronomical sources — NASA eclipse and transit catalogues, opposition and conjunction tables, mutual planetary events, and historical observations (7,384 of them Tycho Brahe’s Mars records). The set is graded by source quality, from 42 highest-quality model start positions and 76 precise transit observations down to 127 older or less precise entries. None of it was used to calibrate the model; it is independent evidence that the orbital mechanics produce correct positions across centuries. Full dataset: verification data reference .

Data sources

Primary sources

Secondary sources

Testable predictions

The model makes six predictions, each naming the observation that would falsify it. Full details: Predictions.

The model would be falsified if:

See Predictions: Verification Pathways for all six predictions with specific values and their verification pathways (the element turns, LOD, the withdrawn glaciation extrapolation, the withdrawn deep-time lunar track, the deep-time obliquity track, the deglacial spin-up).

Uncertainties and limitations

The year-by-year comparison of the published obliquity with La2004 is on Scientific Background §2.

Explicit assumptions

1. Stable solar system — the secular series assume orbital stability over the span they are used; chaotic divergence limits individual-phase predictability beyond ~50 Myr (Laskar et al. 2011).
2. Two-point model — the wobble center and Earth’s perihelion point are mathematical constructs.
3. Mean values exist — the model assumes precession rates oscillate around slowly-evolving means.
4. Bounded harmonic terms only — the fitted ΔT stack’s cycle terms are Fourier terms on named physical periods; no polynomial term is admitted.

What the model does NOT explain

• Why the J2000 rates have the values they do — the general precession rate and the year lengths are cited inputs, not outputs (their evolution through geological time is the model’s, per Expanding Resonance).
• What causes the two precession motions beyond the classical torques and perturbations the model composes.

Reproducibility

The model provides formulas to calculate obliquity, eccentricity, inclination, longitude of perihelion, and day/year lengths at any year — see Formulas Reference for ready-to-use expressions. The published obliquity tracks La2004 to 1.3″ rms over the last 100 kyr and 15.0″ rms over the last million years; the epoch-by-epoch tables are on Scientific Background §2.

Verify against external data

1. JPL Horizons (ssd.jpl.nasa.gov/horizons ) — query Earth’s orbital elements for any date; compare perihelion dates, eccentricity, longitude of perihelion.
2. Laskar’s Tables (A&A 1993 / La2004) — compare obliquity values.
3. Meeus’s Formulas (Astronomical Algorithms, 1998) — verify 1246 AD perihelion-solstice alignment; compare longitude of perihelion progression.
4. 3D Simulation (3d.holisticuniverse.com ) — visual verification of all precession movements; adjust date and observe changes in real-time.

How the model was derived (summary)

1. The seed: one J2000 heliocentric state (JPL Horizons) and the DE440 masses, integrated with the first-order relativistic correction — the planets’ orbits and Earth’s eccentricity, orbit plane and perihelion of date.
2. The clock: the IAU 2006 general precession at J2000, one circle per 25,771.58 years, composed through time from the solar and lunar torques on the measured recession history (the apsidal ratio reads 4.332 at J2000 — a reading, not a constant).
3. The year and day lengths: the IAU sidereal year and the general precession give the J2000 tropical year; the J2000 day is the SI day; the lengths of date follow the clock and the orbit.
4. The obliquity: the spin axis integrated about the series’ orbit plane from the IAU 2006 J2000 obliquity.
5. The check: Meeus’s formula places perihelion at 90° longitude (December solstice) in 1246 AD, and the longitude moves to 102.947° by 2000 AD, 12.947° in 754 years; the series reproduces the alignment, crossing at 1247.2 AD.

All other values (obliquity range, eccentricity range, day/year lengths at any epoch) follow from these inputs.

← Mercury Precession | Supporting Evidence →