Supporting Evidence

The model makes several claims that depart from established theory. This page collects external evidence — published results, unresolved problems, and recent observations — that independently align with the model. None of this evidence was used to develop the model, and the page opens with the invitation to test it.

This page establishes:

• The model is an open framework, with concrete invitations to replicate or contradict it — An open framework, Scope.
• The climate evidence: the 100-kyr band is one broad peak without eccentricity’s split structure, the 405-kyr term is largely absent, and the Cheng 2016 chronology confirms it without orbital tuning — §1, §10.
• The rotation evidence: the 2020-present speedup, the 1-Gyr day-length stall, and the paleo-day counts matched to 0.7 % across 0–380 Ma — §2, §3, §11.
• The dynamics evidence: stable secular eigenfrequencies, the beat structure of the five cycles, Saturn’s window-era retrograde, and the metronome and the obliquity from one N-body seed — §6, §8, §9, §14, §15.
• The whole catalogue in one table — Summary.

An open framework — invitation to test

The model is offered as a framework for testing scientific theories, not as a closed theory. Every observable in the published literature is reproduced (see the next section); the deliberate departures from current consensus are catalogued explicitly, each with a discriminating empirical test.

All data, formulas, and the interactive 3D simulation are publicly available under AGPL-3.0 on github.com/dvansonsbeek/3d . The deep-time chain, climate-formula fits, and the full verification pipeline are scripted end-to-end and reproducible from raw inputs.

We invite independent replication and contradicting evidence. The framework is most useful when others apply it to their own data and report what they find — including findings that falsify any of its claims. Concrete invitations:

• Mercury’s anomaly is RESOLVED to General Relativity — the model carries the relativistic advance as a derived supplement, gated against its own N-body engine; the once-proposed reference-frame reading is closed as a 0.6 % coincidence (see Mercury Precession). BepiColombo (April 2027) remains the shared consistency check and sharpens the solar-J₂ separation.
• Apply the climate formula’s orbital lines (the engine’s 28 beat lines) to a Devonian, Permian, or Cretaceous cyclostratigraphic spectrum. The two-tier prediction (Expanding Resonance) says the precession-band and obliquity lines should sit shortened on Earth’s lengthening clock while the eccentricity lines and the 405-kyr metronome stay at their modern periods; a section violating either half falsifies that tier.
• Reproduce the Wells 1963 / Wu 2024 paleo-day-count match independently — via a different paleontological technique (bivalves, rugose corals, novel growth-band counting), or a different cyclostratigraphic inversion method. The Devonian precession period of 21,699 yr is a one-parameter prediction with no Hadean constraint in the fit.
• Test the Lunar Precession Invariant (Prediction 4) via spectral analysis of tidal rhythmites sensitive to the apsidal period (~9.69 yr at Devonian, ~17.15 yr at −2.5 Gyr). The framework predicts T_apsidal × U ≈ 2,966,728 yr² at every epoch on the era-clock counter U.
• Find a violation of the eccentricity structure — a planetary system whose long-term eccentricity behaviour contradicts what the model’s N-body dynamics give for it.
• Replicate the historical eclipse validation on additional pre-1900 events outside the 26-event canonical set, or extend the 267-event Stephenson 2016 lunar timing analysis with newly recovered primary-source observations.

The framework treats contradicting evidence as the most productive form of engagement. Please open an issue at github.com/dvansonsbeek/3d/issues  or reach out via the Support page with results, questions, or proposed extensions.

Scope: what the model reproduces, where it differs

Before the section-by-section evidence, here is the explicit scope of the model. The first table lists quantities the model reproduces to within stated tolerances — cross-validated against standard ephemerides and published references. The second lists quantities where the model differs from current scientific consensus and proposes a different explanation.

Reproduces standard astronomy (cross-validated)

Differs from current science (the model proposes a different explanation)

Same data, different interpretation

The model uses every observation that Berger, Laskar, Meeus, Chapront, Souami, and others use. Where the model and standard science disagree about a specific value, it is the explanation that differs, not the data. The structure that emerges from this re-interpretation — Earth’s long cycles read against one derived spin clock, the planets on the model’s own N-body engine, and the two-tier deep-time scaling — is what gets new explanations.

Where the model and standard science disagree about a specific future or past value, that’s the falsifiable prediction. Several such discriminating tests are catalogued in Predictions.

Relationship to prior theoretical work

Every observation the model reproduces comes from established astronomy. The model adopts the work of the established authors below in full; where the interpretation of a specific observation differs, that’s flagged in the right column.

Bottom line: every observation is identical to standard astronomy. The reading of Earth’s cycles against one derived clock that lengthens with the tidal history is new, and the explanations for a few specific anomalies (Mercury, Saturn ecliptic-retrograde perihelion, 100-kyr cycle origin) are new. Everything else converges with established science.

1. The 100,000-Year Problem

The dominant ~100-kyr glacial cycle is one of paleoclimatology’s longest-standing open problems; the mainstream puzzle, its proposed mechanisms and the recent reviews are set out once, on Scientific Background §4. The model’s own evidence against direct Earth-eccentricity attribution is three measurements on LR04. The spectral shape is a broad single peak, not eccentricity’s split (95k+125k) structure (Muller & MacDonald 1997 ). The 405-kyr term is essentially absent post-MPT, at an amplitude ratio of 0.12 against the 100-kyr peak. And no bispectral 95k+125k phase coupling is detected: bicoherence 0.507, below the null 95th percentile of 0.555.

The model’s reading is canonical at Climate Formula. The 100-kyr band is one broad peak carried by the planetary eccentricity beats of the model’s own modes — g₄−g₅ (94.9 kyr), g₃−g₅ (98.8 kyr), g₃−g₁ (110.0 kyr) among the formula’s ten eccentricity lines. It is not Earth’s own apsidal precession (~111,635 yr), which is not a line of the formula. The MPT (~1 Ma) is a sensitivity change in the climate system, not a forcing change, quantified there as the obliquity-band ice-share shift.

Ice core dating methodology

The 100-kyr signal claim depends on the timescale being right. Modern ice-core chronologies use several methods that are largely independent of orbital tuning — which is what makes the LR04 vs Cheng2016 cross-check (below) meaningful.

Methods used in modern Antarctic and Greenland chronologies:

1. Annual layer counting — visual stratigraphy, seasonal chemistry, electrical conductivity. Precision: ±1% Holocene, ±2–3% glacial. Resolvable to ~60–100 kyr before layers compact below detection.
2. Volcanic markers — sulfate spikes and tephra (Toba 74 ka, Laacher See 12.9 ka, Campanian Ignimbrite 39 ka) provide absolute tie points.
3. Gas synchronisation — methane is globally synchronous within ~50 yr; links Greenland and Antarctic chronologies (±50–200 yr).
4. O₂/N₂ ratio dating (Kawamura et al. 2007 ) — trapped-air O₂/N₂ correlates with local summer insolation; provides an independent orbital constraint at the precession band (~23 ka), not the 100-kyr band.
5. Radiometric — ¹⁴C to ~50 ka; U-Th on synchronised speleothems to 640 ka (Cheng et al. 2016). Fully independent of orbital assumptions.
6. Orbital tuning — adjusts the chronology to match calculated insolation. Circular if used to test Milankovitch theory.

The circularity test: the orbitally-tuned LR04 stack and the U-Th-dated Cheng2016 speleothem record — chronologies built from completely independent timescales — place the dominant cycle in the same FFT bin (k = 6, centroid ≈ 107 kyr). The 107-kyr centroid is real, not a tuning artifact.

2. Earth’s Rotation Speedup (2020–present)

The model predicts that LOD’s growth is cyclically modulated by the four-line correction stack (Bond + Hallstatt + Jose5 + Jose4), plus the Core-mantle swing episode. LOD itself only grows, but its rate currently runs below the secular baseline — the descending stack phase. The modulation’s next trough falls near ~2,228 AD and its next peak near ~2,752 AD, with short-term fluctuations superimposed on the long-term tidal trend.

Starting in 2020, Earth began rotating faster than IERS predictions. 2020 saw the 28 shortest days since atomic-clock measurements began; July 5, 2024 set the all-time record at 1.66 ms under 24 hours. The IERS Directing Board has acknowledged trouble predicting more than 6–12 months ahead . The short-term speedup is qualitatively consistent with the model’s cyclical-LOD prediction; it is not yet evidence of the long-term trend reversal, which only continued observation over decades can confirm.

3. Day Length Stalled for 1 Billion Years

Mitchell & Kirscher (2023, Nature Geoscience 16, 567)  showed that Earth’s day length stalled at ~19 hours for roughly 1 billion years during the mid-Proterozoic (2.0–1.0 Ga) — atmospheric thermal tides balanced the decelerative torque of lunar oceanic tides. The implication for the model: complex, non-monotonic LOD dynamics are not unprecedented. If atmospheric tides could halt rotational slowing for a billion years, additional mechanisms can produce the cyclical millennial-scale variation the model proposes.

Tidal rhythmites: complementary evidence. Sedimentary records preserving ancient tidal cycles (“tidal rhythmites”) provide independent geological constraints on ancient Length of Day. They show discrepancies with simple tidal-deceleration models — additional evidence that complex rotational dynamics beyond monotonic lunar tidal slowing have shaped Earth’s rotation history. The model embeds a quantitative deep-time LOD evolution via ESSRT Driver 1 (Earth-Moon tidal evolution; Farhat 2022 polynomial — see Expanding Resonance), reproducing the Devonian / Cretaceous tidal-rhythmite day-count record to within ~1 %.

4. Solar Oblateness Uncertainty

The standard Mercury test of General Relativity carries a rarely discussed systematic: the Sun’s quadrupole moment J₂ varies on the solar activity cycle, and an unmodelled periodic component above 0.04 % of J₂ could falsely confirm or contradict GR in BepiColombo’s data. One home: Mercury Precession: Solar Oblateness Uncertainty.

5. BepiColombo: The Second Ranging Epoch

BepiColombo (orbit insertion 21 November 2026, routine science from April 2027) is the consistency check the model shares with General Relativity. The model’s prediction is GR’s, ~575.31″/cy again, since it carries the relativistic advance as its derived supplement. The mission also gives a second precise epoch, bounding any secular change of the advance and sharpening the solar-J₂ separation. One home: Mercury Precession: The BepiColombo Test.

6. Jupiter and Saturn Secular Eigenfrequencies Are Stable

The chain’s window rates for Jupiter’s and Saturn’s perihelion motions are window-epoch readings of the present era’s motion — exact for the window they name, with the true long-term means the secular eigenfrequencies (g₅ ≈ 426″/cy, g₆ prograde). Laskar’s secular theory confirms those eigenfrequencies are stable over ≥50 Myr, and the model’s own N-body engine reproduces them from one cited J2000 state (N-body Exploration). The apparent “pattern changes” that short-window ephemeris tools display are the Great Inequality’s window phase, not a change in the underlying dynamics — the model’s own engine agrees with that mechanism.

The Great Inequality: why short-window tools show “pattern changes”

Jupiter and Saturn sit in a near 5:2 mean-motion resonance, producing the Great Inequality — a ~883-year quasi-periodic oscillation first identified by Kepler and explained by Laplace in 1786. The oscillation amplitude in Saturn’s longitude of perihelion is large enough to dominate century-scale fits:

Saturn’s fitted rate changes sign between the two intervals. JPL’s own documentation cautions: “The elements are not valid outside the given time-interval over which they were fit.” WebGeocalc and similar tools that display the raw ephemeris over centuries will show these Great-Inequality oscillations as apparent trend changes.

Secular theory: the long-term dynamics are stable

Laskar’s orbital solutions (La2004 , La2010 ) decompose planetary eccentricity into eigenmodes g₁–g₈. Each planet’s perihelion motion is a superposition — for Jupiter, g₅ dominates with significant g₆ admixture (~2.8:1); for Saturn, g₆ dominates but g₅ contributes nearly as strongly (~1.4:1), which is why Saturn’s instantaneous rate can reverse sign. Laskar found that g₅, g₆, and s₆ are practically stable over at least 50 Myr — the giant-planet system is far less chaotic than the inner planets, and the average precession trends do not change.

Brouwer & van Woerkom (1950)  showed that eliminating the Great Inequality from the Hamiltonian introduces small correction modes (g₉, g₁₀) that modify but do not destabilise the fundamental rates. Laplace’s 1786 proof that the Great Inequality is truly periodic — averaging to zero over ~10 full cycles — closes the argument: the oscillation does not produce a net secular drift.

Independent cross-validation: Laskar and the model converge

The model’s nodal period and obliquity period are read from its own N-body integration — the dominant nodal mode s₃ of Earth’s orbit from a 20-Myr run seeded by one cited J2000 state, and the obliquity history integrated on it. The derivation makes no use of Laskar’s numerical secular theory or LR04 spectral analysis. A dated record does not see the J2000 beat (~41,098 yr) but the beat of the pole rate’s mean over the record’s window. The spin turned faster in the past, so the mean obliquity was lower and the mean eccentricity higher. Over the LR04 record the pole rate therefore averages 50.517 ″/yr against 50.385 at J2000, and the realized period reads ~40,927 yr. Fitting each obliquity history over that window with the full line set p + sᵢ and one free p, like for like, two independent integrations converge:

The two integrations agree to better than 0.01 % on the nodal mode and to +0.05 % on the realized obliquity period. The empirical LR04 peak lands within one Rayleigh resolution element of both. (Laskar’s precession frequency, 50.476 ″/yr, is a long-window frequency of his solution, not a J2000 rate; a J2000 beat must not be compared with it.) This is convergence, not calibration: the model does not consult Laskar’s eigenfrequencies.

A second independent cross-validation comes from Jupiter’s inclination trend. Using the chain’s window rate for Jupiter’s ecliptic perihelion (a window-epoch reading) produces an inclination-trend error of ~3 arcsec/century vs JPL’s observed values. Using the secular theory’s ~305,000-year g₅ period instead jumps the error to ~8.5 arcsec/century — nearly 3× worse. The window rate outperforms the standard g₅ period on independent JPL data — read as a window-epoch consistency test: the trend baseline is a few centuries, so it adjudicates the present-era rate, not the long-term period (quantity types per 3d doc 109 §9).

7. Obliquity Amplitude: Berger (1978) Dominant Term

The model’s obliquity is derived from one precession equation — the spin axis precessing about the orbit normal taken from its own N-body nodal modes — and reproduces La2004 to 50″ rms over the last 13,000 years with zero fitted constants (Obliquity). Its dominant period is the beat of the precession clock against the nodal mode s₃, and Berger’s standard Fourier decomposition names the same beat as the dominant term.

Berger’s dominant term

Berger (1978) decomposed Earth’s obliquity into 47 quasi-periodic terms. The dominant term (frequency s₃ + k) has:

The derived obliquity carries this amplitude natively, by tracking La2004; it is not a fitted number in the model. The five largest terms:

The dominant term is roughly 3× the next; Berger & Loutre (2001) and Laskar et al. (2004) confirm the value in updated solutions. The frequency s₃ + k matches the same two counter-rotating motions the model identifies: the spin axis’s inertial precession k (50.385 ″/yr at J2000, prograde — the general precession of the equinox plus the planetary precession) and the s₃ eigenmode for Earth’s orbital plane (~−18.85″/yr, retrograde). Because s₃ runs retrograde the two rates subtract, 50.385 − 18.85 ″/yr, which is the ~41,098-yr J2000 obliquity beat directly — the same beat 2π/(ψ̇ − |s₃|) the model carries everywhere else, and whose window mean a dated record sees (~40,927 yr over LR04).

No published work explains from first principles why the dominant amplitude is ~0.684°. It emerges from the full coupled spin–orbit–planetary perturbation solution; without the Moon’s stabilising torque obliquity could vary chaotically between 0° and 85° (Laskar, Joutel & Robutel 1993 ).

The climatic-precession peaks. Berger’s climatic-precession spectrum (e × sin ϖ̄) has multi-peak structure across ~19–24 kyr with no single dominant term; each peak is the precession rate k plus a planetary apsidal mode gⱼ. The shipped climate formula carries those peaks directly as its five p+gᵢ lines (19.0–23.8 kyr), with k the model’s own clock rate and the gᵢ its own apsidal modes (Climate Formula).

References:

• Berger, A. (1978). “Long-term variations of daily insolation and Quaternary climatic changes.” J. Atmos. Sci., 35, 2362–2367.
• Berger, A. & Loutre, M.F. (2001). “Amplitude and Frequency Modulations of the Earth’s Obliquity.” J. Climate, 14(6), 1043–1054.
• Laskar, J. et al. (2004). “A long-term numerical solution for the insolation quantities of the Earth.” A&A, 428, 261–285.
• Laskar, J., Joutel, F. & Robutel, P. (1993). “Stabilization of the Earth’s obliquity by the Moon.” Nature, 361, 615–617.

8. Milankovitch Beat Frequency Structure

Standard orbital mechanics (Vervoort et al. 2022 ) derives two of the five Milankovitch cycles as beat frequencies of the others:

Obliquity period:             1/P_axial  −  1/P_nodal    =  1/P_obliquity
Perihelion precession period: 1/P_axial  +  1/P_apsidal  =  1/P_perihelion

The model states the same two closures as frame arithmetic on four of its dynamical periods. The clock T_p is ~25,771 yr, the equinox along the ecliptic of date. The spin axis’s inertial period T_ψ̇ is 25,722 yr — the pole itself, 0.097 ″/yr faster, which is the planetary precession. The apsidal period T_aps is ~111,635 yr and the nodal period T_s₃ is ~68,751 yr, the last two read from its own N-body modes. The obliquity, a pole-to-pole angle, beats on the pole’s rate; the perihelion of date, a rate along the ecliptic, on the equinox’s:

Three inputs — the derived clock and two orbital modes — give the five Milankovitch periods; the identities hold at every epoch because they are frame arithmetic, with the beats lengthening on the clock at deep time while the orbital modes stay put.

References:

• Vervoort et al. 2022, AJ 164, 130 
• Laskar et al. 2004, A&A 428, 261 

9. Saturn’s Ecliptic-Retrograde Perihelion Precession

A clearly-observed phenomenon: Saturn’s longitude of perihelion moves retrograde in the ecliptic frame at the current epoch, opposite to orbital motion, while secular perturbation theory’s long-term mean is prograde for all planets. The model’s own N-body engine settles the attribution in the standard explanation’s favour: the retrograde is the window phase of the ~900-year Great-Inequality epicycle, and the long-term mean is prograde (g₆ ≈ +2,824″/cy) (N-body Exploration).

The observation

JPL’s WebGeoCalc  shows Saturn’s longitude of perihelion (ϖ = Ω + ω) decreasing over 1900–2000 AD at ~-3,400″/century, and JPL’s Keplerian elements (Standish & Williams 1992 ) give dϖ/dt = −0.419°/century in the 1800–2050 fit. The window dependence sits in ω, not Ω: across JPL’s two fit windows the ascending node stays retrograde (−0.289°/cy over 1800–2050, −0.250°/cy over 3000 BC–3000 AD) while the argument of perihelion reverses (−0.130°/cy against +0.792°/cy), and ϖ̇ with it (−0.419 against +0.542°/cy). The Great Inequality moves the eccentricity eigenfrequencies (g-type, the apsides), not the inclination eigenfrequencies (s-type, the node).

The explanation and the model’s reading

Laplace’s ~900-year oscillation from the near-5:2 Jupiter–Saturn resonance (P_S/P_J = 2.483 against 2.500) is a transient phase on a prograde secular rate of about +19.5″/yr, dominated by g₆. Fitzpatrick  and Murray & Dermott quote +22.44″/yr, while the model’s own N-body reads 28.246″/yr, as do the full eight-planet secular solutions. Those are different truncations, not a disagreement. Prograde is a structural requirement, not a choice: the Laplace–Lagrange coupling matrix has all eight eigenvalues positive (g₁ = +5.59″/yr through g₈ = +0.67″/yr), so no eigenmode can give permanent retrograde apsidal motion, and a retrograde reading must be a periodic perturbation. The model’s own engine reproduces exactly that: the retrograde is the epicycle’s window phase, the long-term mean is prograde, and the fitted window rate is span-dependent (−1,600″/cy over the 300-yr fit against ~−3,400 over century sub-windows). Earth’s obliquity beat is driven by its own orbit’s nodal mode s₃, not by any Jupiter–Saturn coupling. At the milliarcsecond level the modern ephemerides (DE440, EPM2017, INPOP19a) report no significant residual beyond the standard prediction — a different question from the window phase.

The model’s numbers in the two coordinates:

In the observers’ coordinate the chain’s rate -1,576.7″/cy is the number to compare with WebGeoCalc; the Earth-frame RA value is larger by Saturn’s projection factor and is not. The 3D simulation  renders the window-era ecliptic-retrograde motion from the same chain.

10. Cheng 2016 Cross-Proxy Validation

A strong structural test of the climate formula’s orbital line set is whether it fits a paleoclimate record built on a completely independent chronology and recording a different physical mechanism. The Cheng et al. 2016 Asian Monsoon δ¹⁸O record (U-Th-dated speleothems, 0–640 kyr) is the cleanest such test.

The same orbital line set that fits LR04 at R² = 0.780 (post-MPT regime) and 0.92 (stitched three-regime fit across the full 5.3-Myr window) also fits Cheng 2016 Asian Monsoon δ¹⁸O at R² = 0.64 (the shipped 28-line formula, L1 layer only, over the matched 0–640 kyr window). Per-line amplitudes differ between the two records — monsoon strength and ice volume are sensitive to different beats, and inside a 640-kyr window the obliquity and eccentricity clusters are not resolved line by line — so only the fit as a whole is comparable.

The structural agreement is independent of the orbital-tuning question. Cheng 2016’s U-Th chronology is built from radiometric ages of speleothem layers and carries no insolation assumption; if Cheng’s record is well-fitted by the same line set — even with different per-line amplitudes from LR04’s — those line positions cannot be tuning artifacts. The orbital line set is the structure that survives the cross-proxy translation. §1 above made the same point qualitatively for the 100-kyr centroid alone; the R² = 0.64 result extends it to the full line set on a different physical recording mechanism.

Method at Climate Formula. The lines’ time-evolution layer at Expanding Resonance.

11. Paleo-Day-Count Validation

Earth’s precession clock is a spin quantity: the composed lunisolar rate scales with the rotation rate ω(t), so the day-length history and the deep-time clock are one channel (Earth’s Clock). The proper-physics two-layer LOD formula — derived from angular-momentum conservation applied to Farhat 2022’s Moon-distance polynomial fit — predicts days per year at any past geological epoch. The predictions are testable against direct paleontological day-counts (coral growth rings, bivalve daily increments, tidal rhythmites) preserved in the fossil record.

Independent multi-source validation (0–620 Ma)

Phanerozoic match (0–380 Ma): all tabulated points within 0.7 %, with a mean absolute deviation of ~0.3 % — and the flagship Wells 1963 Devonian anchor essentially exact (−0.01 %). The framework reproduces the directly-counted fossil record across the Phanerozoic without any free parameters fit to day-count data. The full Wells 1963 series (extracted via Arbab 2001 review) across nine geological stages 65–600 Ma matches within 0.3 % at every Phanerozoic stage and within 0.7 % at 600 Ma.

The mid-Precambrian record (1–3.5 Ga)

Beyond the cyclostratigraphic era above, the regime-aware recession history (Expanding Resonance §Driver 1½) is tested against the published mid-Precambrian record — cyclostratigraphy, tidal rhythmites and tidal bundles:

Every anchor is reproduced within its published uncertainty. The two solar angular-momentum channels — the ocean solar-tide leak and the insolation-driven thermal-tide pump — are part of this fit; the pump’s mechanism is debated in the literature (Mitchell & Kirscher 2023 vs Zhou et al. 2024) and the fit lets the data decide.

The complete anchor set — 41 published measurements with per-anchor tolerances, including the documented deviations — is machine-checked in the public repository on every change. The Deep-Time Validation Dossier  and its paleo-anchors verification gate recompute every prediction from the engine. A documented miss that silently improves fails the check too, because it would mean the formula changed.

Williams 2000 (620 Ma) discrepancy

The Williams 2000 Elatina tidal-rhythmite measurement (400.3 days/yr at 620 Ma) sits +5.70 % below the framework’s prediction (423.11 days/yr). This is a known small-epoch discrepancy of the smooth two-layer formula: Farhat 2022’s globally-smoothed ocean-tidal-Q curve dips shallower than Williams’s direct rhythmite count suggests, possibly because the Ediacaran-Cryogenian Snowball Earth interval (~720–635 Ma) had unusual ocean-tidal dissipation that the smooth fit averages over. The Mitchell-Kirscher 2023 thermal-tide-lock framework places the transition out of the Proterozoic tidal-resonance regime at this same interval — see §3 above on the 1-billion-year day-length stall.

For Phanerozoic work (≤500 Ma) the proper-physics formula is uniformly better than a linear LOD approximation. For Snowball-boundary epochs the discrepancy is documented honestly rather than papered over.

The proper-physics two-layer formula, its calibration against Farhat 2022, and the full Hadean back-projection (Moon at 1.48 R_E at Patterson’s Pb-Pb Earth age of 4.54 Gyr) are canonical at Expanding Resonance.

12. The Mass Counterfactual

A numerical fit like DE440 and a published integration like La2010 can each tell you what the solar system does — neither can tell you what it would do if a constant were different. This model can, because its planets hang off one causal chain: constants → N-body engine → governed artifact → every rendered and published element. That chain is measured, not asserted.

The experiment (re-runnable in seconds: tools/explore/k6-mass-counterfactual.mjs in the simulation repo ): set GM_Jupiter × 1.01 and integrate the 1800–2100 window fresh, twice — baseline and counterfactual — with an independent readout.

• The baseline closes on the shipped artifact. The freshly integrated window rates reproduce the governed artifact’s banked values (worst case 0.02% relative) — the elements every surface displays are exactly what an integration of the constants gives, with no adjustable layer in between.
• Mercury answers +1.536 ″/cy — first-order secular theory expects ≈ +1.54. One percent of Jupiter’s classical Laplace–Lagrange share of Mercury’s precession, landed on without any fitting.
• The whole system responds in character. Saturn’s rate goes more retrograde (a heavier Jupiter deepens the Great-Inequality window phase), Uranus barely moves (its perihelion rides Jupiter’s own g₅ mode, which shifts with its carrier), and near-circular Neptune’s apse swings hypersensitively — each response the dynamics demands.

The same engine, run against Laskar’s La2010 on Earth’s inclination to the invariable plane over the last 500,000 years, agrees to 0.003° RMS — two independent integrations from nothing but the J2000 state vectors and the DE440 mass ratios. The injection plumbing is enforced by a permanent gate (test:counterfactual): a perturbed artifact must change the evaluated elements, untouched planets must stay bit-identical, and the counterfactual must reproduce.

13. The Rendered Planets vs JPL — Published, Not Tuned

The seven planets in the 3D simulation  are rendered from the N-body element chain alone — zero observation-fitted terms — so comparing the rendered sky against the JPL Horizons ephemeris measures the model’s raw dynamics. The comparison is published either way it falls; nothing is tuned to it. Measured over ~108,000 cached JPL samples (joint RA+Dec RMS, banked as a gate-guarded artifact in the simulation repo):

Two readings. First, the reference-century numbers — tens of arcseconds — come from dynamics with no fitted display layer, and the extrapolation centuries degrade gently rather than unravelling: the chain extrapolates. Second, the residuals that remain are themselves published model content: the difference between the model’s causally-derived system and the fitted ephemeris is part of what the model claims, not something to be calibrated away.

14. The Deep-Time Metronome from Gravity Alone

The geological record carries a famous clock: the ~405-kyr long-eccentricity cycle (the g2−g5 beat), phase-coherent in rocks over hundreds of millions of years. A pre-registered test asked whether the model’s own N-body dynamics — integrated over ±20000000 ÷ 2 years from the JPL J2000 state with zero observation-fitted terms — produce that metronome as their strongest deep-time eccentricity line, with the ~124-kyr and ~95-kyr companions present. The registered acceptance window was 395–415 kyr, a criterion an epoch-local single-line law can never meet.

Measured result: the strongest line of the engine’s deep Earth eccentricity spectrum is 405.6 kyr, with companions at 123.8 kyr and 94.9 kyr, correctly ranked. For reference (labelled as theory, never used as input): the same beat computed from Laskar 2004’s published frequencies is 405.7 kyr; the rock-record value is 405.6 kyr, and the rock value remains the model’s falsification reference. The engine’s g5 (Jupiter’s mode, 4.2574″/yr) and g2 (Venus’s mode, 7.4524″/yr) both match Laskar 2004’s published values (4.257 and 7.452″/yr) to four significant figures. The Moon’s quadrupole term in the integration is what places g2 there: with the Earth–Moon pair merged into a single point mass the mode reads low and the beat long.

The verdict, mode tables and run diagnostics (energy conservation 1.3e-8) are banked as a gate-guarded artifact in the simulation repo; every number on this page reads live from it.

15. Earth’s Obliquity from One Equation

Earth’s axial tilt oscillates — the ~41,000-year cycle that paces the ice ages. The model derives that history from a single averaged precession equation with zero fitted constants. The spin axis precesses about the moving orbit normal, and that orbit normal comes from the model’s own N-body node modes — the same ±10-Myr run as §14. The only anchor is the model’s J2000 axial-precession rate, its certified year laws’ beat, which gives a precession constant of 54.811″/yr against a literature value of ~54.9.

Measured results. The derived obliquity rate at J2000 is -48.00″/cy against the IAU reference of -46.84″/cy — a quantity the simulator’s scene machinery previously had to fit now falls out as a derivation. The dominant obliquity beat emerges at 41.1 kyr — the 41-kyr band — as the difference between the axial-precession rate and the strongest nodal mode (s₃) — the model’s obliquity beat, produced natively by the dynamics. Against Laskar 2004 (labelled as theory): 50″ rms over the last 13,000 years — an order of magnitude better than the fitted harmonic law’s 530″. That is the derivation’s own fit; the published obliquity adds the solar torque at the eccentricity of date and the ice-age oblateness and tracks La2004 to 1.2″ rms over the last 100,000 years and 13.5″ rms over the last million years (Obliquity).

Together with §14 this closes a loop. The deep e-spectrum (the 405-kyr metronome) and the obliquity band (the 41-kyr cycle) are the two orbital pacemakers of the climate record, and both emerge from the model’s own gravity. The model supplies exactly one number each time: the seed’s dynamics, and the derived J2000 precession rate.

16. The Solar System’s Spin Landscape on the Model’s Own Frequencies

The spin histories of the planets hang on a handful of nodal eigenfrequencies — and the model’s own N-body chain derives that frequency table from one J2000 seed, with zero fitted constants. Reading it against the observed spin states (all citations; never model inputs) reproduces the landmark results of the field in one table:

The Earth row carries the deepest point: remove the Moon (its two-thirds of the precession constant) and Earth’s spin rate falls into the band — the obliquity envelope doubles in the model’s own integration. The famous “the Moon stabilizes Earth’s climate” result becomes a statement about this model’s own frequency table — and it is the same frequency table that produces the 405-kyr metronome (§14) and the 41-kyr obliquity band (§15). One engine, one seed: the climate pacemakers and the solar system’s spin-stability map are a single derived object read twice.

(The individual mechanisms are established literature — Ward & Hamilton 2004; Saillenfest et al. 2020; Laskar & Robutel 1993; Margot et al. 2007. What is the model’s contribution is deriving the frequency table they all hang on from one seed with zero fitted constants, inside the same engine that carries the eclipse record and the deep-time climate spectrum.)

Summary

For the model’s specific predictions, see Predictions. For the underlying mechanism, see Climate Formula.

← Mathematical Foundation | Predictions →