arxiv: 2604.03049 · v2 · submitted 2026-04-03 · 🌌 astro-ph.EP

Recognition: 2 theorem links

· Lean Theorem

A 1151-Year Quasi-Commensurability of the Solar System: Empirical Detection, Statistical Characterization, and the Anomalous Exclusion of Uranus

Carlos Baiget Orts

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:00 UTC · model grok-4.3

classification 🌌 astro-ph.EP

keywords quasi-commensurabilitysolar systemplanetary longitudesephemerisUranusVenus returnorbital synchronism

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The pith

Seven planets return to nearly identical positions every 1,151 years while Uranus deviates by a third of an orbit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies 420,403 days, or roughly 1,151 years, as the interval that minimizes the average angular displacement among the heliocentric longitudes of Mercury, Venus, Earth, Mars, Jupiter, Saturn, and Neptune when measured in the DE441 ephemeris. At this specific spacing the seven planets shift by a mean of 13.4 degrees with a standard deviation of only 0.65 degrees that holds across century-scale windows and remains stable when the starting epoch is shifted over 1,200 years. The same interval produces no comparable result for any integer sub-multiple and leaves Uranus displaced by 108.3 degrees, an amount far larger than Neptune's 5.2-degree residue. This sharp difference between the two ice giants is presented as an independent empirical marker consistent with an early impact that altered Uranus's orbital period.

Core claim

An exhaustive search finds T* = 420,403 days as the global minimum of a series-comparison similarity metric applied to daily heliocentric ecliptic longitudes of seven planets over more than 2,600 years of ephemeris data. At T* the mean simultaneous angular displacement of Mercury through Neptune is 13.4 degrees with a standard deviation of 0.65 degrees sustained over a century-long window and stable across 1,200 years of reference epochs. Seven planets participate in the alignment while Uranus alone shows a sidereal residue of -108.3 degrees.

What carries the argument

The series-comparison similarity metric that ranks candidate intervals by the mean simultaneous angular displacement of the seven planetary longitude time series.

Load-bearing premise

The chosen similarity metric and the decision to treat Uranus separately truly isolate a physically meaningful quasi-commensurability rather than a statistical feature of the finite ephemeris dataset.

What would settle it

Compute the actual heliocentric longitudes of the seven planets 420,403 days after a reference epoch and check whether their mean displacement remains near 13.4 degrees while Uranus's displacement remains near 108 degrees.

Figures

Figures reproduced from arXiv: 2604.03049 by Carlos Baiget Orts.

**Figure 1.** Figure 1: Score S(T) for all candidate intervals in [−1,300, +1,300] years. The vast majority of candidates cluster between 50◦ and 140◦ . The two candidates at ±1,151 yr stand far below all others [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: Score distribution. The single candidate with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Daily angular offset δk(t) = λk(t) − λk(t − T ∗ ) for each planet over the 100-year comparison series. Upper panel: Mercury, Venus, Earth, and Mars over 5 years (daily resolution). Dashed lines show the mean offset for each planet. Lower panel: Jupiter, Saturn, and Neptune over 100 years (weekly resolution). Neptune’s offset is nearly constant (σ = 0.10◦ ), consistent with its small sidereal residue of −5.… view at source ↗

**Figure 4.** Figure 4: Convergence of the result with series length. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Heliocentric ecliptic positions at epochs 50, 200, 400, 600, and 800 CE (filled [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

read the original abstract

We report the empirical detection of a multi-planet quasi-commensurability in the Solar System and identify an anomalous exclusion that may bear on the dynamical history of Uranus. An exhaustive search identifies T* = 420,403 days (approx. 1,151 years) as the global minimum of a series-comparison similarity metric applied to daily heliocentric ecliptic longitudes of seven planets -- Mercury, Venus, Earth, Mars, Jupiter, Saturn, and Neptune -- computed from the DE441 ephemeris over +/-1,300 years. At this interval, the mean simultaneous angular displacement of all seven planets is 13.4 degrees, with a standard deviation of 0.65 degrees sustained over a century-long window and stable across 1,200 years of reference epochs. T* ranks first among all 2,600 candidates, with a gap of 1.09 degrees to the second best. No sub-multiple produces a comparable result. Seven of the eight planets participate in the synchronism. The sole exception is Uranus, whose sidereal residue at T* is -108.3 degrees -- nearly one-third of a full orbit -- while Neptune's residue is only -5.2 degrees, one of the smallest among all seven planets after Earth's. This sharp asymmetry between the two ice giants constitutes an independent empirical signature consistent with the hypothesis that Uranus's orbital period was substantially modified by a catastrophic early impact. The interval 1,151 years was identified by Babylonian astronomers as the Venus return period (de Jong 2019); the present work shows it is simultaneously optimal for six additional planets. Source code and data are publicly available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

1151 years is the best alignment period in the DE441 data for those seven planets, but the Uranus exclusion claim needs a null test before it supports an impact story.

read the letter

The main finding is that 1151 years minimizes a series-comparison metric on daily heliocentric longitudes for Mercury through Neptune (skipping Uranus), giving a mean 13.4-degree displacement with 0.65-degree scatter that holds across a century window and stays stable when you shift the reference epoch by up to 1200 years. The search covers 2600 candidates and shows a 1.09-degree gap to the next best period, with no sub-multiple working as well. This builds directly on the known Babylonian Venus return and adds the simultaneous optimality for the other six planets plus the quantified Uranus-Neptune asymmetry at that interval. The public code and DE441 data are a clear plus; anyone can rerun the minimization and check the residues. The asymmetry itself is straightforward in the numbers: Uranus sits at -108 degrees while Neptune is at -5 degrees, one of the tightest in the set. That part of the result is reproducible from the ephemeris. The soft spot is the missing null test. The paper does not show what the distribution of minima looks like under phase randomization or under a metric applied to shuffled longitudes, so the ranking gap could still be an artifact of the 2600-year window and the particular aggregation of angular differences. The early-impact interpretation for Uranus is presented as consistent with the asymmetry but stays speculative without any orbital evolution runs or comparison to other proposed mechanisms. The work is aimed at researchers who track long-term commensurabilities or who already use ephemeris searches for resonance studies. A reader who wants to see whether this period survives stricter statistical controls would find it worth their time. It deserves peer review so the metric definition and significance estimates can be examined in detail.

Referee Report

3 major / 2 minor

Summary. The manuscript reports the empirical identification of a 1,151-year quasi-commensurability T* = 420,403 days in the Solar System. An exhaustive search over 2,600 candidate intervals applied to daily heliocentric ecliptic longitudes from the DE441 ephemeris identifies T* as the global minimum of a series-comparison similarity metric for the seven planets Mercury, Venus, Earth, Mars, Jupiter, Saturn, and Neptune. At this interval the mean simultaneous angular displacement is 13.4° with standard deviation 0.65°, stable across 1,200 years of reference epochs and a 1.09° gap to the second-best candidate. Uranus is excluded from the set because its sidereal residue is -108.3°, interpreted as possible evidence for an early catastrophic impact that modified its orbital period; the interval is noted to coincide with the Babylonian Venus return period. Source code and data are stated to be publicly available.

Significance. If the central claim is robust, the work would constitute a notable empirical detection of a multi-planet synchronism spanning the terrestrial and giant planets, with potential implications for Solar System dynamical history and a link to ancient astronomical records. The public availability of code and data is a clear strength that facilitates verification. However, the absence of a fully specified metric and statistical null tests currently limits the assessed significance, as the result could reflect properties of the finite ephemeris or the chosen subset rather than a dynamical signature.

major comments (3)

[Methods] The exact functional form of the series-comparison similarity metric, including aggregation of angular residues, any normalization, weighting across planets, or invariance properties, is not provided. This definition is load-bearing for the claim that T* is the unambiguous global minimum among the 2,600 candidates and for the reported 1.09° gap.
[Results and Statistical Characterization] No Monte Carlo simulation, analytic significance estimate, or null test against phase-randomized or shuffled ephemeris realizations is supplied to assess the probability that an equally deep and stable minimum arises by chance within the finite ±1,300-year search window. This is required to substantiate that the result is not an artifact of dataset length or metric choice.
[Discussion] The a priori exclusion of Uranus from the seven-planet set and the interpretation of its -108.3° residue (versus Neptune’s -5.2°) as evidence for a catastrophic early impact lack supporting dynamical simulations or tests against selection bias. The claim that this asymmetry constitutes an independent empirical signature therefore rests on an untested assumption about the metric’s behavior under different planet subsets.

minor comments (2)

[Methods] The range, step size, and exact number of the 2,600 candidate intervals should be stated explicitly in the Methods section to allow full reproduction.
[Figures] Figure captions and axis labels for any plots of angular displacement versus epoch or candidate period would benefit from additional detail on the plotted quantity and error bars.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below with point-by-point responses, indicating where revisions will be made to the manuscript.

read point-by-point responses

Referee: [Methods] The exact functional form of the series-comparison similarity metric, including aggregation of angular residues, any normalization, weighting across planets, or invariance properties, is not provided. This definition is load-bearing for the claim that T* is the unambiguous global minimum among the 2,600 candidates and for the reported 1.09° gap.

Authors: We agree that the precise definition of the similarity metric must be stated explicitly. The metric computes, for a candidate interval T, the mean absolute angular difference (modulo 360°) between each planet's heliocentric ecliptic longitude at epoch t and at t+T, averaged uniformly across the seven planets with no differential weighting or additional normalization. The resulting value is invariant to the absolute reference epoch within the tested stability window. We will insert the full mathematical definition into the Methods section of the revised manuscript. revision: yes
Referee: [Results and Statistical Characterization] No Monte Carlo simulation, analytic significance estimate, or null test against phase-randomized or shuffled ephemeris realizations is supplied to assess the probability that an equally deep and stable minimum arises by chance within the finite ±1,300-year search window. This is required to substantiate that the result is not an artifact of dataset length or metric choice.

Authors: We acknowledge the desirability of formal null testing. However, the exhaustive enumeration of 2,600 intervals, the persistence of the minimum across 1,200 years of reference epochs, and the 1.09° gap to the second-best candidate already supply empirical evidence that the result is not a trivial artifact of the finite window or metric. A comprehensive Monte Carlo study with phase-randomized ephemerides lies beyond the scope of the present empirical report and would constitute a separate investigation. We will expand the Results section to discuss these robustness indicators more explicitly while noting the limitation. revision: partial
Referee: [Discussion] The a priori exclusion of Uranus from the seven-planet set and the interpretation of its -108.3° residue (versus Neptune’s -5.2°) as evidence for a catastrophic early impact lack supporting dynamical simulations or tests against selection bias. The claim that this asymmetry constitutes an independent empirical signature therefore rests on an untested assumption about the metric’s behavior under different planet subsets.

Authors: Uranus was excluded only after the metric was evaluated; its inclusion raises the mean displacement by more than 10°. We present the -108.3° versus -5.2° asymmetry as an empirical observation that is consistent with an early impact hypothesis, not as conclusive proof. We will revise the Discussion to frame this strictly as a hypothesis, remove any stronger causal language, and note that dedicated N-body simulations are required to test the impact scenario and to quantify possible selection effects. The uniform application of the metric to all planets is already documented. revision: partial

standing simulated objections not resolved

Full N-body dynamical simulations required to test whether an early giant impact could have altered Uranus's orbital period to produce the observed -108.3° residue.

Circularity Check

0 steps flagged

No significant circularity: empirical minimization on external ephemeris

full rationale

The central claim identifies T* = 420403 days as the global minimum of a series-comparison similarity metric applied directly to daily heliocentric ecliptic longitudes from the independent DE441 ephemeris over a fixed +/-1300-year window for the pre-chosen seven-planet set. This is a numerical search procedure against external astronomical data rather than any derivation, equation, or parameter fit that reduces to its own inputs by construction. No self-citations are load-bearing, no ansatz is smuggled, and the result is not renamed from a known pattern or forced by uniqueness theorems. The a-priori exclusion of Uranus is an explicit search choice whose consequences are then reported, but the minimization itself does not presuppose the outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the accuracy of the DE441 ephemeris over the examined interval and on the unstated definition of the similarity metric; no free parameters are explicitly fitted in the abstract, but the interpretation of the Uranus residue as evidence of impact is an additional assumption without independent corroboration here.

axioms (1)

domain assumption DE441 ephemeris provides accurate daily heliocentric ecliptic longitudes for the seven planets over +/-1300 years
All metric calculations are performed on this dataset

invented entities (1)

catastrophic early impact modifying Uranus's orbital period no independent evidence
purpose: To explain the large sidereal residue of Uranus at T* compared with Neptune
Postulated on the basis of the -108.3 degree versus -5.2 degree asymmetry; no independent falsifiable prediction is supplied in the abstract

pith-pipeline@v0.9.0 · 5611 in / 1576 out tokens · 97650 ms · 2026-05-13T18:00:09.656286+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define a rigorous similarity metric based on series comparison... S(T) = δ(T) + σ_δ(T) where δ is the mean... circular angular distance
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

T* = 420403 days... global minimum... seven planets... Uranus residue −108.3°

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Comet 1P/Halley Completes 15 Orbits in 1,151 Years: Commensurability with the Solar System Quasi-Period and Evidence for Jupiter-Saturn Dynamical Coupling
astro-ph.EP 2026-05 unverdicted novelty 4.0

Comet Halley exhibits a precise 15:1 commensurability with a 1,151-year solar-system quasi-period, with Jupiter and Saturn providing coherent perturbation cancellation over that baseline.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

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