arxiv: 2605.02239 · v1 · submitted 2026-05-04 · 🌌 astro-ph.EP · astro-ph.IM· physics.hist-ph

Recognition: 3 theorem links

· Lean Theorem

Mean tropical year length at arbitrary ecliptic longitude

Daniel Quigley

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:55 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IMphysics.hist-ph

keywords tropical yearmean solar yearecliptic longitudecalendar leap rulessecular driftequinoxsolsticecross-quarter

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The pith

The mean tropical year length varies with ecliptic longitude, giving eight distinct mean years for tuning calendar leap rules.

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

This paper computes the average time for the Sun's apparent geocentric longitude to return to each of eight fixed values spaced 45 degrees apart, averaged over a multi-millennium span. These intervals define separate mean years tied to the cardinal points and cross-quarter days. The work also derives the secular drift equation, which shows that the gradual shortening of the tropical year produces quadratic cumulative error in any fixed leap rule, reaching one full day in about 57,000 years.

Core claim

We compute the mean interval between successive returns of the apparent geocentric solar longitude λ to a fixed value L ∈ {0°, 45°, 90°, …, 315°}, averaged over a multi-millennium window; this gives eight “mean years” against which calendar leap rules can be tuned: four cardinal-point years (equinoxes and solstices); four cross-quarter years. The construction is built on Meeus's low-precision solar theory. We close with a derivation of the secular drift equation, showing that, regardless of how well a leap rule is tuned, the slow shrinkage of the tropical year produces a quadratic cumulative error that reaches one day in ∼57,000 years for any fixed intercalation rule.

What carries the argument

The mean return interval of apparent geocentric solar longitude λ to a fixed ecliptic value L, obtained by averaging the time between successive passages using the polynomial expressions from Meeus's solar theory; this interval varies with L and supplies the input for both the eight mean years and the secular drift derivation.

If this is right

Any fixed leap rule tuned to one of the eight mean years will still accumulate quadratic error from the secular shrinkage of the tropical year.
The eight mean years differ by up to several minutes, so a calendar optimized for the vernal equinox will gradually misalign with the other seasons and cross-quarter points.
The quadratic error term reaches one day after roughly 57,000 years independent of the specific leap interval chosen.
Calendar designers can use the L-dependent mean years as benchmarks to minimize short-term drift for particular observational targets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Dynamic leap rules that slowly adjust their interval could counteract the quadratic drift better than any static rule.
The same averaging technique could be applied to other orbital elements or planets to define analogous mean periods for long-term ephemeris work.
These results highlight why calendars require periodic reform even when initially tuned to high precision.
The multi-millennium averaging window smooths out short-period perturbations but leaves the secular trend as the dominant long-term effect.

Load-bearing premise

The mean intervals and drift rate are computed from a low-order truncation of solar longitude polynomials that must remain accurate when averaged over several thousand years.

What would settle it

Direct numerical integration of the Sun's geocentric longitude over a 5,000-year span using an independent high-precision ephemeris, followed by measurement of the actual mean return times to each L, would test whether the computed intervals and quadratic drift match.

Figures

Figures reproduced from arXiv: 2605.02239 by Daniel Quigley.

**Figure 1.** Figure 1: Vernal equinox at point V, as the primary direction, and the autumnal equinox at point A; geographic poles view at source ↗

**Figure 2.** Figure 2: Precession of the equinoctial points; vernal equinox at point V, as the primary direction, and the autumnal view at source ↗

**Figure 3.** Figure 3: Apparent aberration; the apparent position of a star viewed from the earth can change depending on the earth’s view at source ↗

**Figure 4.** Figure 4: Oscillatory motion of nutation superimposed on precession; nutation is in the direction of the plane defined by view at source ↗

**Figure 5.** Figure 5: Mean year length Y (L) as a function of ecliptic longitude L, computed at 1 ◦ resolution over the window J2000 ± 1500 yr (N ≈ 3000 intervals per longitude). The curve has minimum Y = 365.241618d at L = 107◦ and maximum Y = 365.242748d at L = 288◦ , with peak-to-peak amplitude 97.56s. These extrema lie ∼ 17◦ and ∼ 5 ◦ past the June and December solstices, respectively, because the earth’s perihelion current… view at source ↗

**Figure 6.** Figure 6: Y (L) sinusoids at eight center epochs (−2000 to +5000 CE, half-window ±250 yr each), computed at 1 ◦ resolution. Colors run from blue (epoch −2000) to red (epoch +5000) via a diverging colormap. Dotted verticals mark the minimum of each curve; their progression from left (near L = 45◦–50◦ at epoch −2000) to right (near L = 180◦ at epoch +5000) traces apsidal precession directly. The Y -axis is offset from… view at source ↗

read the original abstract

We compute the mean interval between successive returns of the apparent geocentric solar longitude $\lambda$ to a fixed value $L \in \{0^\circ, 45^\circ, 90^\circ, \ldots, 315^\circ\}$, averaged over a multi-millennium window; this gives eight ``mean years'' against which calendar leap rules can be tuned: four cardinal-point years (equinoxes and solstices); four cross-quarter years. The construction is built on Meeus's low-precision solar theory (Astronomical Algorithms, 2nd ed., 1998), itself a low-order truncation of Newcomb's Tables of the Sun, re-expanded around J2000.0. Where Meeus presents polynomial coefficients without justification, we draw on Smart's Textbook on Spherical Astronomy (6th ed., revised by Green, 1977) for the underlying derivations. Numerical accuracy is validated against the cardinal-point intervals tabulated in Meeus, More Mathematical Morsels, 2002. We close with a derivation of the secular drift equation, showing that, regardless of how well a leap rule is tuned, the slow shrinkage of the tropical year produces a quadratic cumulative error that reaches one day in $\sim$57,000 years for any fixed intercalation rule.

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

The paper adds mean tropical year lengths at the cross-quarter longitudes and a quadratic secular-drift derivation, both from Meeus's model, with validation only on cardinal points.

read the letter

Two things to know about this paper. It gives concrete mean tropical year lengths at the cross-quarter ecliptic longitudes in addition to the usual equinox and solstice points. It also derives a quadratic cumulative error for any fixed leap rule, reaching one day after roughly 57,000 years. The calculations build on Meeus's low-order solar theory and check out against his published cardinal-point intervals. Drawing the secular drift directly from the polynomial coefficients is a clean step that does not require new data. The soft spots are the lack of comparison to higher-precision modern ephemerides and the absence of details on the multi-millennium averaging window or how model truncation errors affect the non-cardinal means. These are moderate issues given the scope, but they limit how far the results can be trusted beyond the model's native accuracy. The stress-test note flags exactly this reliance, and it holds up as a real concern here. The paper is for readers who work on historical calendars or need precise mean-year figures for arbitrary starting longitudes. The thinking is clear and the claims stay within what the model supports. I would send it for peer review to get a technical check on the numerics and to see if the new values justify publication.

Referee Report

2 major / 2 minor

Summary. The manuscript computes the mean interval between successive returns of apparent geocentric solar longitude λ to each of eight fixed values L (0°, 45°, ..., 315°), averaged over a multi-millennium window, using Meeus's low-precision solar theory (a low-order truncation of Newcomb's tables re-expanded about J2000). This yields eight distinct mean tropical years (four cardinal, four cross-quarter) for calendar tuning. Numerical results for cardinal points are checked against Meeus's 2002 tabulated intervals; the paper closes with an analytic derivation of the secular-drift equation showing that any fixed leap rule accumulates quadratic error, reaching one day in ∼57 000 years.

Significance. If the non-cardinal results hold, the work supplies concrete mean-year lengths at cross-quarter longitudes that could inform specialized calendar design and quantifies the fundamental limit imposed by secular shrinkage of the tropical year on static intercalation rules. The explicit numerical validation against existing cardinal tables and the closed-form secular-drift derivation are clear strengths that make the central claims falsifiable and reproducible within the adopted model.

major comments (2)

[§3] §3 (numerical validation): Accuracy is demonstrated only against the cardinal-point intervals tabulated in Meeus (2002). The central claim of eight mean years includes the four cross-quarter longitudes (45°, 135°, 225°, 315°), for which no independent tabulated values or modern-ephemeris cross-checks are provided; the mean return intervals at these L therefore rest entirely on the fidelity of the low-order polynomial without additional verification that neglected periodic or higher secular terms do not bias the multi-millennium average.
[§5] §5 (secular-drift derivation): The quadratic cumulative-error term (one day in ∼57 000 years) is obtained directly by differentiating the time-dependent coefficients of the same Meeus polynomial. Because the underlying model is a low-order truncation whose secular coefficient has not been validated against modern ephemerides over multi-millennial spans, the long-term drift prediction inherits the same untested extrapolation that affects the non-cardinal mean intervals.

minor comments (2)

[Abstract] The abstract omits the precise length of the multi-millennium averaging window and any statement of how uncertainties from the low-precision model are propagated into the reported mean intervals.
[§2] Notation for the averaging operator and the exact definition of the return interval Δt(L) should be introduced with an equation number in §2 to avoid ambiguity when the secular-drift equation is later derived.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the strengths of the cardinal-point validation and the analytic secular-drift derivation. We respond point by point to the major comments below.

read point-by-point responses

Referee: §3 (numerical validation): Accuracy is demonstrated only against the cardinal-point intervals tabulated in Meeus (2002). The central claim of eight mean years includes the four cross-quarter longitudes (45°, 135°, 225°, 315°), for which no independent tabulated values or modern-ephemeris cross-checks are provided; the mean return intervals at these L therefore rest entirely on the fidelity of the low-order polynomial without additional verification that neglected periodic or higher secular terms do not bias the multi-millennium average.

Authors: The Meeus low-precision theory supplies a single polynomial for apparent geocentric longitude λ(t) that applies uniformly to every fixed target L. Reproducing the cardinal-point mean intervals from Meeus (2002) therefore validates the coefficients used for the entire computation. The cross-quarter intervals are obtained by the identical averaging procedure on the same polynomial; no separate tabulated values exist in the source. We will revise §3 to state this uniformity explicitly and to note that the absence of independent cross-quarter checks is a limitation inherent to the low-precision model adopted throughout the paper. revision: partial
Referee: §5 (secular-drift derivation): The quadratic cumulative-error term (one day in ∼57 000 years) is obtained directly by differentiating the time-dependent coefficients of the same Meeus polynomial. Because the underlying model is a low-order truncation whose secular coefficient has not been validated against modern ephemerides over multi-millennial spans, the long-term drift prediction inherits the same untested extrapolation that affects the non-cardinal mean intervals.

Authors: The quadratic drift term is obtained analytically by differentiating the secular coefficients of the adopted Meeus polynomial, exactly as described. The manuscript is framed entirely within this low-precision truncation; the ∼57 000-year figure is presented as the mathematical consequence for any fixed leap rule inside that model, not as an extrapolation beyond it. The closed-form derivation remains reproducible within the stated theory. No revision is required. revision: no

Circularity Check

0 steps flagged

No significant circularity; direct evaluation of external Meeus model

full rationale

The paper's central computation obtains mean return intervals for solar longitude λ = L by numerically solving the Meeus (1998) low-order polynomial expressions for geocentric longitude over a multi-millennium span and averaging the resulting Δt values. This is a straightforward forward evaluation of a pre-existing, externally fitted model (itself a truncation of Newcomb) rather than any re-fitting or self-referential construction inside the present work. The secular-drift equation is obtained by extracting the linear secular coefficient already present in that same external model and integrating it to produce the quadratic cumulative error term; no new parameters are introduced or fitted. Validation is performed only against cardinal-point tabulations from Meeus (2002), but because those tabulations are independent of the current paper there is no reduction of the claimed results to the paper's own inputs. No self-citations appear, no uniqueness theorems are invoked, and no ansatz is smuggled via citation. The derivation chain therefore remains self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of the adopted low-precision solar model and the definition of the multi-millennium average; no new physical entities are introduced.

free parameters (1)

Polynomial coefficients in Meeus solar theory
Coefficients taken directly from Meeus (1998), which are low-order truncations of Newcomb's tables fitted to historical observations.

axioms (2)

domain assumption Meeus's low-precision solar theory accurately represents geocentric apparent solar longitude over multi-millennial timescales
The entire computation of mean return intervals is built on this theory.
domain assumption Averaging return intervals over a multi-millennium window yields the appropriate mean tropical year for each longitude
This averaging procedure defines the eight mean years used for calendar tuning.

pith-pipeline@v0.9.0 · 5523 in / 1672 out tokens · 110197 ms · 2026-05-08T18:55:34.340120+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.

Foundation/AlexanderDuality.lean (8 = 2^3 from D=3, not from longitude partition) alexander_duality_circle_linking unclear

?

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

We compute the mean interval between successive returns of the apparent geocentric solar longitude λ to a fixed value L ∈ {0°, 45°, 90°, …, 315°}, averaged over a multi-millennium window; this gives eight 'mean years' against which calendar leap rules can be tuned

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.

Reference graph

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