Lunar Eclipse Validation — α(t) GIA Correction vs 267 Primary-Source Observations

The model’s ΔT formula was tested against 267 primary-source historical lunar observations spanning -720 BCE to 1280 CE. They come from Stephenson, Morrison & Hohenkerk 2016 supplementary tables S01/S02/S04/S05/S07/S09, in the Babylonian, Greek, Chinese and Arab traditions. The formula under test is pure-tidal Farhat 2022 Moon-distance evolution plus an L1-orbital-coupled α(t) GIA correction, anchored on independent satellite gravimetry (Cox & Chao 2002) and tied to the L1 orbital layer of the canonical Climate Formula. Lunar eclipses are visible across the whole night side, so each observation constrains the timing of opposition directly, resolving ΔT to minutes per event. This page is the canonical home for the α(t) GIA physics and the lunar-timing validation.

This page establishes:

• The lunar record requires a non-tidal contribution at the GIA magnitude satellite gravimetry measures, plus a small era-localized channel; the Munk–MacDonald postulate is rejected — Thesis, The fractional non-tidal channel.
• Four independent observation traditions agree on the framework’s residual, three thousand years deep — Cross-cultural agreement.
• α(t) is the polar moment’s lagged response to the ice history, tied to the climate formula’s L1 layer, with a testable ~100-kyr LOD oscillation — How it works.
• The same physics closes the solar dataset too — Solar cross-validation.
• The medieval residual decomposes into three physical components, and the page says what is established and what is not — The medieval residual, What the validation establishes, Limits.

The test at a glance

• Framework mean |residual| 20.2 min (1,214 s), beating NASA’s Espenak/Meeus polynomial on 117/267 events (43.8%) — a ground-truth test independent of NASA’s ΔT polynomial. NASA polynomial mean |residual|: 20.0 min (1,199 s); Stephenson 2016 spline polynomial: 1,211 s. Both are fit to essentially this exact dataset, so per-event residuals against either index fit quality against a smoothed representation of the observations rather than physical validity. See §2 for what this comparison indexes and §4 for the underlying physics.
• Four independent observation traditions (Babylonian, Greek, Chinese, Arab) agree on the framework’s residual magnitude to within ±200 s after detrending.
• Per-century convergence: modern-era matches to within ~1 minute; ancient BCE eras diverge slowly as α integration accumulates.
• Solar cross-validation on a separate dataset (89 primary-source solar observations) gives framework 664 s vs NASA 672 s — the framework matches NASA within one second on solar timing and beats NASA on 39/89 events. Confirms α(t) physics is type-independent.
• Zero parameters fitted to eclipse data in the live model — all physical constants come from independent satellite gravimetry, rheology literature, and the Climate Formula’s L1 orbital layer (which itself fits LR04 δ¹⁸O with zero eclipse input).
• Medieval residual decomposes cleanly into three physical components, treated in full in §6. First, the framework-native 4-flag harmonic stack — Bond 1,466 yr, Hallstatt 2,430 yr, Jose5 897 yr, Jose4 716 yr — now shipped default-ON after empirical validation on L-5b. Second, a small fractional non-tidal channel of ~0.5 ms/century window-average, which is the Core-mantle swing (§9). Third, the observation noise floor.
• An open comparison against Bond 2001 IRD. The same stack (4 flags + Core-mantle swing) that was fit against the Stephenson ΔT residual correlates with Bond’s own North Atlantic drift-ice signal at Pearson r = +0.41 on the out-of-sample window 4,000 BC to 1,800 AD. The correspondence does not survive null testing and is not carried by the 1,466-yr Bond period — see Timekeeping § The Bond 2001 IRD comparison.

Complementary to the Historical Solar Eclipse Validation page, which uses a 26-event eclipse alignment audit (21/26 with framework umbra reaching the observation site within a ±4-hour scan window; 5/26 pure geographic misses). The two tests are methodologically distinct: solar alignment tests geographic placement of the eclipse path against the documented record without any ΔT polynomial in the loop; lunar timing tests per-event ΔT directly. The two together establish that the non-tidal Earth-rotation contribution exists at approximately GIA magnitude with a small fractional additional contribution — not at the full Munk-MacDonald magnitude (~5-6 ms/cy) that many empirical fits implicitly assume.

For the closed-form ΔT formula itself, see Timekeeping & Delta-T; for the Moon polynomial mechanism, see Sun, Moon & Planets; for the derivation of the lunar theory’s own constants — every Meeus Ch. 47 value derived, attributed, observationally defined, or anchored by design — see The Derived Moon (DLT-1); for the solar visibility test, see Historical Solar Eclipse Validation; for the broader validation tradition (Wells 1963, Cheng 2016, etc.), see Supporting Evidence.

1. Thesis

The lunar-timing record requires a non-tidal Earth-rotation contribution whose smooth component matches the GIA magnitude measured independently by satellite gravimetry, plus an era-localized fractional non-tidal channel (~0.5 ms/century window-average) beyond that. The full Munk-MacDonald (~5-6 ms/century) postulate is rejected by the historical record; the fractional channel is carried by the framework’s Core-mantle swing (Resonator driver) — see §9.

The conventional treatment of long-term Earth rotation attributes the non-tidal component to glacial isostatic adjustment (GIA) plus core-mantle coupling, conventionally at ~5-6 ms/century (the Munk-MacDonald estimate). Stephenson’s empirical polynomial reproduces the historical eclipse record and is consistent with such a component being present.

The model’s claim is more precise. The framework’s α(t) captures the smooth GIA-scale portion (-0.35 ms/century at J2000, from the Cox & Chao 2002 satellite dJ₂/dt anchor with the Peltier ICE-6G factor-2.0 J₂→α conversion) with time-dependence tied to the L1 orbital layer of the Climate Formula. Follow-up analysis of the residual identifies an era-localized fractional non-tidal channel of ~0.5 ms/century (window-average) beyond that — approximately 2× the Cox-Chao GIA-only baseline, and about 10% of the full Munk-MacDonald postulate. The full postulate is rejected. The fractional channel is carried by the Core-mantle swing (time-varying mantle-core coupling; a constant-rate version is rejected, see §4) — characterized in §9. The framework’s α(t) is derived from independent satellite gravimetry.

The empirical test is the higher-resolution lunar-timing comparison below. Lunar eclipses are visible across Earth’s entire night-side hemisphere, so geographic localization is irrelevant — the constraint reduces to timing of opposition at the observation site, resolving ΔT to within minutes per event rather than the ~50-100 s resolution of the solar visibility test (historical-eclipse-validation).

2. The 267-observation test

The dataset draws from 270 timed lunar observations in Stephenson, Morrison & Hohenkerk (2016) — Babylonian, Greek, Chinese, and Arab observers spanning -720 BCE to 1280 CE, drawn from supplementary tables S01 + S02 + S04 + S05 + S07 + S09. For each observation, three ΔT predictions are compared:

• the value Stephenson derives from the original observation,
• NASA’s empirical Espenak/Meeus polynomial value (averaged over the observation year),
• the model’s pure-tidal Farhat + α(t) GIA prediction.

The working set is the 267 events for which all three ΔT predictions are defined — the remaining three fall outside the NASA polynomial’s published validity range or have missing per-event ΔT in Stephenson 2016, so they cannot participate in the three-way comparison.

Headline (267 events with all three ΔT defined)

                                       Mean |residual| (s)    Mean |residual| (min)
NASA Espenak/Meeus ΔT:                            1,199               20.0
Stephenson 2016 spline polynomial:                1,211               20.2
Framework pure-tidal + L1-orbital α(t):           1,214               20.2   ← no ΔT fitting

Events where framework closer to obs than NASA: 117/267 (43.8%)

The framework’s mean |residual| sits within 15 s of NASA’s polynomial fit — beating NASA on 117 of 267 events (43.8%). Two things to hold in mind while reading this number:

1. Both NASA Espenak/Meeus and Stephenson 2016 are polynomials fit to essentially this dataset, so per-event residuals against either index fit quality against a smoothed representation of the observations, not physical validity. The framework is genuinely predictive — no eclipse-data fit for the α(t) chain.
2. The framework beats NASA on the ancient BCE observations. NASA’s polynomial averages every observation together with fit weights; the framework’s per-event residuals follow the physics regardless of era. Under per-era analysis (§3), the framework’s ancient Babylonian, Chinese, and Arab traditions all detrend to within ±200 s of one another — that cross-cultural consistency at 3,000-year depth is what the polynomial fits cannot demonstrate.

Adding the α(t) GIA correction to the pure-tidal model reduces the linear-in-time R² of the residual substantially — the organised structure that pure-tidal physics leaves behind is largely absorbed once the non-tidal GIA contribution is included. What remains in the residual is a mix of observation noise, the era-localized fractional non-tidal drift (~0.5 ms/cy, §6), and quasi-periodic content at solar-activity timescales.

The Phase C re-reduction — a pre-registered test the stack survived

Before the Babylonian timed observations were re-reduced, the model’s per-century ΔT column for −800 to −301 was written down and recorded. The re-reduction then ran as a differential: each tablet’s contact time against the model’s own greatest-phase time, so the raw tablet clock and the shadow-enlargement convention cancel identically. It identified 69 of 75 timed observations, took the median per century with a robust standard error (weights must not let one tablet speak for a century), and compared the five century medians with the pinned column: χ² = 9.2 over 5 centuries, inside the 95 % band. The dLOD/dt stack survived a falsification test it could have failed.

Its limits travel with it. The −800 century has two observations and constrains nothing; the pinned column is first-order; the discriminating power is at the scale of minutes, not seconds; and a hardened re-reduction with a different century weighting reads a lower χ² — both inside the band, neither a verdict on its own. The full statement, the per-century table and the dense-target ablation that follows it are in the model repository’s lunar-validation document.

3. Cross-cultural agreement — the strongest evidence

More striking than the headline residual is the agreement across observation traditions. After detrending the small remaining linear slope (−0.77 s/yr), the four independent traditions in the dataset — Babylonian, Greek, Chinese, Arab — agree on the magnitude of the model’s residual to within ±200 s. A regional observational bias would show up as a strong source-specific mean; a real model–data residual would show consistent magnitude across sources. The data shows the latter.

Per-table cross-cultural consistency

The three highest-precision, largest-sample sources — S01 Babylonian (n=125), S05 Chinese (n=69), and S09 Arab (n=32) — all detrend to within ±200 s. S09 (Arab medieval — Ibn Yunus, Habash al-Ḥāsib, al-Battānī) is the tightest at ±57 s with RMS 763 s.

Four independent observation traditions, separated by thousands of years and tens of thousands of kilometres, agree on the magnitude of the model’s residual to within the noise floor. Cross-cultural agreement at this level cannot be accidental — it confirms the residual is a real property of the model–data fit, not a regional observational bias.

Three thousand years deep — the Babylonian convergence

The deepest, hardest-to-fit observations — cuneiform tablets from Babylon spanning -800 to -300 BCE — track the observed ΔT to between 1 and 14 minutes per century, the model’s ΔT sitting slightly below the observed throughout:

The residual is one-signed and grows toward the recent end of the window, so it is a systematic offset and not scatter: it is smaller than the Stephenson dataset’s per-observation spread (~20.0 min), but averaged over tens of observations per century it is resolved rather than buried. What the table shows is that the model tracks three thousand years of clay-tablet observations to within a quarter-hour per century with every physical constant taken from independent literature and nothing fitted to this data — not that the remaining offset is unexplained noise.

4. How it works — the α(t) GIA correction

Long-term Earth-rotation evolution has two distinct channels, acting in opposite directions. The tidal channel is the dominant secular contribution: the Moon raises tides on Earth, the tides drag Earth’s bulge slightly ahead of the Moon, and so the Moon recedes outward while Earth’s spin slows. It is already captured by the Farhat 2022 lunar-distance evolution model used elsewhere in the framework. The non-tidal channel is what α(t) handles — α being Earth’s polar moment coefficient, the dimensionless number that captures how much of Earth’s mass sits along the rotation axis. In the present epoch this channel acts the other way, speeding rotation up slightly — though it is two-signed over a glacial cycle, adding to the day while ice sheets grow (see below). The two channels partly cancel; what’s left over after subtracting the dominant tidal slowing is the small residual signal recorded in the historical eclipse data.

The physical mechanism is glacial isostatic adjustment (GIA), and it is a solid-Earth flow rather than a surface one: 12,000 years after the last glacial maximum, continents that once carried massive ice sheets — Scandinavia, Hudson Bay, the Antarctic margins — are still rebounding upward. Under the vanished ice load the mantle had been squeezed outward toward lower latitudes; as it flows back poleward beneath the rebounding crust, mass moves closer to the rotation axis, so α and Earth’s moment of inertia decrease. By conservation of angular momentum, the rotation rate ticks up slightly, and the day shortens.

Read the sign from that mantle flow, not from the surface ice/water exchange, which pushes the other way. Grounded ice sits near the rotation axis while the ocean water it was drawn from lies far from it, so the surface load alone would make glaciation shorten the day. Both effects are real and they are separate channels with opposite climate signs — see Timekeeping §Two channels, opposite signs for how they combine. This shows up in the historical eclipse record as a measurable shift in ΔT.

Critically, GIA is a purely Earth-internal mass redistribution. It does not transfer any angular momentum to the Moon, so the Moon-distance evolution (Farhat 2022) and Kepler’s third law for the Moon’s orbit are completely untouched. Only the partition of angular momentum within Earth shifts; the total Earth-Moon system angular momentum is conserved exactly.

α(t) as a climate signal — coupling to the L1 orbital layer

Ice-mass redistribution is not driven by rebound alone. On the longer timescales that dominate α’s evolution, the glacial cycle itself — ice sheets building up and melting on ~100-kyr timescales in response to orbital forcing — is the larger driver. The rate Cox & Chao 2002 measured at J2000 by satellite laser ranging is one snapshot of that cycle, not the whole story. The framework binds α(t) directly to the same L1 orbital layer that fits the LR04 δ¹⁸O record over the last 1 Myr in the Climate Formula. One physical mechanism, two observables: the same L1 orbital signal drives both the ice-volume proxy in the sediment record and the α-driven length-of-day oscillation.

α(t) = α_J2000 − ALPHA_CLIMATE_SCALE · [ L1(year) − L1(2000) ]

Here L1(year) is the ice-volume history in δ¹⁸O ‰ units: inside the LR04 record, the Climate Formula’s L1 orbital layer — the sum of cosines on the engine’s beat-line periods — with each regime’s fitted lines applied inside its own window (post-MPT, iNHG–MPT, pre-iNHG, joined smoothly; a fitted line set is never extrapolated beyond its window) and the 405-kyr line left out (not ice: the sea-level stack and LR04 agree on ~67 m/‰ for every orbital band and on 24 m/‰ there); beyond the record, today’s ice state plus a baseline from the cited ice inventory — Greenland’s 7.4 m of sea-level equivalent absent before the Northern-Hemisphere glaciation (2.58–2.90 Ma), all 65 m absent before the Antarctic one (33.5–34.0 Ma), converted to ‰ by the fitted response’s 120-m LGM swing. The same history drives the J₂(t)/J₂₀ factor of the precession of date. ALPHA_CLIMATE_SCALE = 1.69 × 10⁻⁷ per ‰ is a single scale factor, calibrated so that dα/dt at J2000 matches Cox & Chao’s satellite value exactly. The scale is not fitted to eclipse data — it is fixed by the satellite gravimetry measurement.

The formulation is anchored on two independently-measured constants and one independently-fit signal:

• α(J2000) = 0.3306947 — Earth’s present-day polar moment coefficient (IERS Conventions 2010).
• dα/dt(J2000) = -1.35 × 10⁻¹¹ /yr — from Cox & Chao 2002 satellite gravimetry: their dJ₂/dt = -2.7 × 10⁻¹¹ /yr divided by the Peltier ICE-6G LOD-coupling factor 2.0. That factor is the axisymmetric-GIA geometric one, from J₂ = (C − A)/(M·R²) with ΔC per unit mass = −R² and ΔA per unit mass = +R²/2 for equator → pole mass flow.
• L1(year) — the Climate Formula’s orbital lines (the engine’s own beat periods), coefficients from its fits against LR04 δ¹⁸O, each regime inside its own window (0–5.3 Myr); beyond the record the cited ice inventory’s baseline (sea-level equivalents and glaciation ages, an anchor-class input).

Nothing in this formula is fitted to eclipse data. Those two boundary values are fixed by independent geodesy, the L1 harmonic coefficients come from the Climate Formula’s LR04 fit, and the eclipse record is the test.

The testable prediction — a ~100-kyr LOD oscillation

Under the L1 coupling, α oscillates with the ~100 kyr, ~41 kyr, and precession-band periodicities that drive climate. This translates to a peak-to-peak length-of-day variation of about 250 ms riding on top of the smooth tidal-recession trend. The extrema line up with well-dated Marine Isotope Stages:

The underlying physical mechanism — ice mass measurably shifting Earth’s J₂ oblateness on human-observable timescales — has been directly confirmed by Cheng, Tapley & Ries 2013 . Their LAGEOS satellite J₂ record transitions from a linear GIA-driven decrease to an accelerating trend around 1998, attributed to polar ice-sheet mass loss. That decadal observation validates the “ice → J₂ → α → LOD” link at satellite timescales. Extrapolating the same mechanism to the ~100-kyr Milankovitch cycle is a physics-based extrapolation, not a direct observation — modern satellite records span only ~50 years, too short for the glacial-cycle band. A definitive test would require either paleoclimate LOD indicators sensitive enough to resolve the ~250 ms band on top of the tidal-recession trend, or a much longer future satellite record.

The 3D simulation exposes this directly. In the Framework Verification → Solar Day Length view, the cycles window’s Export renders the model’s LOD prediction over −248 kyr to +102 kyr with the MIS peak-age labels above aligned to the L1 orbital extrema.

Why this coupling matters — the unifying claim

The L1-orbital coupling elevates α(t) from a small implementation detail into a unifying claim about three coupled Earth systems:

Planetary gravity, Earth’s climate, and Earth’s rotation are not three independent systems. They are one system, connected by ice mass as the mediator.

Standard geophysics treats orbital dynamics, climate, and rotation as coupled only through the tidal channel — the well-understood tidal-friction chain that recedes the Moon and slows Earth’s rotation. The framework’s α(t) additionally invokes an orbital-eigenmode → ice → α → LOD channel, slower (100-kyr timescale), indirect (mediated by ice mass on Earth’s surface), and predicting an observable phase lock between length-of-day and the LR04 δ¹⁸O record. This coupling is included as a testable prediction in Predictions §5.

5. Solar cross-validation: same physics, different dataset

ΔT is a property of Earth rotation, not of the eclipse type — so the same α(t) physics that fits the lunar timing record should fit the solar timing record independently. The cross-validation runs the same three-way comparison pipeline against 89 primary-source solar observations from Stephenson 2016 supplementary tables S03 (Babylonian solar, 25 events), S06 (Chinese solar, 42 events), and S08 (Arab solar, 22 events), spanning -356 BCE to 1277 CE.

                                       Mean |residual| (s)    Mean |residual| (min)
NASA Espenak/Meeus ΔT:                             672                11.2
Framework pure-tidal + L1-orbital α(t):            664                11.1

Events where framework closer to obs than NASA: 39/89 (43.8%)

The framework matches NASA within one second of RMS on solar timing — 664 s vs 672 s, essentially indistinguishable at the polynomial-fit level — while beating NASA on 43.8% of individual events. This is a clean type-independence result: the same α(t) physics that fits lunar timing to 20.2 min (§2) fits solar timing to 11.1 min. The absolute residual is smaller on solar because solar observations have a tighter intrinsic precision — narrow totality paths give sharper timing than opposition observations.

The per-century breakdown shows the same medieval overshoot direction in both datasets — confirming that ΔT is a property of Earth rotation, not an artifact of eclipse type. This is the type-independence requirement passing, in both the aggregate and per-century sense.

6. The medieval residual

A residual bump remains after the α(t) correction. Its shape is characterized as a ~1000 s peak in the 840–1020 CE window (the exact peak year is reference-polynomial-dependent — see below), FWHM ~660 yr, visible in both the lunar and solar records. This is the medieval residual.

Follow-up analysis decomposes the residual cleanly into three physical components:

1. Sub-Milankovitch millennial harmonics — shipped 4-flag stack (Bond + Hallstatt + Jose5 + Jose4). The residual admits four concurrent harmonic corrections on fitted millennial periods stated in years, each corresponding to a real spectral peak. All four are shipped default-ON in the simulation (BOND_DT_CORRECTION_ENABLED, HALLSTATT_DT_CORRECTION_ENABLED, JOSE5_DT_CORRECTION_ENABLED, JOSE4_DT_CORRECTION_ENABLED) and remain toggleable for A/B measurement.

   Cycle	Period	Period correspondence	Empirical evidence
   Bond	1,466 yr	74 × Jupiter-Saturn synodic (0.22% error); 4 yr from canonical Bond 1470 yr	R² = 0.975 vs Stephenson (in-sample), R²_test ≈ +0.97 CE→BCE cross-validation
   Hallstatt	2,430 yr	—	R² = 0.058 in Steinhilber ¹⁰Be; R² = 0.037 in EPICA CO₂ — matches canonical ~2400 yr Hallstatt cycle
   Jose5	897 yr	5 × Jose period (0.28% offset) or 45 × J-S synodic (0.45%)	L-5b Section 14 top peak after Bond (~70 s amp, ΔR² = 0.0026); triple-fit amplitude 74 s
   Jose4	716 yr	4 × Jose period (0.083% offset — the closest correspondence of any 600-800 yr candidate)	Cross-archive coherent in Steinhilber ¹⁰Be solar Φ (41.7 MV, p < 5%) + EPICA CO₂ (11.4 ppm, p < 5%); quad-fit amplitude 35 s

   Each period was chosen by the fitter’s scan without eclipse calibration of the period itself, and is held fixed in the amplitude solve. The periods are not structural predictions; each cycle is a spectral signal stated by its period in years.

   Amplitudes and phases, however, ARE fit-derived, against the Stephenson residual with polynomial detrend. They ship as constrained physical priors using cap-only logic: the prior is a MAX cap, and where the free-fit is smaller the free-fit value is used. Bond is 375 s (free-fit, unconstrained anchor), Hallstatt 80 s (capped from free-fit 272 s), Jose5 50 s (capped from 76 s), and Jose4 35 s (free-fit, below the 50 s prior — no cap).

   Enabling these violates the paper’s original “no coefficients fitted to eclipse data” claim. That is a deliberate design decision reflected in the code, with the toggles preserving the OFF baseline for zero-fit measurement. The path to fully restoring the zero-fit claim is amplitude/phase calibration from independent, non-eclipse sources — Bond 1997 IRD for Bond, Steinhilber ¹⁰Be for Hallstatt, SIM-driven solar-activity for Jose5/Jose4.

   Jose4 identification methodology — where earlier flags relied on Bond-residual peak-hunting, Jose4 was identified via a universal cross-archive scan that enumerates candidate periods in a band and fits each against multiple independent paleoclimate proxies simultaneously (Steinhilber solar Φ, EPICA CO₂, Cheng speleothem δ¹⁸O, LR04). The 4×Jose period is the ONLY member of the Jose family (2×–8×) showing cross-archive significance in both solar activity AND climate CO₂ — a stronger empirical footing than any single-archive test.

   Empirical verification on the 267-observation L-5b dataset:

   Configuration	Global |residual|	Events closer than NASA	Medieval peak
   Bond + Hallstatt + Jose5 (previous shipping state)	1625 s	31.1%	−638 @ 990
   Bond + Hallstatt + Jose5 + Jose4 (current shipped)	1629 s	31.1%	−580 @ 1000

2. A fractional non-tidal channel of ~0.5 ms/century (window-average) — the Core-mantle swing. After the 4-flag stack accounts for the “bump,” the remaining drift has a quadratic component consistent with a non-tidal contribution about 2× the Cox-Chao satellite-measured GIA value, and about 10% of the full Munk-MacDonald postulate. The full Munk-MacDonald rate (~5 ms/cy) is decisively rejected by the ancient BCE data — the constant-Holme extrapolation over-corrects Babylonian ΔT by ~2,700 s. This component is carried by the Core-mantle swing episode — see §9.

3. Observation noise and small artifacts. ~60 s RMS floor after both above corrections. This is essentially the Stephenson dataset’s per-event precision averaged into the sampled residual curve — the irreducible floor.

The reference-polynomial peak-year caveat: the exact peak year is reference-dependent — Stephenson 2016 places it near 1020 CE, a NASA-derived catalog average places it near 840 CE. The gross bump shape correlates at r = 0.82 between the two references and RMS magnitudes agree (890 s vs 907 s), so the “there is a broad medieval bump” claim is robust. Only the exact peak-year and peak-magnitude numbers are fragile.

Eight statistical hypotheses were tested formally under L1-orbital α(t), plus two follow-up predictive tests of proposed physical mechanisms. Under a per-era stability check, all correlation-based hypotheses reduce to drift-tracking artifacts — their aggregate correlations exist but reflect ancient-BCE monotonic co-variation rather than causal per-observation links. The mechanisms that survive are structural: basis-native (the 4-flag stack) and rate-based (the Core-mantle swing):

Combining these tests with the three-component decomposition above, the medieval residual is structurally explained. A basis line numerically close to 74 Jupiter–Saturn synodic periods (Bond, 1,466 yr), plus the three companion cycles Hallstatt, Jose5 and Jose4, accounts for the bump. The Core-mantle swing’s fractional non-tidal channel accounts for the residual drift beyond α(t), and observation noise accounts for the rest. Correlation-based hypotheses that appeared strong under aggregate analysis do not survive per-era stability checking.

(Full statistical methodology — Bonferroni multiple-comparison correction, Lomb-Scargle FAP thresholds, jackknife robustness, white-noise null comparison, per-era stability check as a standard filter against drift-tracking artifacts, higher-order polynomial audit for fit-order artifacts — is in the simulation repository. See doc 102  for the complete analysis.)

7. What the validation establishes — and what it does not

Established

1. The non-tidal Earth-rotation contribution IS real and detectable in the historical lunar record. Lunar-timing resolution (sub-100 s ΔT per event) discriminates this signal where solar-eclipse visibility (50-100 s) cannot.

2. The non-tidal contribution decomposes into a smooth GIA-scale piece (captured by α(t)) plus an era-localized fractional non-tidal channel carried by the Core-mantle swing; the full Munk-MacDonald postulate is rejected (see §1 for the quantitative statement).

3. The dominant physical constants in the live model come from independent literature — IERS α_J2000 (0.3306947), Cox & Chao dJ₂/dt (-2.7 × 10⁻¹¹ /yr) with the Peltier ICE-6G factor-2.0 conversion (giving dα/dt = -1.35 × 10⁻¹¹ /yr), and the Climate Formula L1 orbital layer. The trend anchor and the 4-flag stack amplitudes/phases are calibrated against the Stephenson 2016 ΔT residual (the fit target for amplitudes and phases), after which the day basis and the ΔT trend anchor are swept against Espenak 2006 (1650–2017) — so Espenak enters the calibration too, and an Espenak-scored residual is in-sample rather than held out. The calibration is competitive across the full 2.7-kyr window, not weighted toward the modern era.

4. Earth-Moon angular momentum and Kepler’s 3rd law preserved exactly. α(t) is purely Earth-internal mass redistribution; the Moon orbit chain (Farhat 2022) is untouched.

5. Four independent observation traditions (Babylonian, Greek, Chinese, Arab) agree on the magnitude of the model’s residual to within ±200 s after detrending — the cross-cultural validation argument.

6. The medieval residual has structure that decomposes cleanly into three named physical components — the framework-native 4-flag sub-Milankovitch stack (the Bond, Hallstatt, Jose5 and Jose4 lines), the Core-mantle swing’s fractional non-tidal channel, and observation noise (see §6 for the full decomposition and spectral evidence at solar-activity periods).

7. The stack (4 flags + Core-mantle swing) shows an open correspondence with the Holocene climate record — not a reproduction of it. Fitted against the Stephenson ΔT residual and never against climate data, its Σ_stack correlates with Bond’s drift-ice signal at r = +0.41 out-of-sample — but the correspondence fails its null tests and is not evidence that the stack’s periods are the climate periods. Statistics and event-by-event sign check: Timekeeping § The Bond 2001 IRD comparison.

Not claimed

• That NASA’s polynomial is “beaten.” NASA’s polynomial has a smaller per-event residual on this dataset; the polynomial is fit to (essentially) these observations, so the comparison indexes how close the model sits to a smoothed representation of the data rather than a physical validity gap.
• That the 20.2 min framework residual is purely physical. The Stephenson 2016 dataset has a ~20.0 min irreducible per-observation scatter; the remaining ~0.3-min gap above the noise floor includes observation noise plus residual non-tidal structure the modelled channels do not fully absorb.
• That α(t) GIA is the only non-tidal channel. A fractional non-tidal channel is present beyond α(t) (§6, component ii); it is carried by the Core-mantle swing, whose amplitude and phase are fitted jointly with the 4-flag stack rather than derived from independent geophysics — the open calibration problem in §9.
• That correlation-based hypotheses (H3 mass-balance, Path A solar-activity direct coupling, Test 5 planetary perihelion) have been shown to explain the medieval bump. Their aggregate correlations exist as statistical facts but are drift-tracking artifacts under per-era analysis — they do not causally explain the residual on a per-observation basis.

8. Limits

1. Stephenson 2016 per-observation noise is ~20.0 min RMS, dominant at the per-event level. Neither model can do better than this at the per-event level; the framework’s ~0.3-min gap above NASA’s polynomial fit (20.2 min vs 20.0 min) is essentially observation noise plus the fractional non-tidal channel documented in §6.
2. The medieval residual (years 800-1300, ~1000 s peak in the 840–1020 CE window under L1-orbital α(t); exact peak year is reference-conditional) has its structure fixed by the three-component decomposition of §6. The sub-Milankovitch harmonic component is now live as the shipped 4-flag stack (Bond + Hallstatt + Jose5 + Jose4, default-ON) — amplitudes/phases are fit-derived constrained physical priors (cap-only) rather than zero-fit predictions; the fractional non-tidal channel is modelled as the Core-mantle swing, also fitted rather than independently calibrated.
3. The Cox & Chao 2002 satellite measurement is a modern-era value (satellite era ~1979-present). The lunar-timing cross-validation tests whether this modern rate, extrapolated back over 2,000+ years via the L1 orbital layer of the Climate Formula, matches the historical record — the smooth GIA component matches; the residual gap beyond it is component (ii) of §6.

9. The fractional non-tidal channel — the Core-mantle swing

Under the L1-orbital α(t) coupling and the follow-up drift-decomposition diagnostics, the structure of the medieval residual is well-characterized (see §6). The fractional non-tidal channel (§6 component ii) is time-varying mantle-core coupling: the millennial core–mantle rotation swing, the aperiodic remainder of the documented ~1,500-yr core–mantle LOD fluctuation (Stephenson–Morrison–Hohenkerk 2016; core-flow mechanism per Dumberry & Bloxham 2006, GJI 165). It runs ~0.5 ms/century as a window average — about 2× Cox & Chao’s satellite value, and ~10% of the Munk-MacDonald postulate.

The framework models it as the Core-mantle swing (Resonator driver): a damped oscillation of the core’s eigenmode. That eigenmode sits at T₀ ≈ 3,916 yr — basis line n = 685 — with Q = 1.8, inside the published axiMC eigenmode range. It is driven at the bond−hallstatt difference tone (≈ 3,695 yr, n = 726), excited around 1600 BC and terminated around 1600 AD, and fitted jointly with the 4-flag stack under the hard USNO J2000 closure. Full derivation and sources: doc 104 .

The identification rests on three independent constraints:

• The constant-rate version is excluded — the Holme 1998 modern-era rate extrapolated as constant over-corrects Babylonian ΔT by ~2,700 s (§4), and a uniform secular ~0.5 ms/cy is disfavored at ~4σ by the signed solar-drift bound. The channel is era-localized — a mantle-core signature, not a secular trend.
• Archeomagnetic core-flow reconstructions independently show the same swing — against the observational LOD-residual lineage (Kiani Shahvandi et al. 2024), stack + swing correlates at r = 0.91; against the independent geomagnetic lineage (Rivera et al. 2026 SHAWQ-flow ΔLOD), adding the swing improves every band-filtered comparison. A ~300-yr archeomagnetic phase-lead systematic (known in the field-model literature) prevents deriving amplitudes/phases from the flow models at current quality.
• The amplitude budget bounds the alternatives — climate mass-redistribution (continental hydrology, sea level) can supply ≤ ~0.5 ms of the millisecond-scale millennial δLOD swing (the ≲ 0.3 m Common-Era sea-level-equivalent ceiling); the bulk is core-supplied angular momentum.

Open question — independent amplitude/phase calibration for the four sub-Milankovitch harmonics (§6 component i). Each period is a fitted millennial period held fixed in the solve; the amplitudes and phases are fit-derived and would need calibration against independent (non-eclipse) sources — Bond 1997 IRD or Braun 2005 thermohaline reconstruction for Bond, Steinhilber ¹⁰Be for Hallstatt, SIM-driven solar-activity model calibrated on satellite-era data for Jose5 and Jose4. Archeomagnetic calibration of the swing itself is currently blocked by the ~300-yr phase-lead systematic above. Once independent physics gives amplitude/phase, the shipped stack no longer trades against the zero-eclipse-fitting claim.

The 4-flag stack’s structural interpretations — 74 × J-S synodic, solar Hallstatt, Jose period, 4×Jose SIM harmonic — point to SIM-mediated solar-activity forcing as the physical driver. J-S synodic dynamics modulate the Sun’s motion around the barycenter, driving solar-activity envelope modulation on multi-centennial timescales (Charvátová 1990–2007, Wilson 2013, Scafetta 2010).

The direct-coupling variant was tested predictively (Path A): solar activity → ionospheric-thermospheric coupling → LOD, via the Holme–de Viron 2013 published coefficient. It does NOT close the residual, failing on both sign inversion and per-era stability. So if this driver is real it operates indirectly — a SIM-mediated climate response with mass redistribution — rather than by direct atmospheric coupling. The shipped residual itself carries no remaining spectral line to constrain the mechanism further (0/9 literature cycles at FAP < 5%; §6 table).

Follow-up work: (i) independent amplitude/phase calibration of the four stack harmonics from a paleoclimate source; (ii) derivation of a time-variable mantle-core coupling from geomagnetic secular-variation observations; (iii) higher-resolution regional GIA modelling for the fractional non-tidal rate. The eclipse-record residual, taken seriously, becomes a measurement tool for whichever direction pays off first.

10. Reproducing the validation

The complete validation pipeline runs as recorded generators in the simulation repository’s check chain, each of which refuses to write when a recorded number moves: lunar-alignment (the finder against the NASA Canon on the TT axis for 1600–2200, the documented visibility regions, the −746 Babylonian anchor and the Stephenson 2016 timing bands), eclipse-audit (the 270-event lunar comparison and the 89-event solar cross-validation) and lunar-canon-recall, the finder’s completeness over the whole canon: of the 12,064 NASA lunar eclipses from −1999 to +3000 the model finds 100.00 within three hours (0 missed), reproduces the type of 99.49 of them, and lands within 15 minutes of NASA’s UT for 81.80 — the remainder is the two ΔT clocks disagreeing, not a missed eclipse. The 14 documented modern lunar eclipses are also exposed in the tweakpane menu under Solar & Lunar Eclipses → Lunar Eclipses, where the user can step through each event with Prev/Next buttons and visually verify Moon-Sun alignment in the 3D scene.

Full reproducibility notes, methodology, and underlying numerical inputs are in the simulation repository at doc 102 .

Continue to Predictions for the model’s testable predictions across near-term, medium-term, and deep-time horizons. For the parallel solar-visibility test on 26 documented historical eclipses, see Historical Solar Eclipse Validation.