L1 Attribution Reference — Berger vs Holistic Model

This page summarises the per-integer mapping; the complete reference with all 20+ candidate combos per integer and rank-ordered tables is at 3d repository doc 93 . For the underlying L1 lattice structure and the 32-integer set: Climate Formula. For the synthesis statement this finding supports: Climate Summary. For the standard secular-theory framing being compared against: Eigenfrequencies §The eigenmodes are real.

The L1 lattice itself (the set of integer divisors of 8H) is identical under both attributions — the disagreement is about which planet–planet gravitational coupling produces each peak. Standard cyclostratigraphy attributes each peak to a single Berger / Laskar eigenmode beat (“the 95-kyr peak is g₄−g₅ Mars–Jupiter eccentricity”); the Holistic framework shows every one of those peaks has an equally valid alternative attribution as an Earth-planet PLANET_CYCLES beat, usually naming a different planet. This is not a contradiction of secular theory — it is a structural alternative that the eigenmode enumeration does not surface, because secular theory enumerates g_j, s_j rather than individual planet cycles (Axial, Peri_ecl, ICRF, AscNode, Obliq, Ecc).

Summary table — Berger vs Holistic top-1

For each of the 32 L1 integers: period in kyr, canonical Berger / secular-theory label, the Holistic model’s top-1 Earth-planet beat from PLANET_CYCLES, and the agreement status.

32 L1 components total. 0 agree with Berger on both planet and mechanism; 1 has same planet but different mechanism; 26 name a different planet entirely; 5 are not predicted by Berger at all.

Three example disagreements

n = 120 — the 22.4 kyr peak: Venus or Jupiter?

Berger labels n = 120 the “k+g₂ Venus climatic precession” peak (period 22.4 kyr). The Holistic top-1 is an exact 2-term beat:

Earth.Axial(104) + Jupiter.Obliq(16) = 120

The simplest possible structure — no 3-term combinations needed. Earth’s axial precession at 8H/104 plus Jupiter’s obliquity cycle at 8H/16 sums to 8H/120 = 22.4 kyr. Under the Holistic framework the 22.4 kyr LR04 peak is driven by Jupiter’s obliquity coupling to Earth’s spin axis — not by Venus’s climatic precession.

n = 28 — the 95 kyr eccentricity peak: Mars+Jupiter or Mars+Saturn?

n = 28 (95 kyr, Berger’s famous eccentricity peak) is labelled g₄−g₅ Mars-Jupiter ecc in classical secular theory. The Holistic top-1 is:

Earth.Axial(104) − Mars.Ecc(52) − Saturn.Obliq(24) = 28

Both attributions involve Mars; the secondary partner differs — Berger says Jupiter (via the g_j eigenmode), Holistic top-1 says Saturn (via Saturn’s obliquity cycle). The amplitude (0.754) is the highest in the post-MPT spectrum after n=22.

n = 65 — the 41 kyr obliquity peak: Earth or Jupiter?

n = 65 (41.3 kyr, the canonical obliquity peak) is Berger’s “k+s₃ Earth obliquity” beat. The Holistic top-1 is:

Earth.Axial(104) − Jupiter.Peri_ecl(39) = 65

A clean 2-term difference. Under Holistic, the 41 kyr LR04 obliquity signal is driven by Jupiter’s perihelion-ecliptic coupling to Earth’s spin axis — not by Earth’s own nodal precession via the s₃ eigenmode.

Earth’s spin axis is present in all 32 top-1 attributions

Every one of the 32 Holistic top-1 beats uses one of Earth’s two spin-axis cycles as the base term: Earth.Axial(104) (period 25,794 yr, precession of the equinoxes) in 30 of 32, and Earth.Obliq(64) (period 41,915 yr, obliquity oscillation) in the remaining 2 (n=9 and n=22). The remaining 30 integers are produced by modulating Earth.Axial with one or two planet-cycle harmonics; the two Earth.Obliq integers (n=9, n=22) are modulated similarly. Earth’s spin axis is the universal carrier frequency for the L1 lattice.

Planet participation as the modulating term in the top-1 attributions:

Jupiter dominates as the secondary modulator — present in three-quarters of the L1 beats. Consistent with the Jupiter–Saturn resonance lock found in Law 6 (see Fibonacci Laws) and with Jupiter carrying most of the solar system’s angular momentum.

Lattice tests — Test C-Balance and Test C-Libration

Two pre-registered tests on the published Laskar 2004 nominal solution (INSOLN.LA2004.BTL.ASC) characterise the 8H lattice at the lattice-wide level — independent of the per-integer attribution above. Test C-Balance asks which planets dominate the most dynamically stable beats in LA2004; Test C-Libration asks whether observed periods drift away from the framework’s lattice positions or oscillate symmetrically around them.

Test C-Balance — Does the stability sub-lattice reflect the balance laws?

Method. Scan every integer n ∈ [5, 200] (period range 13 to 537 kyr). For each integer, compute the minimum drift across LA2004 eccentricity and obliquity 5-Myr sliding windows (measures stability in whichever proxy the cycle is most naturally expressed). Tag the integer with its nearest Laskar simple beat and the planets involved, then rank by drift. The top 30 most stable beats = the stability sub-lattice. Compare its planet composition against a binomial null built from the full-scan base rate.

Dominant pattern in the stability sub-lattice: k + g₆ (Earth-spin coupled to Saturn’s perihelion), concentrated at 13–18 kyr, with secondary clusters at s₆+s₇ (Saturn-Uranus nodal sum, ~44 kyr) and k−s₃ (~18 kyr). This is why both Saturn and Earth-spin (k) emerge enriched in the table below — they typically appear paired in the most stable beats, not as independent contributions.

Saturn appears in 80 % of the most stable beats (versus 40 % of the full scan) at p = 4 × 10⁻⁴. Mercury and Mars are completely absent from the stability sub-lattice (each 0/30, p < 0.05 vs base rate). Earth-spin enrichment reflects that most chaos-resistant cycles are precession sidebands rather than long-period secular modes.

This independently corroborates the framework’s balance laws (Fibonacci Laws). Laws 3 and 5 identify Saturn as the anti-phase anchor — the planet that cancels against the seven others in the inclination and eccentricity balance partitions. Test C-Balance arrives at the same conclusion from an entirely different direction: it ranks dynamical stability in LA2004, not balance-partition amplitudes. Both methods point at Saturn. The Mercury / Mars absence independently confirms Laskar 1989’s inner-planet chaos prediction.

Two caveats. The Saturn-Jupiter pair (Law 6’s specific prediction of an 8H/65 resonance lock) is absent from the stability sub-lattice — stability picks out Saturn ALONE (typically paired with Earth-spin), not Saturn-Jupiter. And the L1 ∩ stability-sub-lattice intersection is small (only n = 21, the Venus-Uranus nodal sum at 127.7 kyr): the climate-driven L1 selection and the stability-driven sub-lattice selection are near-orthogonal projections of the same underlying 8H lattice, identifying different physically meaningful subsets.

Test C-Libration — Is Laskar’s “drift” actually libration around the lattice?

Insight motivating the test. LA2004 starts from J2000 initial conditions — an arbitrary snapshot of where the planets happened to be on 1 January 2000. There is no physical reason this snapshot sits exactly at the framework’s balanced equilibrium. If the 8H lattice IS the equilibrium, then integrating outward from J2000 should produce symmetric libration (oscillation around 8H/n), not monotonic drift. Test C-50 showed precession-sideband drift of 16–24 % over 50 Myr; this test asks whether that drift is chaos, libration, or something structured.

Method. For each of the 32 L1 integers, track the nearest spectral peak in LA2004 across 48 sliding 4-Myr windows (1-Myr step) of both eccentricity and obliquity. Per integer, compute bias (mean observed period − predicted period, in %), trend (total period change across 50 Myr from linear regression), and residual_std (std of detrended residuals around the trend).

Aggregate result — the lattice IS approximately the equilibrium.

The aggregate bias is statistically indistinguishable from zero. On average, observed LA2004 periods sit on the framework’s lattice, not systematically away from it. The hypothesis that J2000 is an off-equilibrium random phase and the system oscillates symmetrically around the lattice is supported.

Two structured per-integer patterns:

• Pattern A — pure libration around the lattice (long-period secular beats with no Earth-spin involvement). Example: n = 21 (s₂+s₇ Venus-Uranus, bias −1.33 %, trend 0.19 %, residual std 0.87 %); n = 28 (g₅−g₂ ≈ 95-kyr eccentricity, bias +0.88 %).
• Pattern B — structured shift in k-involving beats. Every k-involving precession beat (n = 48 to 152) shows trend dominating residual_std by 15–27× — a smooth structured shift, not chaos (which would give trend ~ residual_std). Examples: n = 48 (s₆−s₇, trend 4.9 %, residual 0.18 %, ratio 27×); n = 65 (k+s₃ obliquity main, trend 3.78 %, residual 0.25 %, ratio 15×); n = 120 (k+g₂, trend 3.4 %, residual 0.25 %, ratio 14×). This is the LOD evolution effect: as Earth’s spin rate k changes across 50 Myr, every k+s_j and k+g_j beat period shifts proportionally. The proper-physics correction applied in Test C-50 captures this directly.

Together with Test C-50, this resolves what was previously called “drift” or “chaos” in LA2004’s precession band into a structured signal: pure orbital beats libreate around the framework’s lattice equilibrium; k-involving beats undergo LOD-driven evolution that the framework’s proper-physics formula predicts. Scripts: scripts/l1_libration_test.py.

Scope and the n = 7 exclusion

This page covers L1 only. The canonical climate formula’s other two layers — L2 (3-line off-lattice 405-kyr carbon thermostat) and L3 (6 Heaviside step components at PETM, EOT, Mi-1, MMCT, iNHG, MPT) — are not orbital beats and carry no Berger-vs-Holistic attribution comparison. Both are documented at Climate Formula.

The LR04 full-record spectrum has a 4σ peak at T = 383.22 kyr (= 8H/7). The well-established 405-kyr eccentricity line sits 21.8 kyr away and is off the 8H lattice — its spectral energy leaks into the nearest lattice bin (n=7), which the divisor-spectrum scan detects but classifies as “Unpredicted” (no family-level beat predicts 383 kyr exactly). Including n=7 in L1 would double-count with L2; the 32 integers tabulated above are the complete L1 set.

See also

• Climate Summary — the synthesis statement this dual-attribution result supports
• Climate Formula — the canonical L1+L2+L3 architecture
• Insolation Null Test — empirical demonstration that adding Berger insolation features to L1+L2+L3 yields ΔR² = 0
• Related Work — position relative to recent peer-reviewed literature
• Eigenfrequencies — the g_j and s_j fundamental frequencies used by Berger
• Fundamental Cycles — the 8 × 6 PLANET_CYCLES period table
• Full per-integer reference (3d repo)  — all 20+ candidate combos per integer, rank-ordered