Diophantine Astrophysics arithmodynamics · a candidate field at the boundary of number theory and celestial mechanics

Which orbital architectures observed in the universe are forced by number-theoretic constraints, rather than selected by initial conditions or dynamics?

draft · day 2 · 2026-05-20 data: NASA Exoplanet Archive 5933 planets · 1037 multi-planet systems

Contents

Why this is a real gap
The conjectures (with one major reformulation)
Test 1 — partial-quotient distribution vs Gauss-Kuzmin
Test 2 — dynamical-age stratification
Test 3 — RV-only control for selection bias
Test 4 — Conjecture II at Farey-9
Test 5 — Khinchin and Lévy constants flips Conj I
Test 6 — Conjecture II with larger alphabets and Markov-chain test
Test 7 — Mixture fit and pile-up structure quantifies Conj I″
Synthesis and what to do next
Reproduction

1. Why this is a real gap

Three established fields each see one face of the same elephant and don't talk to each other.

Celestial mechanics catalogs mean-motion resonances case-by-case — Jupiter–Saturn 5:2, the Trappist-1 chain 8:5:3:2, the Kirkwood gaps in the asteroid belt — without a unifying theory of which integer ratios are admissible.
Diophantine approximation (Khinchin, Roth, the Markov spectrum) classifies irrationals by how badly they resist rational approximation.
Exoplanet statistics has accumulated ~5500 confirmed planets in ~1000 multi-planet systems — but no theory predicts the distribution of their period ratios from first principles.

The bridge: the observed period-ratio distribution is the pushforward of the cosmic initial-condition measure through some sequence of number-theoretic filters. Day-1 assumed the filter was KAM stability (favouring noble irrationals). Day-2 falsifies that and identifies the dominant filter as resonance trapping during migration (favouring low-order rationals).

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2. The conjectures

Conjecture I (original) — Khinchin Selection falsified, see Test 5

Long-lived multi-planet systems concentrate on period ratios whose continued-fraction expansion has bounded partial quotients — i.e. more noble, more KAM-stable irrationals.

The per-ratio Khinchin and Lévy constants both shift in the opposite direction at z ≈ +10 (Test 5).

Conjecture I″ — Resonance Excess strongly supported

Observed period ratios show systematically larger partial-quotient geometric mean (Khinchin) and faster-growing convergent denominators (Lévy) than the Gauss-Kuzmin null. The 1-D marginal density of {r} = r − ⌊r⌋ deviates from the Gauss measure at χ² = 111 on 19 dof (Test 7), with localizable pile-ups at 3:2 and just-below-integer (Lithwick–Wu). The dominant filter on period ratios is migration-driven resonance trapping, not KAM-stability suppression of resonant tori.

Conjecture II — Resonance Lattice Rigidity supported across Farey-9, 12, 15

For an n-planet system, the resonance chain — the tuple of nearest low-order commensurabilities — concentrates on a sparse subset of the combinatorially possible chains. Concentration is robust to the size of the alphabet (Tests 4 and 6). Consecutive resonances in a chain are non-independent at coarse resolution.

Conjecture III — Arithmetic Hair not yet tested

Galactic disk substructure (moving groups, phase-space spirals) is the image, under the galactic potential's integrable flow, of rational points on a lower-dimensional resonant submanifold of action space.

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3. Test 1 — partial-quotient distribution vs Gauss-Kuzmin

Take all 1037 multi-planet systems from the NASA Exoplanet Archive, sort each by period, compute adjacent ratios r = P_i+1/P_i, and for each r compute the continued-fraction expansion to depth 8. Null: Gauss-Kuzmin, P(a_k = m) = log₂(1 + 1/(m(m+2))).

a_k	empirical	Gauss-Kuzmin	ratio
1	0.430	0.415	1.04
2	0.167	0.170	0.98
3	0.088	0.093	0.94
4	0.056	0.059	0.95
5	0.040	0.041	0.99
6	0.028	0.030	0.95
7	0.019	0.023	0.83
8	0.017	0.018	0.92
9	0.015	0.014	1.02
≥ 10	0.140	0.137	1.02

χ² = 21.5 on 9 dof. The excess at a_k = 1 looked like evidence for noble-irrational favouring, but Test 5 shows that's overwhelmed by the heavy tails.

Empirical vs Gauss-Kuzmin distribution — Empirical (red) vs Gauss-Kuzmin (dark) over 10924 a_k samples.

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4. Test 2 — dynamical-age stratification

N_orbits = stellar_age × 365.25·10⁹ / P_{inner_days} spans log₁₀ 5.8 to 13.0 across 689 ratio-pairs with measured age.

quartile	median log₁₀ N_orbits	frac(a_k=1)	frac(a_k≥4)
Q1	10.43	0.430	0.328
Q2	11.31	0.428	0.299
Q3	11.64	0.437	0.315
Q4	12.04	0.434	0.321

Quartile trend in dynamical age — No monotonic trend across 1.6 decades of dynamical age.

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5. Test 3 — RV-only control for selection bias

sample	n systems	n a_k	frac(a=1)	χ² / dof
all multi	1037	10924	0.431	21.5 / 9
RV-only	208	2072	0.435	7.4 / 9
Transit-only	683	7214	0.434	14.9 / 9

RV vs Transit vs Gauss-Kuzmin — All three samples deviate from Gauss-Kuzmin with the same per-sample magnitude. Selection bias is not the explanation.

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6. Test 4 — Conjecture II at Farey-9

Resonance alphabet Σ = {(p,q) : gcd=1, p > q ≥ 1, p+q ≤ 9}, |Σ| = 13. Each adjacent ratio gets the nearest symbol.

L	n systems	distinct observed	uniform expectation
2 (3-planet)	219	79	~123
3 (4-planet)	81	70	~79

3-planet systems in the period-ratio plane — Three-planet systems in the (P₂/P₁, P₃/P₂) plane. Grey lines = low-order Farey ratios; Trappist-1 marked in red.

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7. Test 5 — Khinchin and Lévy constants flips Conjecture I

Two universal constants govern continued-fraction expansions for almost every real (Lebesgue):

Khinchin K₀ ≈ 2.6855. Geometric mean of partial quotients converges to K₀; E[ln a_k] = ln K₀ ≈ 0.988.
Lévy L₀ = π²/(12 ln 2) ≈ 1.187. (1/n) log q_n converges to L₀.

Both are smaller for noble-ish irrationals (KAM-favoured). Conjecture I predicted period ratios shift downward.

quantity	theoretical	empirical	SE	z
⟨ln a_k⟩	0.988	1.194	0.020	+10.09
(1/n) log q_n	1.187	1.428	0.022	+10.95
K = exp ⟨ln a⟩	2.685	3.301	—	—

Khinchin constant histogram — Per-ratio Khinchin mean over 1564 adjacent pairs. The empirical centre is decisively to the right of K₀.

Levy constant histogram — Per-ratio Lévy slope. Empirical mean > L₀; the golden ratio (ln φ) is far to the left of the bulk.

The flip: period ratios are closer to low-order rationals than random, not further. Both Khinchin and Lévy at z ≳ 10 on the opposite side of the Gauss-Kuzmin null.

Bonus: conditional dependence of consecutive partial quotients

Under Gauss-Kuzmin the (a_k) are asymptotically i.i.d. Contingency table on buckets {1, 2, 3, 4, ≥5}:

a_k \ a_k+1	=1	=2	=3	=4	≥5
=1	1933	882	519	331	1681
=2	850	352	172	125	558
=3	519	190	93	66	235
=4	348	115	57	45	154
≥5	1520	526	309	172	722

χ² = 173.2 on 16 dof (critical@p=0.01 ≈ 32).

Conditional dependence heatmap — (obs−exp)/√exp from independence. (1,1) and (≥5,≥5) cells deviate negatively; (1,≥5) and (≥5,1) deviate positively — small/large alternation is favoured over repetition.

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8. Test 6 — Conjecture II with larger alphabets and Markov-chain test

alphabet	\|Σ\|	L=2 distinct	uniform exp.	entropy (bits)	uniform entropy
Farey-9	13	79	123	5.90	7.40
Farey-12	22	131	176	6.73	7.78
Farey-15	35	166	201	7.18	7.78

Top L=2 chain types, Farey-12

chain	count
2:1 — 2:1	14
2:1 — 7:4	6
2:1 — 11:1	5
2:1 — 7:3	4
5:3 — 7:4	4
7:3 — 7:3	4

Markov-chain independence test

alphabet	dof	χ² raw	χ² merged	verdict
Farey-9	144 / 100	259.9	142.0	significant
Farey-12	441 / 196	493.4	210.6	marginal
Farey-15	1156 / 256	864.6	247.8	underpowered

Markov-chain heatmap — Deviation from independence in (s₁, s₂) for Farey-12 chain symbols (top-12 shown). Diagonal excesses dominate — consecutive identical resonances cluster.

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9. Test 7 — Mixture fit and pile-up structure quantifies Conj I″

Take the fractional part {r} = r − ⌊r⌋ ∈ [0, 1) of every adjacent ratio (n = 1564). This pools over integer parts and gives a single 1-D density to test.

Non-parametric chi-square (20 equal bins)

baseline	χ²	dof	critical @ p=0.001
Gauss measure 1/((1+x) ln 2)	111.2	19	43.8
uniform on [0,1)	50.5	19	43.8

Both nulls are decisively rejected. The data is closer to uniform than to Gauss (the Gauss measure itself tilts toward x=0, and the data does not), but is non-uniform with specific identifiable structure.

Top bin-by-bin deviations from Gauss

{r} window	observed	Gauss expected	z	physical interpretation
[0.50, 0.55)	114	74.0	+4.65	3:2 pile-up
[0.10, 0.15)	63	100.3	−3.72	gap just past integer (Lithwick–Wu)
[0.90, 0.95)	88	58.6	+3.84	pile-up just below integer (Lithwick–Wu)
[0.05, 0.10)	71	105.0	−3.32	same gap, inner edge
[0.85, 0.90)	83	60.2	+2.94	broad 2:1 pile-up shoulder

The pile-up/gap pattern at {r} ≈ 0.90–0.95 (excess) and {r} ≈ 0.05–0.15 (deficit) is the well-known Lithwick & Wu (2012) asymmetric structure around first-order mean-motion resonances: systems pile up just inside the resonance (period ratio slightly less than integer) and leave a gap just past it. Conjecture I″ predicted this pattern from the resonance-trapping framework, and it appears here at z > 3 in two independent bin pairs.

Narrow-window pile-up at ±2% of each Farey-9 center

center {p/q}	label	observed	Gauss expected	ratio	z
0.000	integer (2:1, 3:1, …)	44	67.4	0.65	−2.85
0.250	5:4	55	72.2	0.76	−2.03
0.333	4:3	71	67.7	1.05	+0.40
0.500	3:2	84	60.2	1.40	+3.07
0.667	5:3	66	54.2	1.22	+1.61

The narrow ±2% windows confirm: only 3:2 shows a sharp pile-up at the exact resonance. The "near-integer" excess from the bin-chi-square test is asymmetric (Lithwick–Wu) and lives at {r} ≈ 0.90–0.95, not at {r} = 0 itself — which actually shows a slight deficit (z = −2.85) at the ±2% scale.

Narrow-window pile-up bar chart — Observed vs Gauss-expected mass within ±2% of each Farey-9 center. Only the 3:2 pile-up is sharp at the exact-resonance scale; the 2:1 / 3:1 / … pile-ups are broad and offset (see bin table above).

Mixture-model fit: α·Gauss + (1−α)·Farey

We model the empirical density as a mixture of the Gauss measure and a KDE-style mass concentrated at Farey-N centers with bandwidth σ:

p({r}) = α · 1/((1+{r}) ln 2) + (1−α) · (1/Z) ∑_{(p,q) ∈ Σ_N} w_{p,q} · K({r}; {p/q}, σ)

MLE over a 51×41 (α, σ) grid:

Farey alphabet	\|Σ\|	α	σ	ΔLL vs pure Gauss	2·ΔLL
Farey-9	5	0.02	0.500*	+28.5	57.1
Farey-12	8	0.02	0.500*	+28.6	57.2
Farey-15	12	0.02	0.500*	+28.6	57.3
Farey-20	22	0.00	0.500*	+28.7	57.3

*σ hits the upper grid boundary at 0.5 in every run. This is informative: the data prefers very broad Gaussian bumps, which means the "Farey trapping" picture should not be read as razor-sharp peaks at low-order ratios but as broad enhancements with width ~20–50% of the period-ratio scale. Sharp-peak mixture models are non-identifiable here.

Mixture density vs data — Empirical histogram of {r} (grey), pure Gauss measure (dark), Farey-12 component at MLE σ (red), and the best mixture (green). Vertical lines mark Farey-12 centers, with line width proportional to weight.

Profile likelihood for α with σ fixed at the MLE (Farey-12). The MLE favours α ≈ 0; 95% CI extends only to about 0.27. Most of the {r} measure is captured by Farey neighborhoods, not by the Gauss component.

Alphabet ablation — Farey fraction (1 − α) and ΔLL vs pure Gauss across Farey-N alphabets. The Farey weight monotonically increases as the alphabet grows (the model approaches a KDE in the limit).

What the fit tells us, cleanly:

The {r} distribution is decisively non-Gauss (χ² = 111 on 19 dof).
The deviations are localized at 3:2 (sharp pile-up) and at the Lithwick–Wu shoulder/gap around 2:1 (broad asymmetric structure).
A simple α·Gauss + (1−α)·Farey mixture model is over-parameterized at large σ: the Farey weight 1 − α is near-unity at every alphabet size. The data's structure is broader than the sharp-peak ansatz; a more honest model would parameterize the resonance width per (p,q).

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10. Synthesis and what to do next

Falsified: Conjecture I in its original "noble irrationals dominate" form (Test 5).
Supported, with structure now mapped: Conjecture I″ — period ratios are closer to low-order rationals than random (Tests 5, 7), with the specific deviation pattern being 3:2 pile-up plus the Lithwick–Wu asymmetric structure at 2:1 (Test 7).
Supported: Conjecture II at multiple Farey alphabet sizes; consecutive resonances non-independent at Farey-9 (Tests 4, 6).
Open: Conjecture III (Gaia substructure).

Open problems (ordered by accessibility)

Build a physical mixture model: instead of constant σ, give each (p,q) its own width σ_p,q ∝ μ^2/3/(pq) (the resonance libration scale for mass ratio μ). The two-parameter fit becomes a one-physical-parameter fit (μ-distribution + a trapping-efficiency function).
Decompose the Lithwick–Wu asymmetry: fit the "pile-up − gap" signature near each first-order resonance (4:3, 3:2, 2:1, 5:4) separately and check whether the asymmetry magnitude scales as predicted by capture-from-disk theory.
Extend the Conjecture II Markov-chain test with a sample large enough to resolve Farey-15 structure — needs PLATO or the next Kepler reanalysis.
Test Conjecture III on Gaia DR3 phase-space spirals: are spiral windings rational multiples of local vertical/radial epicycle frequencies?

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11. Reproduction

# Pull data
curl -sG "https://exoplanetarchive.ipac.caltech.edu/TAP/sync" \
  --data-urlencode "query=SELECT hostname, pl_name, pl_orbper, st_age, discoverymethod FROM ps WHERE default_flag = 1 AND pl_orbper IS NOT NULL" \
  --data-urlencode "format=csv" -o ps_dm.csv

# Day-1 tests
python3 analyze.py
python3 analyze_dynamical.py
python3 analyze_rv_and_c2.py

# Day-2 tests
python3 analyze_conjecture_I_deep.py
python3 analyze_conjecture_II_extended.py
python3 analyze_mixture_v3.py

Dependencies: numpy, matplotlib.

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