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?

v1.5 · 2026-05-21 draft · day 3 data: NASA Exoplanet Archive · Gaia DR3 · JPL SBDB (asteroids + TNOs) 5933 planets · 115 k stars · 100 k asteroids · 5 k TNOs

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).

2. The conjectures

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))).

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.

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.

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

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.

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.

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}:

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

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

Top L=2 chain types, Farey-12

Markov-chain independence test

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)

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

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

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.

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:

*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.

10. Test 8 — Physical mixture and Lithwick–Wu asymmetry physical scaling fails; LW signature confirmed

Test 7 left a clean open problem: the constant-σ ansatz over-parameterized the data (σ hit the upper grid boundary). The natural fix is a physical width — each resonance gets its own σ tied to its libration scale. For a coprime resonance (p, q) with p > q ≥ 1, define order = p − q. First-order mean-motion resonances (2:1, 3:2, 4:3, 5:4) have order 1 and the widest libration; higher orders are narrower.

We test three width-scaling laws, all parameterized by a single σ₀:

We also stop collapsing integer ratios (2:1, 3:1, 4:1, …) into a single center: each gets its own bump at {r}=0 with its own width.

Model comparison

*σ hits the upper grid boundary in M0 (the data still wants very broad bumps even with per-(p,q) centers).

Why this matters: the simplest physical picture — each (p,q) trap has a Gaussian-shaped basin of attraction with width set by single-resonance libration theory — does not generate the empirical {r} distribution. Either (a) the population is dominated by interaction effects between multiple resonances, (b) selection effects mimic broader pile-ups, or (c) the relevant width is set by the disk-migration history, not the late-time libration scale.

Lithwick–Wu asymmetric pile-up test

Lithwick & Wu (2012) showed that Kepler near-resonant pairs concentrate just outside first-order MMRs (period ratio slightly greater than p/q), with a depleted region just inside. We test this for the four canonical first-order resonances by counting ratios in three signed bands around each center:

• below:   r ∈ (p/q − W, p/q − 0.02)
• near:   r ∈ (p/q − 0.02, p/q + 0.02) — exact-resonance region
• above:   r ∈ (p/q + 0.02, p/q + W)

Window W = 0.05 (the narrow LW pile-up scale):

Widening the window to ±10% or ±15% the asymmetry washes out (z < 1 for all resonances), because we then include the broader r-distribution that surrounds each resonance. The LW effect lives on the 2–5% scale, exactly where libration theory predicts.

11. Test 9 — Asymmetric LW pair model fit LW asymmetry detected at z ≈ 2.7

Test 8 confirmed a Lithwick–Wu asymmetry at 2:1 in narrow-window counts. Test 9 generalises that to a parametric model that fits all four first-order resonances simultaneously with a shared asymmetry parameter γ.

Model

For each first-order MMR center c ∈ {0, 1/2, 1/3, 1/4} (for 2:1, 3:2, 4:3, 5:4 resp.), define an asymmetric Gaussian pair:

f_LW(x | c) = (1/(1+γ)) · N(x; c−ε, σ) + (γ/(1+γ)) · N(x; c+ε, σ)

where ε is the pile-up offset, σ the pile-up width, γ the above/below amplitude ratio. The total model is

p({r}) = α · g({r}) + (1−α) · (1/4) Σ_res f_LW({r} | c_res)

where g is the Gauss measure. Free parameters: α, ε, σ, γ.

We constrain the search to the LW-physical scale identified by Lithwick & Wu (2012): ε ∈ [0.5%, 4%] (offset of 1–4% from exact resonance), σ ∈ [0.3%, 2%] (sharp peaks). Without this constraint the model is non-identifiable in the broad-σ limit (Test 7 showed why).

MLE results

Significance of the asymmetry

Profile 68% CI on γ: [3.5, 5.0+]; 95% CI: [1.8, 5.0+]. The upper bound is poorly constrained because γ saturates the grid — γ = 5 and γ = 10 produce nearly identical LW pair shapes (one component shrinks to zero). The robust conclusion is γ > 1.8 at 95% confidence — asymmetry strongly favouring the "above" position.

Universality across resonances

Per-resonance γ fit (others held at γ = 1):

All four first-order MMRs independently prefer γ ≫ 1 in the same direction. The joint 4-γ fit converges to γ ≈ 4 for each. Fitting per-resonance γ (4 extra parameters) gives ΔLL = −0.30 relative to shared γ — worse, so the shared model is favoured. The LW asymmetry is universal across first-order resonances at the fit-detectable level.

12. Test 10 — Joint LW + broad-Farey mixture Gauss baseline rejected; LW at z ≈ 3.3

Tests 7–9 left two complementary observations:

• The {r} distribution has broad pile-ups around Farey-12 centers (Test 7: best constant-σ mixture at Farey-12 / Farey-20 with σ → grid boundary 0.5 captured ΔLL ≈ +28 vs pure Gauss).
• It also has a narrow asymmetric Lithwick–Wu pair just outside each first-order resonance (Test 9: ε ≈ 1.7%, σ ≈ 0.4%, γ ≈ 5, significant at z ≈ +2.7 in isolation).

These look like two physically distinct components — a broad shoulder (mig­ration-induced bulk) plus narrow asymmetric peaks (LW pile-ups). Test 10 fits them jointly.

Model

p({r}) = α₁ · g({r})                                  (Gauss baseline)
       + α₂ · f_LW({r} | ε=0.017, σ_LW=0.004, γ=5)    (Test-9 LW pairs at 2:1, 3:2, 4:3, 5:4)
       + α₃ · f_Farey({r} | σ_F)                       (Farey-12 broad bumps)

with α₁ + α₂ + α₃ = 1 and all ≥ 0. The LW pair shape is held at the Test-9 MLE; broad-Farey uses per-(p,q) centers (22 of them) with free bandwidth σ_F. Free parameters: α₁, α₂, σ_F.

MLE

LL = +5.28; ΔLL vs pure Gauss = +34.01. (Positive LL means each data point's log-density is on average above 0, i.e. above uniform — the empirical density is above uniform on net.)

Significance of the LW component

In the joint fit (against a calibrated broad-Farey background) the LW component is detected at z = 3.26 — substantially stronger than the z = 2.69 of the marginal Test 9 fit. The LW contribution carries 1.0–4.0% of the {r} mass at 95% confidence.

13. Test 11 — LW pair test at 2nd-order MMRs same direction, ε scaled by ~⅓

Test 9 found the LW asymmetric pair structure at first-order MMRs (2:1, 3:2, 4:3, 5:4) with offset ε ≈ 1.7%. Does the same signature exist at second-order resonances (p − q = 2: 5:3, 7:5, 9:7, 11:9)? If yes, capture-from-disk theory predicts ε should scale with the resonance libration width, which for second-order goes as μ instead of μ^2/3 — i.e. much smaller.

Narrow-window counts at 2nd-order centers (no structure visible)

No individual 2nd-order resonance shows narrow-window asymmetry above noise (|z| < 0.6 for all four). The signal, if it exists, lives in the parametric fit.

Parametric LW pair fit at 2nd-order centers (shared ε, σ, γ)

Per-resonance γ scan (others held at γ = 1):

14. Test 12 — Conjecture III on Gaia local sample naive Farey-ratio form not supported

Conjecture III predicted that galactic-disk substructure (moving groups, phase-space spirals) consists of the image of rational points of the local frequency lattice under the galactic potential's integrable flow. The clean operational reading is: the V_φ positions of moving groups in the solar neighbourhood should cluster at Farey-low fractions of the circular velocity v_circ.

Data

Pulled 115 145 Gaia DR3 stars within 100 pc with full 6-D phase space (parallax/parallax_error > 10, RV present, RV uncertainty < 5 km/s, RUWE < 1.4). Converted ICRS astrometry to LSR galactic Cartesian velocities (U, V, W) via astropy, with the Schoenrich-Binney-Dehnen (2010) solar peculiar velocity. After a sane kinematic cut (|U|, |V| < 200 km/s, |W| < 100): 114 711 stars.

Farey-ratio test on canonical moving-group V_LSR

For each known moving group, we compute V_φ/v_circ (taking v_circ = 232 km/s) and look up the nearest Farey-15 fraction in (0.5, 1.5):

Two of seven groups (Wolf 630 at 6:7 and Arcturus at 4:7) land within 0.3% of a Farey-15 ratio — striking. But the remaining five sit at typical random-spacing distances (|Δ| = 0.013–0.039). Under the uniform null, the expected number of groups within 0.01 of a Farey-15 ratio is ~ 7 × 0.40 = 2.8, very close to the observed 2 (Wolf 630 only at ±0.01; Arcturus is at ±0.0025, also passes). So this is not a clear excess.

Background-population test

For 5000 random local stars within V_φ/v_circ ∈ (0.5, 1.5):

The local-stars population is, if anything, less concentrated near Farey-15 ratios than uniform would predict. There is no population-level Farey trapping in V_φ/v_circ.

Caveats and the steel-man

This test is much weaker than the analogous exoplanet tests, for identifiable reasons:

1. Wrong observable. The conjecture as I stated predicts rational frequency ratios (Ω, κ, ν_z), not rational velocity ratios. For a flat rotation curve, κ/Ω = √2 — an irrational. The frequency lattice at the LSR is already irrational, so the conjecture has to be read as something like "moving groups occur at solutions of m·Ω + n·κ = constant for small integers (m, n)" — which is the standard literature claim about bar/spiral resonances and does NOT predict Farey ratios of V_φ.
2. Wrong sample. The 100 pc cube is dominated by thin-disk stars with V_φ near v_circ. A test of Conjecture III is more naturally posed in action space (J_R, J_z, L_z) using a deeper halo + thick-disk sample, or at the Antoja-2018 vertical phase-space spiral specifically.
3. Wrong null. The Farey-15 alphabet is dense enough in (0.5, 1.5) that random V_φ/v_circ values land within 0.01 of some Farey ratio ~40% of the time. The signal must be very tight to stand out.

The honest conclusion: Conjecture III as a Farey-ratio claim about V_φ/v_circ is wrong. The conjecture can probably be reformulated as a resonance-condition claim that is consistent with the literature (Hercules ↔ bar OLR, etc.), but in that reformulation it stops being a new mathematical claim and becomes a restatement of existing dynamics.

15. Test 13 — Asteroid belt as parallel population Kirkwood gaps recovered; population-contrast result

The exoplanet tests find resonance trapping (pile-ups at low-order rationals). The asteroid belt is the canonical opposite — its Kirkwood gaps are depletions at low-order rationals with Jupiter, the textbook KAM-stability filter. We apply the same framework to 100 000 main-belt asteroids (2.0 ≤ a ≤ 3.5 AU, from JPL SBDB) to see whether the framework discriminates between trapping and filtering as populations.

Sanity check: Kirkwood gaps appear in the right places

The script's gap-finder lists:

Good — the framework is using the right data.

Test A: partial-quotient distribution vs Gauss-Kuzmin

χ² = 8 170 on 9 dof (cf. 21.5 for exoplanets). Per sample, the deviation is stronger than the exoplanet deviation (we expected scaling ≈ 21.5 × 100 000 / 1564 ≈ 1 400 if effect sizes were equal; observed 8 170 is ~5.8× larger per sample).

Tests B & C: Khinchin and Lévy — same direction as exoplanets

Test D: the cleanest discriminator — {r}-bin distribution

20-bin chi-square on {r} = (P_J/P_a) − ⌊P_J/P_a⌋:

Both decisively reject Gauss and uniform. The top bin-deviations from Gauss tell the discriminating story:

Reformulated conjecture (informal)

We can now state a cleaner population-comparison conjecture:

16. Test 14 — Kuiper belt (TNOs) as mixed-mode population Plutinos at z=+23; first noble-leaning sample

The Kuiper belt is a deliberately mixed-mode test of Conjecture I‴. Three sub-populations coexist:

• Plutinos: ~250 known TNOs resonance-trapped at 3:2 with Neptune (P_TNO/P_N = 1.5; a ≈ 39.5 AU). Migration-trapped during Neptune's outward migration.
• twotinos: small population trapped at 2:1 (P_TNO/P_N = 2.0; a ≈ 47.8 AU).
• Cold & hot classical KBOs: ~thousands of TNOs in the smooth band a ≈ 42–47 AU, mostly between Neptune MMRs. The dominant population by count.

Predictions before the test: trapping pile-ups (positive {r}-deviations) at 3:2 and 2:1; otherwise smooth or KAM-filtered-like. Data: 5 184 TNOs with 30 ≤ a ≤ 60 AU from JPL SBDB.

The Plutino pile-up jumps out

Narrow-window pile-up test at known Neptune MMRs (±2% in r):

Plutinos confirmed at z = +23.4 — a clean trapping pile-up, as Conjecture I‴ predicted. The 7:4 excess is the cold classical KBO bulk at a ≈ 44 AU: not a sharp resonance but a population of stable, smooth orbits that happen to sit there.

Surprisingly, twotinos at 2:1 do not show as a clear pile-up (z = −2.06) — the canonical "twotinos" are too sparse relative to the local classical-belt background to register in this sample. The 4:3, 5:4 deficits reflect the under-populated inner edge of the Kuiper belt (most discovered TNOs have a > 38 AU).

Khinchin and Lévy: first noble-leaning population

Implication: the framework needs at least two observables

Khinchin alone is degenerate: trapping and local filtering at the same resonance can both push it up (because partial quotients near rationals are large in both cases — pile-up at p/q gives a_k = q + small, gap at p/q means surviving asteroids accumulate at p/q ± ε, also a_k = q + small). Only when the sub-population sits at non-resonant noble locations does Khinchin drop below K₀ — and that's the TNO classical belt's signature.

The minimum sufficient discriminator is the signed {r}-bin profile: where pile-ups are, where gaps are, and where the distribution looks noble. This is what Conjecture I‴ should be about going forward.

17. Synthesis and what to do next

• Falsified: Conjecture I in its original "noble irrationals dominate" form (Test 5).
• Falsified: the naive resonance-libration ansatz σ_p,q ∝ 1/order. Order-dependent widths fit worse than constant widths (Test 8). Single-resonance libration theory is not the operative width-setting mechanism.
• Supported, with the Lithwick–Wu signature now confirmed and parameterised: Conjecture I″. Bin pile-up at 3:2 (z = +4.65, Test 7); narrow-window asymmetry at 2:1 (z = −3.08 at ±5%, Test 8); parametric asymmetric LW pair fit gives γ ≈ 4–5 with z_eff ≈ +2.7 (Test 9), universally across 2:1, 3:2, 4:3, 5:4 at ε ≈ 1.7% offset and σ ≈ 0.4% width — quantitatively matching the LW (2012) prediction.
• Supported: Conjecture II at multiple Farey alphabet sizes; consecutive resonances non-independent at Farey-9 (Tests 4, 6).
• Falsified in naive form: Conjecture III. The Farey-ratio prediction for V_φ/v_circ of moving groups fails on the local Gaia sample (Test 12).
• New result, supported: the framework discriminates populations (Test 13). At {r}∈[0.50,0.55), exoplanets show z = +4.65 excess (3:2 pile-up) and asteroids show z = −11.4 deficit (5:2/7:2 Kirkwood gaps).
• New result, supported but richer than predicted: Test 14 on 5 184 TNOs confirms the Plutino 3:2 pile-up (z = +23.4 narrow-window) — same sign as exoplanets at the same resonance. But the Khinchin mean drops below the Gauss-Kuzmin null (z = −7.4): TNOs are the first noble-leaning population because the classical-KBO bulk sits between MMRs. Conjecture I‴ revised: the discriminator is the signed {r}-bin profile, not Khinchin alone.

What changed our understanding in Test 8

The natural physical ansatz — that each Farey symbol carries a Gaussian basin with width set by single-resonance libration theory — does not generate the observed distribution. The data is broader, more asymmetric, and more strongly weighted toward the lowest-order resonances (2:1, 3:2) than libration theory predicts. The right physical model is not "resonances act independently" but something coupling multiple resonances through migration history or secular dynamics.

Open problems (revised after Test 8)

1. Replace the per-(p,q) Gaussian ansatz with the Lithwick–Wu pile-up model directly: each first-order resonance contributes a pair (depleted-near, enhanced-above) component instead of a symmetric bump. Fit the asymmetry magnitude per resonance.
2. Connect the asymmetry pattern to disk-migration history: for a given gas-disk lifetime and surface-density profile, what is the predicted (below, near, above) distribution? Compare to the table above.
3. Identify which low-order chains (from Conjecture II Test 4) correspond to Lithwick–Wu populations vs Farey-trapped chains — this distinguishes "frozen-in migration" architectures from "tidally evolved" ones.
4. Extend the Conjecture II Markov-chain test with a sample large enough to resolve Farey-15 structure (needs PLATO or the next Kepler reanalysis).
5. Test Conjecture III on Gaia DR3 phase-space spirals: are spiral windings rational multiples of local vertical/radial epicycle frequencies?

18. 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
python3 analyze_mixture_physical.py
python3 analyze_lw_pair.py
python3 analyze_joint.py
python3 analyze_lw_2nd_order.py

# Gaia DR3 pull (200+ MB) and Conjecture III test
curl -sG "https://gea.esac.esa.int/tap-server/tap/sync" \
  --data-urlencode "REQUEST=doQuery" --data-urlencode "LANG=ADQL" \
  --data-urlencode "FORMAT=csv" \
  --data-urlencode "QUERY=SELECT source_id,ra,dec,l,b,parallax,parallax_error,pmra,pmdec,radial_velocity,radial_velocity_error FROM gaiadr3.gaia_source WHERE parallax > 10 AND parallax_over_error > 10 AND radial_velocity IS NOT NULL AND radial_velocity_error < 5 AND ruwe < 1.4" \
  -o gaia_100pc.csv
python3 analyze_gaia_c3.py

# Asteroid belt pull and Diophantine analysis (Test 13)
curl -sG "https://ssd-api.jpl.nasa.gov/sbdb_query.api" \
  --data-urlencode 'sb-cdata={"AND":["a|GE|2.0","a|LE|3.5"]}' \
  --data-urlencode "fields=full_name,a,e,per,H" \
  --data-urlencode "limit=100000" -o asteroids_mb.json
python3 analyze_asteroids.py

# TNO pull and Kuiper-belt analysis (Test 14)
curl -sG "https://ssd-api.jpl.nasa.gov/sbdb_query.api" \
  --data-urlencode 'sb-cdata={"AND":["a|GE|30.0","a|LE|60.0"]}' \
  --data-urlencode "fields=full_name,a,e,i,per,H" \
  --data-urlencode "limit=20000" -o tnos.json
python3 analyze_kuiper.py

Dependencies: numpy, matplotlib, scipy (for filters), astropy (for ICRS → galactic).

19. Sources & code

All analysis scripts and the cached data dump are served from this site, so the entire pipeline is reproducible offline:

Original data source: NASA Exoplanet Archive Table Access Protocol (TAP), https://exoplanetarchive.ipac.caltech.edu/TAP/sync. The exact query used to produce ps_dm.csv is in §13 above.

The plots embedded in this page (partial_quotients.png, c1_mixture_fit.png, …) are produced as side-effects of these scripts and can be downloaded by right-clicking each figure.

20. Bibliography

Observational astronomy

• Lithwick, Y. & Wu, Y. (2012). Resonant repulsion of Kepler planet pairs. ApJL 756, L11. arXiv:1204.2555. The asymmetric pile-up signature tested in §10 and §11.
• Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. (2014). Architecture of Kepler's multi-transiting systems. II. New investigations with twice as many candidates. ApJ 790, 146. arXiv:1202.6328. Population-level pile-ups at 3:2 and 2:1 mean-motion resonances in Kepler.
• Gillon, M., Triaud, A. H. M. J., Demory, B.-O., et al. (2017). Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1. Nature 542, 456–460. arXiv:1703.01424. Source of the TRAPPIST-1 8:5:3:2 resonance chain used as benchmark.
• NASA Exoplanet Science Institute. NASA Exoplanet Archive — Planetary Systems table (ps). https://exoplanetarchive.ipac.caltech.edu/. DOI: 10.26133/NEA12. Primary data source for Tests 1–11.
• Gaia Collaboration (2023). Gaia Data Release 3. Summary of the content and survey properties. A&A 674, A1. arXiv:2208.00211. Source of the 115 k local sample used in Test 12.
• Dehnen, W. (2000). The effect of the outer Lindblad resonance of the Galactic bar on the local stellar velocity distribution. AJ 119, 800–812. arXiv:astro-ph/9911161. Standard interpretation of the Hercules moving group as the bar's outer Lindblad resonance — the canonical physics that Conjecture III's "Arithmetic Hair" framing tried to generalize and that Test 12 instead recovers as a single resonance condition rather than a Farey-ratio law.
• Antoja, T., Helmi, A., Romero-Gómez, M., et al. (2018). A dynamically young and perturbed Milky Way disk. Nature 561, 360–362. arXiv:1804.10196. Discovery of the Gaia phase-space spiral; the canonical object Conjecture III would predict structure for, though Test 12 used the simpler (U, V) plane.
• Schönrich, R., Binney, J., & Dehnen, W. (2010). Local kinematics and the local standard of rest. MNRAS 403, 1829–1833. arXiv:0912.3693. Solar peculiar velocity (U, V, W) = (11.10, 12.24, 7.25) km/s used in Test 12.
• Kirkwood, D. (1866). On the theory of meteors. Proc. American Association for the Advancement of Science, 1866 meeting. Original observation of the asteroid gaps at low-order resonances with Jupiter — the phenomenon Test 13 recovers as the discriminating signature against the exoplanet pile-ups.
• Wisdom, J. (1982). The origin of the Kirkwood gaps: A mapping for asteroidal motion near the 3:1 commensurability. AJ 87, 577–593. Modern dynamical explanation of the 3:1 Kirkwood gap via chaotic eccentricity excitation — the canonical KAM-stability removal mechanism.
• NASA/JPL Small-Body Database (SBDB). SBDB Query API. https://ssd-api.jpl.nasa.gov/doc/sbdb_query.html. Source of the 100 k main-belt asteroid orbital elements used in Test 13 and the 5 k TNO records used in Test 14.
• Malhotra, R. (1995). The origin of Pluto's orbit: Implications for the Solar System beyond Neptune. AJ 110, 420. arXiv:astro-ph/9504036. Resonance-trapping (migration capture) origin of the 3:2 Plutinos — the precise mechanism producing the z = +23 pile-up Test 14 detects.
• Chiang, E. I. & Jordan, A. B. (2002). On the Plutinos and twotinos of the Kuiper Belt. AJ 124, 3430–3444. arXiv:astro-ph/0210440. Characterization of the 3:2 and 2:1 resonant populations in the Kuiper belt; relevant for the asymmetric Test 14 detection (clean Plutinos, weak twotinos).

Celestial mechanics

• Wisdom, J. (1980). The resonance overlap criterion and the onset of stochastic behavior in the restricted three-body problem. AJ 85, 1122. Foundational on chaotic dynamics near MMRs — relevant background for §10.
• Murray, C. D. & Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press. Reference for libration-width formulas σ ∝ μ^2/3 tested as M1/M2 in §10.

Diophantine approximation and continued fractions

• Khinchin, A. Ya. (1935, English transl. 1964). Continued Fractions. University of Chicago Press. Original statement and proof of Khinchin's theorem on the geometric-mean limit K₀ ≈ 2.6855; tested in §7.
• Lévy, P. (1929). Sur le développement en fraction continue d'un nombre choisi au hasard. Compositio Mathematica 3, 286–303. Original derivation of the Lévy constant L₀ = π²/(12 ln 2); tested in §7.
• Roth, K. F. (1955). Rational approximations to algebraic numbers. Mathematika 2, 1–20. Theorem giving the irrationality measure of algebraic numbers — the asymptotic notion behind Conjecture I.

KAM theory and Hamiltonian stability

• Kolmogorov, A. N. (1954). On conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR 98, 527–530.
• Arnold, V. I. (1963). Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Russian Mathematical Surveys 18, 9–36.
• Moser, J. (1962). On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1962, 1–20. The KAM trio: foundational results on the survival of quasi-periodic motions under perturbation. The qualitative motivation for Conjecture I — now reformulated as Conjecture I″ after §7.

Statistical methods

• Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Math. Stat. 9, 60–62. Theorem behind the 2·ΔLL ~ χ² nested-model tests in §11.
• Harris, C. R., Millman, K. J., van der Walt, S. J., et al. (2020). Array programming with NumPy. Nature 585, 357–362. doi:10.1038/s41586-020-2649-2.
• Hunter, J. D. (2007). Matplotlib: A 2D graphics environment. Computing in Science & Engineering 9, 90–95. doi:10.1109/MCSE.2007.55.

21. Changelog

• v1.5 — 2026-05-21
  • Test 14: Diophantine analysis on 5 184 TNOs (JPL SBDB, 30 ≤ a ≤ 60 AU) against Neptune. Plutino 3:2 pile-up detected at z = +23.4 (narrow-window) — clean trapping confirmation. Other MMRs (4:3, 5:4, 5:2, 7:3, 2:1) under-represented; twotinos at 2:1 surprisingly show z = −2.06 deficit. Classical-KBO bulk at a ≈ 42–47 AU dominates the {r} distribution.
  • First noble-leaning population: TNO Khinchin K = 2.585 < K₀ = 2.685 at z = −7.44, the opposite direction from exoplanets and asteroids. Driven by the classical-KBO bulk between MMRs, the closest empirical analogue to "noble irrationals" in this analysis.
  • Conjecture I‴ refined: the discriminator is the signed {r}-bin profile, not the Khinchin mean alone (Khinchin is degenerate between trapping and local KAM filtering; only "between-resonance" populations show K < K₀).
  • Three-population Khinchin/Lévy comparison plot.
  • Bibliography expanded with Malhotra 1995 (Plutino origin via Neptune migration) and Chiang & Jordan 2002 (Plutinos & twotinos). 23 references total.
• v1.4 — 2026-05-20
  • Test 13: parallel Diophantine analysis on 100 000 main-belt asteroids (JPL SBDB). Kirkwood gaps recovered at 4:1, 3:1, 5:2, 7:3, 2:1 with their canonical a values. χ² vs Gauss = 10 443 on 19 dof. Khinchin and Lévy shift in the same direction as exoplanets (toward rationals) but with much smaller magnitude.
  • Population-contrast result: at {r} = [0.50, 0.55), exoplanets show z = +4.65 excess (3:2 pile-up); asteroids show z = −11.4 deficit (5:2/7:2 Kirkwood gaps). Same {r}-location, opposite sign — the framework discriminates resonance-trapping from KAM-stability filtering as populations.
  • New conjecture: Conjecture I‴ (Population-Sign Reversal) — the sign of the {r}-bin deviation from Gauss-Kuzmin at low-order rationals identifies which filter dominates: positive (pile-up) for trapping, negative (gap) for KAM stability.
  • Bibliography expanded with Kirkwood 1866, Wisdom 1982 (3:1 chaos), JPL SBDB API. Total: 21 references.
• v1.3 — 2026-05-20
  • Test 12: Conjecture III tested on 115 145 Gaia DR3 stars within 100 pc with full 6-D phase space. Computed (U, V)_LSR via astropy; ran Farey-ratio test on canonical moving-group V_φ/v_circ. Naive Farey-ratio form fails: only Wolf 630 (→ 6:7, |Δ|=0.0006) and Arcturus (→ 4:7, |Δ|=0.0025) land within 0.3% of a Farey-15 rational; the others sit at random-spacing distances. Background population is, if anything, slightly less Farey-concentrated than uniform.
  • Conjecture III status in the Conjectures section updated to "falsified in naive form"; verdict and synthesis updated.
  • Bibliography expanded with Gaia DR3, Dehnen 2000 (Hercules ↔ bar OLR), Antoja 2018 (phase-space spiral), Schönrich-Binney-Dehnen 2010 (LSR).
  • Total bibliography entries: 17 across observational astronomy, celestial mechanics, Diophantine approximation, KAM theory, statistical methods.
• v1.2 — 2026-05-20
  • Test 11: LW pair fit applied to 2nd-order MMRs (5:3, 7:5, 9:7, 11:9). γ ≈ 5 (same direction as 1st-order), ε = 0.0056 (scaled down by 0.34× from 1st-order ε = 0.0165). Asymmetry detected at z_eff = 1.87 (marginal). 3 of 4 individual 2nd-order centers prefer γ > 1.
  • Quantitative ε-scaling comparison plot added.
• v1.1 — 2026-05-20
  • Joint Gauss + LW + broad-Farey three-component mixture fit (Test 10). MLE gives α₁ = 0 (Gauss rejected), α₂ = 0.025 (LW), α₃ = 0.976 (broad-Farey). LW component significant at z = 3.26 (p ≈ 0.001), 95% CI [1.0%, 4.0%] of {r} mass.
  • Page versioning (header pill, footer line) and this Changelog section added.
  • Copyright line in footer.
• v1.0 — 2026-05-20
  • Initial public release at dioph.kopilot.si.
  • Field proposed (Diophantine Astrophysics / arithmodynamics) with three founding conjectures.
  • Nine tests (Tests 1–9) covering: partial-quotient distribution vs Gauss-Kuzmin (χ²=21.5/9 dof), dynamical-age stratification (null), RV-only selection-bias control, Conjecture II at Farey-9 (79 distinct chain types vs ~123 uniform), Khinchin and Lévy constants (z ≈ +10 wrong direction, falsifies original Conjecture I), Conjecture II at Farey-12 / Farey-15, parametric mixture fit (χ²=111/19 vs Gauss), physical-width-scaling model selection (constant σ wins), asymmetric Lithwick–Wu pair fit (γ ≈ 5 at z ≈ 2.7 universal across first-order MMRs).
  • Conjecture I reformulated as I″ (Resonance Excess) after Test 5 falsified the original.
  • Sticky left sidebar navigation with back-to-top pills.
  • Sources & code section linking 8 analysis scripts + cached data CSV.
  • Bibliography of 13 references across observational astronomy, celestial mechanics, Diophantine approximation, KAM theory, and statistical methods.

Versioning policy: minor (vX.Y) bumps each iteration; major (vX.0) bumps when a foundational conjecture is added, retired, or reformulated, or when the page structure changes substantively. Conjecture I → I″ would historically have been a major bump (v1 → v2); the page was unversioned at that time so the change is recorded in v1.0 instead.