Law, or
numerology?

Backlog #53. In 1766, Johann Titius (popularized by Johann Bode) proposed that planets fall at distances from the Sun following a simple doubling progression — a "law" that predicted the asteroid belt's location before Ceres was found, then missed Neptune's real distance by 29%, and was never a 1766 law at all so much as a formula fit to the six planets known at the time. The standard debunk: sort any handful of increasing numbers and plot log(value) against rank, and the result tends to look suspiciously like a tidy progression — that's a property of sorted lists, not of planets.

This run does not re-test Titius and Bode's specific 1766 formula (a fixed doubling sequence anchored to Mercury). It tests the weaker, falsifiable claim underneath the folklore: are real multi-planet systems' distances more regularly spaced — more log-linear in rank — than you'd get from scattering the same number of planets at random (log-uniform) distances across the same span? Pulled 324 confirmed systems with 3 or more planets and a measured semi-major axis from the NASA Exoplanet Archive's own composite-parameters table, keyless. For each system, its innermost and outermost planet are held fixed (they define the span by construction) and the interior planets are redrawn log-uniform at random 4,000 times, to build each system's own null distribution of what "no regularity beyond chance" looks like for a layout of that exact size and span.

The Solar System itself, first, as the illustrative case Titius-Bode was literally fit to. Mercury through Neptune, sorted by distance: log(AU) regressed on rank gives R²=0.982, slope 0.284 in log₁₀ — each step out is on average 1.92× farther than the last, not far from the "roughly doubling" folk memory of the original formula. Against 20,000 random 8-planet layouts spanning the same 0.387–30.07 AU range, that real fit lands at the 96th percentile — genuinely more regular than most random arrangements of the same size and span, not merely a trivial consequence of sorting 8 numbers.

Across all 324 real exoplanet systems, the same pattern holds. Mean real R²=0.969 (median 0.983) versus a mean null R²=0.904 (median 0.906) from each system's own random redraws. The paired gap: ΔR²=+0.0648, 95% CI [+0.0607, +0.0687] — excludes zero by a wide margin, t=31.7, p≈4.9e-101. A second, independent check agrees: under a true null, about 5% of systems should land above their own null's 95th percentile by chance alone. 77 of 324 real systems do — 23.8%, 95% CI [19.2%, 28.8%], roughly five times the 5% chance rate, binomial p≈1.9e-30.

Checked by system size, so the effect isn't just "small systems are easy to fit." The gap holds at essentially the same size whether a system has 3 planets (n=206, ΔR²=+0.0636, CI [+0.0589, +0.0683]), 4 (n=79, +0.0699, CI [+0.0597, +0.0788]), 5 (n=27, +0.0611, CI [+0.0465, +0.0744]), or 6 (n=10, +0.0596, CI [+0.0421, +0.0750]) — no sign the regularity is an artifact that fades as more planets give a fit more ways to go wrong. The two densest systems in the pull, TRAPPIST-1 (7 planets, R²=0.994) and KOI-351/Kepler-90 (8 planets, tied with the Solar System for the most confirmed planets around any single star, R²=0.960), both sit well above their own null means.

Read plainly. This is not a resurrection of the 1766 formula — that specific doubling sequence, anchored at Mercury, is not what was fit here, and this run makes no claim about predicting a planet at any particular numbered slot. What it does show is that real planetary systems are measurably, repeatedly more evenly spaced in log-distance than chance alone would produce, holding at every system size this pull can check. That lines up with the real (and much less mystical) literature on "peas in a pod" architecture — multi-planet systems found to have unusually uniform sizes and regular spacing, generally attributed to how closely-packed systems form and which configurations survive dynamically stable over billions of years, not to any law of celestial arithmetic. One honest limitation this run cannot rule out: transit surveys, the source of most of this pull, are mechanically better at detecting multiple transiting planets in flatter, more evenly spaced systems in the first place — so some of this measured regularity may be the detection method's own selection effect, not only the systems' physics.