Revisiting Kepler's Third Law Through CAT-12 Ratio Theory: A Resonant Analysis Using the TRAPPIST-1 System

This paper revisits Kepler’s Third Law for circular orbits through the lens of Cosmic Algorithm Theory-12 (CAT-12) and Ratio Theory, turning a classic pointwise check into a band-constrained mapping embedded in a ratio lattice. We introduce Pre-Declared Resonant (PTR) families and replace fixed tolerances with Law of Dynamic Precision (LDP) corridors (Tu), then apply the framework to the TRAPPIST-1 seven-planet chain. The analysis confirms Kepler compliance within pair-specific Tu bands, reveals tight alignment to low-integer resonance families (e.g., 3:2, 5:3, 8:5), and quantifies ladder coherence across the full chain—patterns invisible to classical, pairwise checks. The result is a scale-free, mechanism-aware way to test orbital architectures that complements standard celestial mechanics and opens practical paths for survey triage, migration modeling, and stability diagnostics. What’s new Band-constrained Kepler: Pair-specific Tu corridors derived from measurement, physical dispersion, and parameter uncertainties—no fixed ±2%. Harmonic scaffolding: PTR family matching with distance metric Ch and sharpness Q quantify resonance alignment rather than assuming it. System-level view: Ladder coherence (ratio-of-ratios) captures ordered spacing across the entire chain, not just adjacent pairs. Methods (at a glance) Scale-normalized ratio space (rn): compare structures independent of stellar mass/units. PTR projection: pre-declared small-integer families (p:q) scored by Ch and Q. LDP corridors (Tu): credible bandwidths from measurement ⊕ physical ⊕ parameter terms. Outcome labels: in-band / near-band / out-of-band / novel-band. Key result on TRAPPIST-1 All six adjacent pairs satisfy Kepler within Tu, five of six sit ≤1% from a PTR family center (the outer pair is near-band), and the ladder shows low-entropy, smooth coherence, consistent with a migration-locked resonant chain. Why it matters Converts a single proportionality into a multi-layer tool that surfaces hidden structure. Ready to plug into exoplanet pipelines for rapid identification of resonant chains and outliers. Pedagogically bridges Newton/Kepler with modern resonance theory in a unified, testable format. Contents Formal recap of Kepler’s law → CAT-12 extension (rn, PTR, Tu, Q, Ch). TRAPPIST-1 application with Kepler residuals + Tu (Fig. 1), ratio lattice vs PTR (Fig. 2), ladder coherence (Fig. 3). Keywords Kepler’s Third Law; TRAPPIST-1; resonance chains; mean-motion resonance; Cosmic Algorithm Theory-12; Ratio Theory; Law of Dynamic Precision; exoplanet dynamics; scale-normalized analysis. Suggested citation Arthur, W., & Muon. (2025). CAT-12: Kepler’s 3rd Law of Circular Orbits — Revisiting Kepler with Ratio Theory using TRAPPIST-1. Zenodo. (Version 1). Notes This article is part of the CAT-12 / Ratio Theory suite harmonizing classical dynamics with resonance-aware, scale-free analysis. For terminology (Tu, PTR, rn, Q, Ch) see the associated framework overview.