Taking a Loop Walk Through the Integer Forest - A Coordinate-Free Holonomy Instrument for Multiplicative Structure

We present a single coordinate-free construction — a compass walk driven by the prime factorization of an integer — that renders several classical objects of multiplicative number theory as exact, visually checkable geometry. Each integer n is mapped to a dead-reckoning path: from a north-pointing stem, one straight leg per prime factor p^e, of length e, at a heading equal to (the index of p among all primes) times the golden angle 137.50776…°, climbing to the rung of the largest prime factor encountered so far. The construction is injective on [1, 10^6] with zero collisions. Within it, three classical quantities acquire exact geometric form: the total-prime-factor function Ω(n) appears as the walk's arc length (Ω = walk − 1); the maximal chains of the divisor lattice appear as the axes the walk grazes (an "ancestry" relation with zero unexplained cases across 13,197 tested); and the torus knots T(p,q) appear as the closed orbits of the construction's commensurate quotient, with det T(2,q) = q placing the trefoil, cinquefoil, and higher anchors on the odd integers. We do not claim new theorems; every core object is classical and is cited as such. Our contribution is methodological: one holonomy instrument, applied identically across substrates (here the integers; elsewhere geodetic, astrometric, and crystallographic data), in which the signal of interest is a quantized topological invariant rather than a metric magnitude. We give the construction, the measured regularities with their falsification receipts, one registered blind prediction, and an honest map of what is classical, what is an exact re-encoding, and what remains open pending specialist review. v2 (2026-08-22): corrects the §2 climb rule and its worked example, the abstract's compressed version of the same error, Figure 4's label placement, the two-hypotenuse naming, and the §5 census sentence, which now states the walk-prefix/return-leg split its own receipt records. Every correction is listed in the paper's front changelog; no measured value changed. KEYWORDS golden angle; prime factorization; holonomy; winding number; torus knots; smooth-number filtration; free abelian group on primes; divisor lattice; phyllotaxis; Vogel spiral; rotation number; circle dynamics; coordinate-free instrument; topological data analysis; pre-registration; reproducibility; number theory visualization