Instruments for the Magnetic Forest: A tool suite for the integer-forest lattice model, and what it verifies

This is a methods and verification companion to The Magnetic Forest, and the first of several planned follow-ons to Jellyfish: The Holonomist of the Sea. It houses the fourteen-instrument suite that measures the flux spectrum of the integer-forest lattice, states plainly what each tool reads, and asks a single question of the physics they return: what is the Hofstadter butterfly's structure actually made of? The answer is textbook, and the paper claims nothing else — the fractures are commensuration, a whole number of phase periods fitting a whole number of lattice rows, and commensuration requires closure, a loop the count can come back around. What the suite adds is a direct, reproducible measurement of that known physics on a lattice built from arithmetic rather than from a crystal, together with three results worth naming as treasure and an honest record of three framings tested in discussion and set aside. It is a checkpoint: not peer reviewed, with a limited literature pass, and every reproducing instrument shipped with its launcher, ledger, data table and figure panels.

Keywords

holonomy; topological invariants; Hofstadter butterfly; commensuration; Aubry-Andre-Harper; Berry curvature; magnetic flux lattice; integer forest; smooth numbers; reproducible instruments; verification