A Rider in Two Waves: Different music, the same moves

A damped pendulum pushed by a wave is one of the oldest objects in physics. This book reads it with a stroboscope that counts, and with a method in which every prediction is written into a dated record before its run, with a bar it can fail and usually a named rival. Over the record to 5 November 2026, 527 tests were graded against predictions written first: in 448 at least one prediction landed, in 223 every one did, and 79 missed outright. The count was made by a script that read the whole record, then checked by reading entries by eye, every test from 23 October on among them; the scripts and the readings by eye are published here, and give the count from the public extract alone. The book reports 386 landings by name and names 173 misses or voids, 95 of which were redesigned into predictions that then landed. Every number names its run, and the record’s registrations, verdicts and results are published here as its public extract. What it found: A law with nothing fitted. When an extra tone is added, a locked pair of dances moves by an amount that falls as one over the tone’s harmonic number squared. A first-order theory reproduced all 56 measured moves; the tenth and twelfth harmonics, written down first, landed 14 of 14 as close as the runs print; and a second engine written from scratch finds the same moves and carries the law to the thirtieth harmonic (§4.28). One gearbox in two media. A dance’s grip multiplies step by step by the same law in the toy and in the standard model of a real device, the Josephson junction, and predicted settings no run had visited, in both, within 0.017 (§4.28). A selection rule and its break. A rule says which of two mirrored dances survives a touch of weight. Counting the radio wave’s quanta predicts where a second tone breaks it, and says one tone alone never can. At the break the hold passes through a hand-off window that weight narrows for one pair of dances and widens for another; the widening was written down before its run and came within four millionths of the second tone’s strength (§4.23, §4.28). Exact counts across media. A count of the dances built on the toy agreed with 205 of 205 junction rows (§4.25). Bear’s coin. A rider lands heads or tails by where it is set down and nothing else, along an edge between the faces that stays ragged however closely it is examined: a fractal edge with an uncertainty exponent of 0.592. A second engine, written from scratch and sharing only the equation, flips it the same way for all 1,800 riders of record, and the toy that ships with the book lets any reader flip it (§4.17). The grip forgets and the slip remembers. A rider locked in a gear forgets where it started and one slipping between gears keeps it, as written down before the read, for 99.1% of 110 locked runs and all 101 slipping ones (§2.6). A two-tone rectifier, mapped knob by knob for engineers (Part 6). The lopsided step the device literature reports as a tunable Josephson diode, followed across the rider’s weight; a setting where its window stops depending on the weight, with a one-number rule for how far that holds, confirmed blind at four frequencies within 1.1%; a dark point placed by a symmetry to seven parts in a million; and a weight meter that a drifting drive cannot fool, reproduced by a second engine written independently. A manual. Every knob of the rectifier is turned and written up, with step-by-step instructions for driving it and checking the bench. Much of the physics underneath is known, and the book says where: Arnold tongues and the devil’s staircase, phase locking and Shapiro steps, Floquet multipliers, the resistively and capacitively shunted junction model, harmonic mixing, rocked ratchets and current reversal, and the superconducting diode effect under a biharmonic drive. Its claims are the measured map, the laws that held blind on it, and the record of how each was found. It is not a propulsion book: nothing here pushes itself, and the book says so (§1.2). A Rider in Two Waves is one of five related releases made together in autumn 2026: Held, Rocking, Locked, a mini-book on the three-body problem; The Loop Atlas, on the loops that let you keep the change; this book, its spin-off; version 7 of Loop Walking: A Field Guide to Hacking the Universe with Holonomy, much enlarged by these projects; and Does Claude Dream of Holonomic Sheep?, the collected no-physics reads. A no-physics read is a habit from the start of Bear and Claude’s pattern-recognition work: once per exchange, Claude reads the newest data with its labels and laws taken away and tells what the numbers are doing as a story. More than 900 were written across these projects; seven of this book’s eight parts open with one. Published with the book: its Markdown source; the record’s public extract, every registration, verdict and result; play_the_toy.py, the toy in one page of plain Python for any reader or AI to run, with Try the toy, a two-page card that walks through six settings, and the toy’s fidelity check against 1,111 riders of record; depth_axis_explorer.html, the live bench in one web page, whose rectifier bench runs the junction with every knob of Part 6 and checks itself against runs of record; and the counting kit, the scripts and readings by eye that give the book’s count from the extract. This is a preprint from a part-time, volunteer research team that works in the open and gives its tools away. It has had no peer review, and its literature pass is limited: the text marks each outside work as read, read in part, or checked against an abstract or an index record. The runs, engines, figures and much of the prose were made with Claude (Anthropic); responsibility for the work rests with the named human author. CC BY 4.0.