The Hand That Seeds the Field

Seven pointers, driven from nothing but the published tooth counts of the surviving gears, each one multiplied out as a pure product of driven-over-driver.

Five of those errors are the same number. That is not coincidence and it is not wear: nineteen tropical years against two hundred and thirty-five synodic months is off by -12.48 ppm, and every dial that divides by 235/19 inherits it. One gear in the machine has 127 teeth. 127 is 254/2, and 254 is the sidereal months in nineteen years. The identity is cast in the bronze.

One correction I had to make against the reference tables I was reading: two gears in the lunar train, n2 and n3, are listed with their tooth counts swapped. With 57 and 15 in the published order the sidereal month comes out wrong by thousands of ppm; swapped back, it lands at -14.4.

The Venus train uses the period relation 462 synodic cycles in 289 years. It is a famously strange choice, so I searched every alternative. Ranked on accuracy alone, 462/289 comes 8th out of 231. Ranked on buildability — can you factor it into gears a bronzesmith can actually cut — it comes 3rd out of 120. Neither explains it.

Then I optimised the five planets jointly for the fewest distinct gear sizes across the whole machine, and 462/289 wins outright. Seven gear sizes — 17, 21, 22, 31, 37, 40, 79 — serve every planetary train. The designer was not maximising accuracy. He was minimising the number of different things he had to cut. That is a manufacturing decision, recoverable two thousand years later from the numbers alone.

The pin-and-slot is the mechanism's one nonlinear element — it fakes the Moon's varying speed. I rebuilt it three ways in C and pushed a phase through 200,000,000 ticks, which is 195,312 simulated years, then rewound the whole thing.

Rewound 200 million ticks out and back, the double leaves a residue of 1.7347e-10 turns — the state does not come home. The integer accumulator comes home exactly. The structural map comes home exactly and in constant time, because there is nothing to walk back. Both accumulators are O(n) in the distance travelled. The ratio is O(1) forever.

Measured on this machine with clang -O2 and inline-assembly barriers so the optimiser cannot delete the loops. Wall-clock timings move a few percent run to run; these are from one run, unedited. The integer path is float-free end to end — the sine is a Q30 CORDIC I wrote for it, whose worst error over a full turn is 1.814e-08 rad, against roughly half a degree of backlash on a hand-cut bronze tooth.

The mechanism does not gear the anomalistic month directly. It runs the sidereal month on one shaft, precesses the slot frame on another, and takes the beat between them.

Two clocks beat one by 51x on the same tooth budget. And the two gears that set it are 50 and 50 — a ratio of exactly one — so the entire lunar anomaly falls out of the difference between two things running at the same speed in different frames. That is the whole idea in one sentence.