Reversibility of exact calculation by saturation and an incommensurability budget for coprime rings in 7 bits

We present a system of exact, reversible calculation —the Fractal SaturationGrammar (FSG)— that operates with integer arithmetic, without floating pointand without positional zeros. The engine processes numbers of arbitrarylength with a bounded register (7 bits per cell) and no drift: in a ten-millionstep reversibility test, engine drift is zero, against 3.61×10⁻⁹ for the64-bit float (measured up to 10 M steps; beyond that, extrapolation). Atheorem bounds the simultaneous return of coprime rings: **T ≤ 20·M(N)**; inthe N=99 case, five one-byte cells determine a cycle of **82,645,608,260 exactbeats** — literal incommensurability produced by integer mathematics, verifiedwithout simulating. The same framework re-expresses the round of the Mayacalendars (759,200 days) as a case of the theorem, without attributing ourvocabulary to their record. The human-AI collaboration that produced theseresults is described in Methods.