The Hebel Sparse Exact-Scale Method: Exact Division of Numbers Whose Parts Lie Millions of Decimal Orders Apart

Exact rational arithmetic stores a number as one dense numerator and denominator, so a value such as 5×10^k + 3×10^(-k) costs about 2k decimal digits although four small integers describe it. Multiple-precision floating point avoids those digits only by rounding, and the tiny part is lost unless the precision grows with k. We describe the Hebel Sparse Exact-Scale Method (HSES): a number is kept as a sparse sum of exact integer coefficients times powers of ten, and a quotient as a pair of such sums. Division only shifts exponents and removes the common integer factor of the coefficients; it never writes out the zeros between far-apart pieces and never rounds. We give an eleven-line Python divider with a correctness proof and benchmark it against GMP rationals, Python's Fraction, MPFR, mpmath and SageMath for scale gaps from 10^(+-10) to 10^(+-10^7). Every HSES result was exact and took 0.48-0.90 microseconds regardless of the gap, while GMP rational division took up to 581 ms at the largest gap. Worked examples from physics (relativistic velocity composition and a LIGO arm stretched by a gravitational wave) show IEEE double arithmetic, and in one case 113-bit arithmetic, silently dropping the small term while HSES stays exact; at these realistic gaps of 14-35 decimal orders GMP is still 3-6 times faster. We also report where the method loses.