Incoprea hypothesis tests · volume II
Error in large reductions switches from benign to catastrophic at a critical reduction depth rather than growing smoothly, so reproducibility is a threshold property and the threshold shifts predictably with precision and ordering.
Error should show a sharp increase beyond a critical reduction depth, and reordering below that depth should be harmless.
Falsified by: Error that grows smoothly across depths, with no threshold, would return the problem to the random-walk account.
Sequential summation in float16, float32 and float64 of N uniform(-1,1) draws, N from 100 to 10^7, twelve seeds, median. Doubles as reference. Error is measured in units of unit-roundoff times the sum of absolute values, so the amplification factor is directly comparable across precisions. Local log-log slope over an eleven-point window tests for a knee. Reordering tested at N = 10^6.
code/tests/floating-point-phase-transition.py from the recorded run.| Amplification exponent | 0.03 |
| Amplification at N = 100 | 0.09 |
| Amplification at N = 10^7 | 0.27 |
| Largest local slope jump | 0.19 |
| Reordering change (rounding units) | 0.05 |
Rounding amplification in float32 is flat at about 0.06 to 0.30 across five decades (fitted slope 0.03): absolute error grows only in proportion to the sum of magnitudes, not superlinearly with depth. The local log-log slope never jumps by more than 0.19 around a median of 0.10. Relative to the answer, error still grows as N^0.5, because a cancelling sum is a random walk of size sqrt(N) while the error is fixed to the magnitudes. Reordering 10^6 terms changes the result by 0.05 rounding units — small, and equally smooth in N. There is no critical depth.