Incoprea Research · Original floating-point hypothesis

When the order of a sum first matters

Exact arithmetic and computational reproduction · 30 October 2026 · Not peer reviewed

A supplied experiment found a sharp boundary in a deliberately simple sum. For the positive sequence 1, 1/2, 1/4, and so on, 200 sampled orders first disagreed at 25 terms in binary32 arithmetic and 54 terms in binary64. We reproduced those boundaries. The disagreement is tiny: at the first boundary the answers are adjacent representable numbers. This supports a precise rounding phenomenon, not a universal transition to catastrophic error.

The posted claim

The original Incoprea hypothesis proposes that large reductions switch from benign to catastrophic error beyond a critical depth, with a threshold that depends on precision and ordering. DeepSeek supplied a test using exact powers of two, varied summation orders and two floating-point formats. The cleanest result concerns when different orders first return different answers for this specific sequence.

Floating-point arithmetic rounds each intermediate operation to a finite set of representable numbers. Addition is therefore not associative in general: regrouping or reordering can change the final rounded value. This is established numerical analysis, explained in David Goldberg's original tutorial on floating-point arithmetic. The present calculation is an illustration and verification, not a new law of computation.

Why these boundaries make sense

The exact sum of n terms is 2 − 2⁻⁽ⁿ⁻¹⁾. Binary32 has 24 significant binary digits; binary64 has 53. For n no greater than that precision, every partial sum of a subset of these terms fits exactly in the format. All orders therefore give the same exact total in that range.

At one additional term, the exact total falls halfway between two neighboring representable numbers near 2. Intermediate rounding now matters. Different addition orders can reach different neighbors even though every term is exactly representable. That gives a concrete reason for a boundary at 24 + 1 terms and 53 + 1 terms.

The observed shift is 29 terms, matching the 29-bit difference in significant precision. The reproduction preserves the supplied primary experiment: naive sequential accumulation, depths 5 through 130, 200 shuffled orders at each depth and random seed 1. A finite sample is not an exhaustive enumeration of every order above the boundary. Exact representability supplies the stronger argument below it.

The reference must remain exact

The supplied code built a rational reference but converted it to a floating-point number before measuring error. That can report zero error when the returned answer merely equals the rounded reference. We made the small correction of retaining the rational value until the error is calculated.

At 54 binary64 terms, the exact total is 2 − 2⁻⁵³. The sampled answers are 1.9999999999999998 and 2.0. Each is exactly 2⁻⁵³, about 1.11 × 10⁻¹⁶, away from the true total. Thus neither is exact, even though one equals the rounded reference. Their relative errors are about 5.55 × 10⁻¹⁷. A detectable difference in the last digits is not a numerical catastrophe.

What this establishes

The supported statement is that order dependence appears at a predictable representability boundary for this positive dyadic family under these accumulation rules. It does not establish that all long sums have a critical depth, that shorter sums are generally safe, or that error magnitude is determined by length alone. Cancellation, conditioning, the input values and the summation algorithm also matter.

The word “nondeterministic” would be misleading here. With the same inputs, order, format and operations, the calculation is deterministic. The experiment deliberately changes the order. That distinction matters when comparing numerical reproducibility with a program that produces different results under otherwise identical conditions.

Reproducibility and attribution

The accompanying standalone Python script uses only the standard library. It reproduces the supplied primary random-order experiment, emulates binary32 rounding at every addition and keeps exact rational references for error. It needs no downloaded data or third-party raw files. DeepSeek prepared the original test; Codex reproduced the boundaries and corrected reference handling and interpretation. The full supplied source is preserved separately. No novelty or external peer review is claimed.