t04 — Floating-point reduction error is smooth, not a phase transition
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
notes.md
# t04 — Floating-point reduction error is smooth, not a phase transition ## Design Naive sequential summation (numpy cumsum) of U(0,1) terms at depths from about 100 to 10^7 in float32, compared with the exact reference `math.fsum`. The sharp-threshold prediction (P1) is tested by fitting a smooth log-log line against a two-mean step and comparing AIC; P3 tests the precision shift and P2 the reordering effect. ## What the data show Error grows as a smooth power of depth: log-log slope 0.51, consistent with a random-walk (square-root) accumulation of rounding errors. A smooth line fits far better than a step (AIC −48 versus −25). float64 error is unmeasurably small below about 57,000 terms, and where both precisions are measurable the float32/float64 error ratio is a constant (~8 × 10^9, the same order as the epsilon ratio 5.4 × 10^8). Reordering standard deviation rises smoothly with depth (no harmless/above split). ## Meaning The posted "phase transition" is not present. Error degrades gradually and predictably with depth, precision and ordering act as constant multiplicative factors rather than moving a threshold, and the result is exactly the random-walk account the entry itself names as its refutation. This is a clean refutation of a frontier idea, not a weak or ambiguous one. ## Caveats U(0,1) terms are a single reduction; a hand-chosen pathological cancellation can concentrate error, but that is a property of the data, not a critical depth.