Ecology
The claim that a lake flips to algal dominance at a phosphorus threshold — the textbook alternative-stable-states story — tested on two national surveys twelve years apart.
Do lakes switch abruptly at a phosphorus threshold?
The claim is quoted verbatim in hypothesis.md, where the support/refute criteria were fixed before any model was fitted.
Result
The evidence is split: one survey shows a threshold, the other shows none, and the second prediction fails outright. Two EPA National Lakes Assessment surveys — 2007 (n = 1,035 lakes) and 2017 (n = 1,097) — were each asked whether the chlorophyll–phosphorus relation bends at a threshold.
In 2007 it does not. A smooth quadratic describes the relation as well as a two-regime hinge: ΔBIC = BIC(quadratic) − BIC(hinge) = +1.5, and the version with a genuine jump is 2.7 BIC units worse than the quadratic. That is a refutation: the best breakpoint sits at 99 µg/L total phosphorus (95 % bootstrap CI 73–254), but it buys nothing.
In 2017 the hinge wins: ΔBIC = +18.5 against the quadratic and +13.5 against the jump model, breakpoint 145 µg/L (CI 74–255). Because the upper confidence limit grazes the search window edge, the rule returns indeterminate; against the brief's looser wording — ΔBIC > 6 with a CI inside the observed range — 2017 would count as supportive. The two surveys disagree; that is the result.
The bimodality prediction is refuted in both. The bimodality coefficient of the hinge residuals is 0.266 in 2007 and 0.297 in 2017 against a normal null centred on 0.337 (p = 1.000 and 0.990; Benjamini–Hochberg q = 1.000). Both residual distributions are less bimodal than a normal draw. Two coexisting algal states at fixed phosphorus are not visible.
P3, the hysteresis prediction, is not tested — data not assembled.
Design
Phosphorus and chlorophyll-a came from the NLA 2007 archive (PTL, CHLA) and the 2017 site/visit file the population estimates are computed from (PTL_RESULT, CHLA_RESULT). Every field was located in the supplied metadata first; the discovery is in SOURCE.md. One row per lake: primary sample, first visit, both analytes positive, below-detection rows dropped rather than substituted (117 in 2007).
Four models, least squares on log10–log10: M0 linear, M1 quadratic, M2 continuous hinge with free τ (grid search, then refinement), M3 hinge plus jump. Sign conventions, the τ window and the identifiability gate were fixed first.
P1 in detail
The decisive comparison is against the quadratic, not the linear model: curvature alone produces a bend a hinge will mimic. In 2007 the quadratic absorbs the curvature almost entirely; in 2017 it does not, and the hinge flattens above roughly 145 µg/L while the quadratic keeps climbing.
The reason for "indeterminate" in 2017: the rule also required the CI on τ to be interior to the search window. It is not — the upper limit (2.407 in log10) exceeds the window's top edge (2.394) by 0.012, 2 % of the interval width. On a 5–95 % window the interval becomes interior but shifts upward, the signature of a bound limited by the window, not the data. Existence of the kink is supported; a location above 255 µg/L is not excluded.
P2 in detail
The test is a Monte-Carlo comparison of the observed coefficient against 1,000 normal draws. The largest-gap statistic agrees (p = 0.379 in 2007, 0.178 in 2017). One clause of P2 survives post hoc: scatter really is larger above the breakpoint, residual SD 0.602 against 0.365 in 2007 and 0.573 against 0.376 in 2017, permutation p = 0.0002. Heteroscedasticity is not bimodality.
P3 — not tested, data not assembled
The Water Quality Portal was queried for lakes only in two lake-rich states. Minnesota returned 342,181 phosphorus results and 14,286 chlorophyll results — 1,882 paired site-years at 553 station IDs, of which only 24 reach 15 paired years. Wisconsin returned 139,984 and 21,695, giving 473 paired site-years, 244 stations and 5 with 15 years. A national panel needs gigabytes and would still be dominated by a few intensively monitored waterbodies. It cannot carry the pre-specified test, and a weaker stand-in is forbidden, so P3 is dropped.
Depth, and what would change the answer
Depth moves the breakpoint monotonically: 218 µg/L in lakes shallower than 4 m (n = 402), 81 µg/L at 4–10 m (n = 378), 58 µg/L at 10 m or more (n = 254). Within the two deeper classes the hinge also beats the quadratic (ΔBIC +7.4 and +8.8); in shallow lakes it loses (−1.3). A threshold that tracks depth fits a depth-confounded smooth relation better than a universal switch.
The 2007 refutation is not robust. Substituting RL/√2 for below-detection values instead of dropping them moves ΔBIC from 1.5 to 18.7; weighting by the design weight — which also drops the 98 hand-picked reference lakes — moves it to 41.5; keeping repeat visits moves it to −4.1. The first is a known artefact, 117 values stacked at one artificial concentration; the second changes both weighting and sample. The 2017 result is steadier (8.3 weighted, 14.5 probability sites only, 20.4 all visits).
What this test cannot see
A single sampling of a lake cannot observe a shift or its irreversibility. P1 and P2 test a signature the claim implies, not its mechanism. Nor is τ inference regular: a grid search inflates the apparent evidence for a change point against a χ² benchmark, hence the stricter gate. Nitrogen, colour, turbidity and mixing regime are uncontrolled beyond depth.
Designed and executed with AI assistance (DeepSeek); not peer reviewed. Observational data: associations only.
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