Weights ≠ Function · 4 of a series
Can your measurement hide what’s really happening?
I pre-registered a prediction about how big the merging barrier would be. It failed. The reason is a lesson about metrics: some measurements saturate and stop telling you anything. Smallest case: one hidden layer, sigmoid units.
Pre-registered scoreboard (Study 2, 2% inclusion)
| Hypothesis | Statement | Result |
|---|---|---|
| H4a | Exact-search residual: upper quartile ≤ ε | ✓ all 5 tasks |
| H4b | Permutation-only median above 2 points in flip pairs | ✓ all 5 tasks |
| H4c | Weight matching with flips within ε of exact in ≥80% of pairs | ✓ all 5 tasks |
| H5 | Permutation-only worse than flips in ≥90% of flip pairs, under 4 measures | ✓ all 5 tasks |
| H1 | All pairs learn the same lines | ✗ quadrilateral (2 of 32 pairs), hexagon (6 of 29) |
| H7 | The weights predict the size of the barrier (error) | ✗ ρ = 0.007 |
The failed test, and the exploratory re-test
Spearman ρ between M = Σ|sᵢ|/2 over flipped units and the barrier left without flips (129 pairs), 95% bootstrap intervals. Circle: pre-registered, error barrier. Diamonds: exploratory loss re-tests, decided after seeing the failure.
Why it failed: saturation
Schematic, not data. Once a substantial unit is erased at the midpoint, the blend predicts “negative” almost everywhere, so the error barrier caps near the task’s positive rate (observed: median 17–22 points against positive rates of 18.8–22.1%). Loss has no such ceiling. Orange: error barrier. Cyan: loss barrier.
Technical statement
H7 was tested as ρ(M, BF), where BF is the training-error barrier with the exact_sign permutation applied without its flips. The pre-registered rule labels the result “weak” because the interval includes zero; the paper reports it as not supported. Sensitivity: ρ = −0.16 [−0.35, 0.03] at 1% and −0.21 [−0.37, −0.05] at 5%. The loss re-test (decision D8) gave raw ρ = 0.86 [0.79, 0.90] and calibrated ρ = 0.65 [0.54, 0.75]; raw losses grow with weight scale, so part of the raw correlation may be scale. A confirmatory test needs a fresh pre-registered run.
Evidence label: scoreboard and intervals are transcribed from the preprint (Sec. 5.1, 5.4); the saturation curves are a schematic. Scope: one hidden layer, sigmoid. Source: Lakhanpal (2026), preprint v2.