The claim
The note’s second half, “Smoothies, Milk, and Signal Detection Theory”: “Adding a little milk: increases creaminess, boosts the perceptual distinctiveness of the smoothie relative to milk (higher μ₁ − μ₀), so d′ improves — More Good.” And: “Your smoothie only tastes like a smoothie so long as its distribution is discriminable from milk.”
Discriminability from milk only falls as milk is added. Let taste be a point in some perceptual space, s for the other ingredients and m for milk. Under linear mixing the blend at milk fraction f tastes t(f) = (1 − f)s + f·m, and its distance from milk is |t(f) − m| = (1 − f)|s − m|. That is strictly decreasing in f, from the first drop. With equal noise, d′ against milk is proportional to it, so it never rises. It was checked numerically: for s = (1, 0.3) and m = (0, 1), the distance at f = 0, 0.2, …, 1 is 1.221, 0.977, 0.732, 0.488, 0.244 and 0. “Less Good” as milk dominates is consistent with falling d′. “More Good” at first is not.
What the line is about. The note’s first model already has it: goodness U = I(V)·Q(f), with Q single-peaked at a preferred fraction f*. That is a preference over composition. It is a value, and the taster can find it only because the blends at different f are discriminable. The two come apart in both directions: two blends can be easy to tell apart and equally good, or nearly indistinguishable and on opposite sides of the peak. The ridge f = f* that “more of all the other stuff” stays on is a ridge of preference. That it is also a ridge of constant d′ against milk is a coincidence of linear mixing, not the reason the smoothie stays good.
The pattern. MOM-CLM-166 found the formal reading of the owner’s LEGO parable stated with its sign inverted. A criterion for detection was substituted for one about retrieval, so that the parable’s rule (set an item aside when it does not stand out) was written as its opposite (CLAIM-307). Here a criterion for discrimination is substituted for one about value, and the rising half of the owner’s More Good, then Less Good is reversed. In both, a signal-detection quantity is put in place of the owner’s quantity, and the owner’s own words, kept beside the formal reading (in the same file here, a link away there), contradict it.
Why it is Active
The monotonicity is one line under the stated mixing model and was checked numerically. That the note asserts the rising half is a matter of its text, quoted above.
What it does not say
- That signal detection has nothing to say about smoothies. Whether the taster can find f* depends on telling nearby blends apart, which is a question of d′ between blends, not against milk.
- That perception mixes linearly. A nonlinear map could break monotonicity, and the defeat condition says what would have to be shown.