MMP.16:5.1 - The largest response difference disappears in the readout
A team has two candidate response laws for a calibrated device assumed valid over the input range 0 to 2:
- H1: y=x;
- H2: y=x^2.
Both agree at the previously inspected inputs 0 and 1. The team needs to know whether the modeled response at x=2 is below or above 3. The laws answer differently: 2 and 4. Their applicability across the range is an assumption supplied for this case.
With a numerical readout r=y+e and a known error bound -0.1 <= e <= 0.1, the design x=2 gives:
| Design | H1 records | H2 records | Consequence |
|---|---|---|---|
| x=2, numerical readout | [1.9,2.1] | [3.9,4.1] | Disjoint intervals separate the alternatives. |
Now recover a missed feature of the actual instrument: it saturates at 1. Its record is r=min(1,y+e). At x=2 both alternatives always record 1. The large latent response difference gives no distinction at all.
Changing the input to x=0.5 yields:
| Design | H1 records | H2 records | Consequence |
|---|---|---|---|
| x=0.5, saturating readout | [0.4,0.6] | [0.15,0.35] | Both ranges lie below saturation and are disjoint. |
This design separates the supplied alternatives without replacing the instrument. A record 0.27 retains H2; a record 0.52 retains H1. A record 0.37 fits neither under the given error bound, so it reopens the response or observing assumptions rather than forcing a label.
The result at x=0.5 supports the answer at x=2 through the supplied response families. If those families were justified only up to x=1, this design would not establish the requested extrapolation. The next work would concern that range extension or access to a suitable direct observation.
If the only available inputs were 0 and 1 and the record error law were the same under both accounts, the available designs would not separate them. That conclusion concerns the stated access; it does not say the response question is unanswerable under every possible instrument.