| Side | channel 1 | channel 2 |
|---|---|---|
| N | K = 2.4955, A = 0.4965, b = 0.2479 | K = 0.1021, A = 11.3142, b = 5.9507 |
| D | K = 0.1549, A = 9.1834, b = 106.5935 | K = 0.4234, A = 3.9872, b = 0.3411 |
with the standard channel form K + A·bP/(1+bP) and α(P) = N(P)/D(P). Two channels on each side — the minimum the model allows.
| Quantity | Value |
|---|---|
| α(0) = K_N/K_D | 3.8560 |
| α(∞) = (K_N+A_N)/(K_D+A_D) | 1.0480 |
| Peak α | 3.8560 (at P → 0) |
| Crossings | 2, at 0.0030 bar and 0.2955 bar |
Parity check. Both limits exceed 1, so they lie on the same side and the theorem requires an even number of crossings. Two is even. The construction is parity-consistent, as any valid design must be.
What the device does. The selectivity starts high (3.86), falls below 1 between 0.003 and 0.296 bar, and returns above 1 at higher pressure. That is a window of reversed preference spanning two decades — the complement of the earlier designs, which switched on rather than off.
A device with two channels per side has four free constants plus two half-saturations: six numbers. A three-channel design has nine. The minimal construction is the one a fabrication effort should attempt first, because every additional channel is an additional material requirement.
The earlier rule ("three channels minimum") would have sent a fabrication programme after a harder target for no reason. That is the practical cost of an untested design rule: it misdirects effort, not just belief.
| Design | Channels/side | Shape | Window |
|---|---|---|---|
| Page 12, design 1 | 3 | switches on | 0.016 – 0.43 bar |
| Page 12, design 2 | 3 | switches on, peak 2.0 | 0.0024 – 2.8 bar |
| Page 12, design 3 | 3 | switches on | 0.14 – 0.78 bar |
| This page | 2 | switches off | 0.003 – 0.30 bar |
Two shapes exist: a window where the preferred gas reverses (α > 1 inside) and one where the normal preference fails (α < 1 inside). Both are two-state devices; they differ in which state is the exception.
A switch-off design is more useful for sensing than for separation — the exception window is narrow and the background state is unambiguous. That is a device-selection rule that did not exist before this construction.
| # | Lead | Disposition |
|---|---|---|
| N-41 | Two-channel designs come in switch-on and switch-off shapes, separable by whether α(0) is above or below the crossing band. | test now |
| N-42 | Switch-off windows are narrower than switch-on windows at equal channel count, which makes them better sensors and worse separators. | test now |
| N-43 | The minimal channel count for k switches is k+1, not 2k−1; test for k = 3. | test now |
| N-44 | Any realisable membrane has K values set by its support layer, so the achievable α(0) range is narrow and constrains the design. | note |
| N-45 | A fabrication programme should target a two-channel switch-off design first, because it needs the fewest material properties. | note |
| N-46 | The six constants of a two-channel design map onto measurable quantities: two permeances and two Langmuir pairs. | test now |
If one sentence survives: the minimal double switch needs two channels per side, not three, and it exists in a switch-off form whose narrow window makes it a sensor rather than a separator.