The parity theorem says a crossing count is even when α(0) and α(∞) agree, odd when they differ. That rules out "at most two" and permits exactly two — a selectivity that switches off and back on as pressure rises. Whether such a design exists is a separate question from whether it is permitted. It exists.
A parameter search over three channels per side (equal counts) found three designs with two crossings, all with same-side limits as the theorem requires.
| Side | channel 1 | channel 2 | channel 3 |
|---|---|---|---|
| N | K=0.117, A=8.07, b=99.6 | K=1.276, A=1.40, b=5.95 | K=0.104, A=0.46, b=0.98 |
| D | K=0.414, A=4.31, b=0.33 | K=0.149, A=3.52, b=46.1 | K=2.087, A=4.10, b=89.3 |
α(0) = 0.584, α(∞) = 0.784, peak 1.006. Crossings at 0.0156 bar and 0.431 bar. The selectivity starts below 1, rises just above it, and returns — a two-state window roughly two decades wide.
α(0) = 0.846, α(∞) = 0.797, peak 2.001. Crossings at 0.00237 bar and 2.80 bar. This is the strongest of the three: a three-decade window in which the preferred gas reverses, with the ratio doubling at the peak.
α(0) = 0.250, α(∞) = 0.644, peak 1.032. Crossings at 0.142 bar and 0.782 bar.
This is the batch's first constructible output. The parameters are a blueprint for a membrane whose selectivity reverses twice. Whether any real material realises these Langmuir constants is a separate, material-specific question — but the target is now a number, not a hope.
The fireball tail index came out 0.32 (PWM) or 0.78 (Hill) on the same data, a factor 2.4. Any field quoting a tail index has at least this spread hidden. The design is a reporting standard:
| Required item | Why |
|---|---|
| Three estimators (Hill, MLE, PWM) | their spread is the methodological uncertainty |
| Threshold-stability plot | a drifting estimate signals an incomplete catalogue |
| Catalogue snapshot (n, date) | catalogues change; the index does not |
| Derived-column flag | a derived column makes the index non-independent |
Design rule: if the three estimators disagree by more than the effect being claimed, the index cannot support the claim. On CNEOS that threshold is a factor 2.4.
I2's four literature cases span 75.8× in reported conductivity purely through λ. The design is the same shape as branch A's:
A constructive inversion: a substrate-dominated measurement (λ ≫ 1) is a poor κ measurement but an excellent substrate measurement. Methods that fail at their intended quantity can be repurposed at the quantity they actually measure.
The CNEOS catalogue contains two columns correlated at 0.99992 in log–log with a residual of 0.0064 dex. That is not two measurements; it is one measurement and a recalibration. The detector:
if log-log R² > 0.999 and residual sd < 0.02 dex:
flag the pair as derived, not independentApplying it to any catalogue takes seconds and prevents an entire class of double-counting errors. The rule generalises: before using two columns as independent evidence, check whether one is a function of the other.
| # | Lead | |
|---|---|---|
| B1.1 | Materialise design 2: find a channel pair with the required b spread (three decades). | note |
| B1.2 | Two-switch selectivity is a new device class; the design space is the (K ratio, A ratio, b spread) cube. | test |
| B1.3 | The number of switch windows is bounded by the number of distinct b clusters per side. | test |
| B1.4 | A three-switch design requires opposite limits and unequal counts; construct one. | test |
| B1.5 | Switch window width is predictable from the b spread alone, without simulation. | test |
| B1.6 | The peak ratio is set by A_N/A_D and is independent of the crossing positions. | test |
| B1.7 | Any two-state membrane can be inverted to a two-state sensor. | note |
| B1.8 | Three-channel designs are the minimum; two cannot produce a double crossing. | test |
| B1.9 | The design cube has a feasible region that can be mapped and reported. | test |
| B1.10 | Membranes with two switches would outperform single-switch membranes in pressure-swing separation. | note |
| B1.11 | Parity constrains every proposed switch design, so a design claim can be falsified before fabrication. | test |
| B1.12 | A switch with a peak ratio of 2 is the best case for three channels; more channels raise it. | test |
| # | Lead | |
|---|---|---|
| A1.1 | Three-estimator reporting would change the published tail index in most fields. | test |
| A1.2 | The estimator spread is itself a completeness diagnostic for the catalogue. | test |
| A1.3 | PWM is the stable estimator; Hill is the drifting one; this ordering is general. | test |
| A1.4 | Any index quoted to two decimals is over-precise by a factor of ten. | note |
| A1.5 | Estimator spread scales as 1/√n and can be predicted in advance. | test |
| A1.6 | Fields that fit tails in log-log space systematically use Hill. | note |
| A1.7 | A tail index and its estimator spread should be reported as a pair, like value and uncertainty. | note |
| A1.8 | Derived columns are common enough that a repository audit would find several. | test |
| A1.9 | Threshold-stability curves could be a required figure in any tail-fitting paper. | note |
| A1.10 | The 0.32 value is a defensible reference for CNEOS and can be quoted with its stability. | test |
| A1.11 | Estimator dependence is worst where the data are most truncated. | test |
| A1.12 | A standard would make half of tail-based claims unpublishable as stated. | note |
| # | Lead | |
|---|---|---|
| D1.1 | λ should be a reported quantity, like an uncertainty. | note |
| D1.2 | A two-spot protocol with one spot below λ = 1 is the minimum viable measurement. | test |
| D1.3 | Substrate-dominated measurements can be repurposed as substrate measurements. | test |
| D1.4 | The 75.8× spread is computable from reported spot sizes alone, so any reader can check it. | test |
| D1.5 | Three of four published 2D conductivities are substrate-limited. | test |
| D1.6 | The same λ structure exists for any measurement with a substrate term. | test |
| D1.7 | A reference ratio can be designed into a device to make it reference-insensitive. | note |
| D1.8 | Reference-dominated methods should report their reference, not their sample. | note |
| D1.9 | The two-spot rule generalises to two-condition protocols in other fields. | test |
| D1.10 | Inter-laboratory disagreement would halve on re-expression in one convention. | test |
| D1.11 | A reference-free measurement exists wherever the reference enters linearly. | test |
| D1.12 | Reporting the mechanism of failure (λ > 10) is more useful than reporting the number. | note |
| # | Lead | |
|---|---|---|
| E1.1 | Half of publicly checkable claims change status on re-analysis; test on five more. | test |
| E1.2 | Derived-column detection automates the first check. | test |
| E1.3 | Snapshot dates should be mandatory for catalogue-based results. | note |
| E1.4 | Conditional claims survive re-analysis better than unconditional ones. | test |
| E1.5 | A five-catalogue audit would establish the base rate. | test |
| E1.6 | Replication takes an hour per claim once the data are public. | test |
| E1.7 | Most catalogues have at least one derived column; find them. | test |
| E1.8 | Re-analysis should be citable, and is not. | note |
| E1.9 | The residual of a derived relation recovers the hidden variable. | test |
| E1.10 | Public APIs are under-used relative to their accessibility. | note |
| E1.11 | Volume incompleteness is detectable from tail drift alone. | test |
| E1.12 | A replication template halves the time to a verdict. | note |
| Rank | Lead | Why | Cost |
|---|---|---|---|
| 1 | B1.2 / B1.9 | A new device class with a mapped design cube. First constructible output of the batch. | days, simulation |
| 2 | A1.1 | Three-estimator audit of twenty published indices could move a field. | days, no lab |
| 3 | D1.5 | If true, a large literature is measuring substrates. | hours, reanalysis |
| 4 | E1.5 | Establishes the replication base rate across five catalogues. | days, APIs |
| 5 | B1.4 | Three-switch design: tests the parity limit at its edge. | hours, simulation |
If one sentence survives: the parity constraint turned a screen into a design rule, and the rule produced three two-switch membranes with full parameters — the first thing in this batch that could be built rather than checked.