I2's strict test quotes four literature cases with λ = 0.0625, 10.0, 19.2 and 30.6. The bias factor is
kappa_app / kappa = R(lambda) = 1 / (e^lambda * E1(lambda))with E1 the exponential integral, computed here by direct quadrature (no lookup table). Evaluating R at the four quoted λ values:
| Case | λ | R = κ_app/κ |
|---|---|---|
| A MoS₂/SiO₂ | 19.20 | 20.15 |
| B graphene/SiO₂ | 30.60 | 31.57 |
| C Si nanofilm | 10.00 | 10.92 |
| D Si nanofilm, small spot | 0.0625 | 0.4162 |
Spread across the four cases: 75.8×. Three of the four sit in the substrate-dominated regime where the reported conductivity is set by the substrate, not the film; one sits below the crossover and reads low.
What this adds. The I2 record already said the crossover is at λ* = 0.4348 rather than 1, and that three of four cases are unidentifiable for a single spot. This re-scoring puts a single number on the practical consequence: the same method, applied to four published cases, reports κ_app/κ spanning 75.8×. That is the spread criterion's answer to "is the method measuring the material?" — it is measuring λ.
The hub's simulation table gives percent errors for three layouts on two fields:
| Field | edge | grid | interleaved | spread |
|---|---|---|---|---|
| boundary-null | 23.2% | 0.83% | 2.22% | 28.0× |
| rim | 37.8% | 0.90% | 5.57% | 42.0× |
Overall spread across layouts: 45.5×. The claim under test is that interleaved witness patches beat edge blanks — and they do, by a factor of about ten. But the layout choice spans 45.5×, which is four to five times larger than the effect being claimed.
Interpretation. The original claim is directionally supported, but its magnitude is not stable: it is a claim about a ratio that itself ranges over a factor of ten across field type and layout. The honest statement is that interleaved beats edge on boundary-structured fields by roughly 10×, and no layout wins everywhere — which the hub's record already says in words, now with a number attached.
Earlier, the maximum number of crossings of α(P) = 1 was 2 in every stress test, including runs with four saturating channels where a bound of S−1 would have allowed three. That is a discrepancy worth resolving, so it was attacked directly.
| crossings | 0 | 1 | 2 | 3+ |
|---|---|---|---|---|
| count | 281 | 103 | 16 | 0 |
Most parameter draws produce no crossing at all; one crossing is common; two is rare; three was never observed.
The obvious way to manufacture three crossings is to stagger the half-saturation constants so that each channel turns on in sequence. Half-saturation constants were spread over two or three decades, with the numerator's set shifted by a constant factor:
| Construction | crossings |
|---|---|
| S = 2, two decades | 0 |
| S = 3, two decades | 0 |
| S = 4, three decades | 0 |
| S = 3, Hill exponent 2 | 0 |
| S = 3, Hill exponent 4 | 0 |
The staggered constructions produced no crossings at all — the shifted numerator stayed above the denominator throughout. This is a failure to find a counterexample, not evidence that none exists: the construction was too crude, because a constant shift of all half-saturations preserves the ordering rather than interleaving it.
Conjecture (stated, not proved). For α(P) = N(P)/D(P) where N and D are sums of Langmuir-type saturating terms K + A·bP/(1+bP) with at least one pressure-independent term on each side, the equation α(P) = 1 has at most two solutions on P > 0.
Evidence for: 1,000+ random draws at S ≤ 4 with no more than two crossings; no construction yet produces three.
Evidence against: none, but the search is weak. A proper attack requires interleaving the half-saturations (b1 < b2 < b3 on one side, between them on the other) with amplitudes chosen to force alternation.
Why it matters. If true, the screening rule becomes: more than two crossings is impossible for this class, so any reported third crossing indicates a model outside the class — a stronger and more useful statement than the S−1 bound.
| # | Hypothesis | Disposition |
|---|---|---|
| N-16 | The two-crossing bound holds for interleaved half-saturations; a constructed alternation with amplitudes chosen to force sign changes will either produce three or reveal the structural reason it cannot. | test now — pure computation |
| N-17 | Any measurement method whose reference is a substrate will show a reported-property spread of at least 10× across published cases unless λ is reported. | test now — scan published datasets for λ |
| N-18 | The layout spread (45.5×) exceeding the claimed effect (10×) is the general signature of a design claim that is really a field-type claim. | note — needs a second dataset |
| N-19 | In I2, the single case below the crossover (λ = 0.0625) is the only one where a single-spot measurement carries material information, so published single-spot values are systematically the ones with the smallest λ. | test now — selection-bias check on cited values |
If one number survives: the same thermal method, applied to four published cases, reports a conductivity spanning 75.8× — and the largest term in that spread is λ, not the material.