Part A · I2 and H0199 re-scored under the spread criterion
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.
Part B · H203 re-analysed from live catalogue data
Source: ssd-api.jpl.nasa.gov/fireball.api, fields energy (radiated, kt) and
impact-e (estimated impact energy, kt), plus date, location, altitude and velocity. Saved
locally as data/fireballs.json (44 KB).
| Quantity | Value |
|---|---|
| Events | 602 |
| Radiated energy range | 1.24 – 37,500 kt |
| Impact energy range | 0.048 – 441 kt |
| Date coverage | 1997 – 2026 |
This is a real catalogue, and it is not volume-complete. Its lower cut-off is set by detection, so no statement here is about the true impact rate — only about what this catalogue contains.
The hub's H203 record concluded that the impact energy is a deterministic function of the radiated energy (reported r = −0.995), which is why the result was not promoted. That is confirmed here, and more sharply than reported.
| Statistic | Value |
|---|---|
| Pearson r, log₁₀ energy vs log₁₀ impact energy | 0.99992 |
| Spearman ρ | 0.99950 |
| Residual sd about a straight log–log line | 0.0064 dex |
| Median ratio impact/radiated | 0.03154 |
| Ratio range | 0.0118 – 0.0387 (3.29×) |
Interpretation. Over four and a half decades of energy, the two columns agree to 0.0064 dex — about 1.5%. The impact energy carries almost no information independent of the radiated energy. Any analysis that treats them as two measurements is using one measurement twice.
The 3.29× ratio spread is the one caveat: the relation is tight but not exactly proportional, so "deterministic" is right to a few per cent, not to machine precision. The hub's own conclusion — that this makes H203's tail result conditional rather than physical — stands and is strengthened.
Hill estimator of the tail index above several thresholds:
| Threshold (kt) | Exceedances | Hill α |
|---|---|---|
| 0.2 | 602 | 0.257 |
| 0.5 | 602 | 0.337 |
| 1.0 | 602 | 0.440 |
| 2.0 | 590 | 0.619 |
| 5.0 | 369 | 0.722 |
The hub reports b = 2.65–2.82 and α = 1.33–1.41 for thresholds 0.2–2 kt. This re-analysis gives α = 0.26–0.62 over the same thresholds — a factor of two to five lower, and in the opposite direction from the reported range.
A 40-point threshold sweep gives α from 0.24 to 0.69, median 0.37, a spread of 2.86×. The estimate is strongly threshold-dependent, which is the usual signature of a Hill estimator applied below its valid region.
Three explanations are possible, and I cannot separate them with this data alone:
The honest statement: the tail-index number in H203 does not reproduce from the current catalogue under a Hill estimator, and the discrepancy is large enough that the number should not be quoted until the estimator and the catalogue cut are both specified.
| H203 element | Status after re-analysis |
|---|---|
| Impact energy is a recalibration of radiated energy | Replicated, and sharper (r = 0.99992, 0.0064 dex) |
| "Not promoted until a second catalogue exists" | Correct call — the dependency is confirmed |
| b ≈ 2.65–2.82, α ≈ 1.33–1.41 | Not reproduced; α = 0.26–0.62 here |
| α = b/(3γ) with predicted 1.35 | Cannot be tested until b is re-derived consistently |
| # | Hypothesis | Disposition |
|---|---|---|
| N-20 | Published impact-loss tail indices are estimator-dependent, and the spread across estimators exceeds the difference between competing physical models. | test now — three estimators on this catalogue |
| N-21 | A catalogue whose impact energy is calibrated from radiated energy cannot test a hypothesis about impact energy, because the two are one measurement. This applies to any catalogue with a derived column. | test now — check other catalogues for derived columns |
| N-22 | The 3.29× ratio spread is not noise but encodes the velocity and altitude dependence, so the residual of the energy relation recovers the entry parameters. | test now — regress the residual on velocity and altitude |
If one sentence survives: the deterministic-recalibration objection to H203 is confirmed at 0.0064 dex, and the tail-index number does not reproduce from the current catalogue — so the record should keep the objection and withdraw the number until the estimator is specified.
Incoprea Research · batch 1 · 3 November 2026. Computational model results and one real-catalogue re-analysis. No new physical experiment was performed. No novelty claimed; no systematic literature search. Code, data and JSON accompany this page set.