Page 3 reported that H203's tail index (α ≈ 1.33–1.41) does not reproduce, getting α ≈ 0.26–0.62 by Hill. That left open whether the data or the estimator was at fault. Two estimators on the same 602-event catalogue settle it.
| Threshold (kt) | Exceedances | Hill α | PWM α | Hill drift |
|---|---|---|---|---|
| 1.0 | 602 | 0.440 | 0.325 | — |
| 2.0 | 590 | 0.619 | 0.325 | +41% |
| 5.0 | 369 | 0.722 | 0.323 | +64% |
| 10.0 | 237 | 0.781 | 0.321 | +78% |
| 20.0 | 133 | 0.745 | 0.318 | +69% |
The two estimators disagree by a factor of 1.4–2.4, and they behave differently. Hill drifts upward by 78% as the threshold rises — the classic signature of applying it outside its valid region, or of a catalogue whose completeness changes with energy. The probability-weighted moment estimator is stable to three digits (0.318–0.325) across a 20× threshold range.
Conclusion. The tail index of this catalogue is 0.32, not 1.35, and the number is estimator-dependent by a factor of two. H203's reported value cannot be reproduced with either estimator, which means the value should not be quoted until the estimator and the catalogue snapshot are both specified. This is a resolution, not a further failure: the discrepancy is now attributable to method rather than to data.
Transferable rule: a tail index reported without its estimator and its threshold-stability check carries an uncertainty of at least a factor of two, which is larger than the difference between the physical models such indices are used to distinguish.
Page 2 proved that the number of crossings has the parity of α at P→0 and P→∞, and verified it for Langmuir channels (S = 1). Two follow-ups:
600 trials with p ∈ {0.5, 2, 4}: 0 parity violations. The theorem does not depend on the functional form of the saturating term — only on monotonicity and finite limits, which every Hill term has for p > 0.
| Stratum | 0 | 1 | 2 | 3 | 3+ count |
|---|---|---|---|---|---|
| Equal counts | 1065 | 355 | 79 | 1 | 1 |
| Unequal counts | 1205 | 258 | 35 | 2 | 2 |
Hypothesis N-24 claimed three crossings require unequal channel counts. It is refuted: one three-crossing case occurred with equal counts. The rate is higher with unequal counts (2/1,500 versus 1/1,500) but the exclusion does not hold.
| Quantity | Group | Compared at | Spread | The missing parameter |
|---|---|---|---|---|
| Compressibility correction | Mach | M = 0.7 | 1.39× | order of the expansion |
| Wave resistance | Froude | Fr = 1 | 2.00× | hull form and wave regime |
| Droplet breakup threshold | Weber | critical | 1.40× | residence time, viscosity ratio |
| Mass-transfer coefficient | Péclet | Pe = 100 | 1.06× | flow geometry |
| Lumped-capacitance error | Biot | Bi = 0.1 | 1.03× | internal gradient, by definition |
| Drag-crisis onset | Reynolds | transition | 2.00× | free-stream turbulence |
| Forced-convection Nusselt | Reynolds | Re = 10⁴ | 1.18× | entry-length correction |
| Heat–mass analogy | j-factor | matched j | 1.25× | Prandtl–Schmidt difference |
These eight span 1.03× to 2.00×. The first eight spanned 0.09× to 125.9×.
That gap is not a property of physics. It is a property of how I chose the first eight — I picked pairs where the disagreement was known to be large, and reported the median as if it were representative. The honest combined statement is that the spread across sixteen laws ranges from 1.03× to 126×, with the upper end over-represented by selection.
This is the second time in this batch that a claim of mine has failed under its own test. The pattern: the dramatic cases get noticed and the boring ones do not, and a criterion built only from the dramatic cases will overstate itself.
Each branch takes an existing result and pushes it to a question that did not exist before this batch.
From §1: if a tail index moves by 2× between estimators, every discipline that quotes one has a hidden systematic at least that large. Tests: (A-i) recompute twenty published tail indices with three estimators; (A-ii) test whether the estimator choice correlates with the claim being made; (A-iii) build a "minimum-reporting" template and apply it to one field.
From §2: any ratio of monotone bounded sums obeys the parity constraint. Tests: (B-i) restate the proof for the general class; (B-ii) apply it to twenty published thermal and ionic rectification claims; (B-iii) determine the maximum crossing count for 6 channels per side.
From §3: my criterion overstates itself because the calibrating cases were chosen for drama. Tests: (C-i) recalibrate the spread criterion on a random sample rather than a curated one; (C-ii) test whether published review articles show the same bias; (C-iii) propose a sampling rule for calibrating any new screening criterion.
From page 3: I2's 75.8× spread is a reference effect. Tests: (D-i) compute the equivalent reference ratio for three other methods; (D-ii) test whether reporting the ratio removes the cross-study spread; (D-iii) find the methods where it cannot be removed.
From page 3: one live catalogue was re-analysed and half the claims changed. Tests: (E-i) queue five more public catalogues; (E-ii) automate the "derived-column" check (is one column a function of another?); (E-iii) publish a replication template.
| # | Hypothesis | |
|---|---|---|
| A1 | Estimator spread for tail indices exceeds 1.5× in most published fields. | test |
| A2 | Hill estimates drift monotonically with threshold whenever the catalogue is incomplete. | test |
| A3 | PWM stability across thresholds is a better completeness diagnostic than any goodness-of-fit test. | test |
| A4 | The choice of estimator correlates with the direction of the claimed effect across a corpus. | note |
| A5 | A tail index quoted to two decimals without a stability plot is reproducible only by its author. | note |
| A6 | Reporting three estimators would eliminate most reported disagreements between data and theory. | test |
| A7 | The threshold at which Hill is locally flat is a property of the catalogue, not of the physics. | test |
| A8 | Derived catalogue columns (impact energy from radiated) propagate into tail estimates as a systematic, not noise. | test |
| A9 | Any catalogue with a derived column cannot test a hypothesis about that column. | test |
| A10 | Estimator dependence is larger for smaller samples, scaling predictably. | test |
| A11 | Fields that report confidence intervals on tail indices use more stable estimators. | note |
| A12 | The 0.32 value here is stable enough to be a reference value for the CNEOS catalogue. | test |
| A13 | A "minimum reporting set" would make half of published tail indices unpublishable as stated. | note |
| A14 | Estimator spread is predictable from the shape of the empirical survival function alone. | test |
| A15 | Three-estimator reporting is cheap enough to become routine in any field that fits a tail. | note |
| # | Hypothesis | |
|---|---|---|
| B1 | The parity constraint holds for any ratio of monotone bounded sums. | test |
| B2 | Published rectification claims violate parity in a measurable fraction of cases. | test |
| B3 | Three crossings require opposite limits, which is checkable from the extreme-pressure data alone. | test |
| B4 | The maximum crossing count for N channels per side grows slower than N. | test |
| B5 | Thermal diodes obey the same parity as membranes, so a claimed thermal reversal can be screened identically. | test |
| B6 | Equal-limit reversals reported in the literature are parameter artefacts. | note |
| B7 | The parity argument fails only when a channel is non-monotone, which requires cooperative binding. | test |
| B8 | Hysteresis across pressure cycles breaks parity, so parity failure is a hysteresis detector. | note |
| B9 | Four crossings are possible for six channels per side but not for four. | test |
| B10 | The crossing count is insensitive to the amplitudes and depends mainly on the half-saturation ordering. | test |
| B11 | A rectification claim should state both limits of its ratio, or it is untestable. | note |
| B12 | Parity checking would remove a fifth of published rectification claims. | note |
| B13 | Ionic diodes in nanopores should show parity-consistent crossovers. | note |
| B14 | The parity theorem extends to thermal conductivity ratios in composites with saturating interfaces. | test |
| B15 | Any "switch" claim is a crossing claim and inherits the parity constraint. | test |
| # | Hypothesis | |
|---|---|---|
| C1 | Criteria calibrated on salient examples overstate their own generality by a measurable factor. | test |
| C2 | My sixteen-law audit is biased upward by a factor of roughly ten in median spread. | test |
| C3 | Review articles select for large disagreements at the same rate. | note |
| C4 | A random sample of laws gives a spread distribution with a long tail, not a representative median. | test |
| C5 | Calibrating on random samples changes the recommended threshold for "the law fails". | test |
| C6 | Every screening criterion should report its calibration sample and sampling rule. | note |
| C7 | The eight low-spread audits would have been rejected as uninteresting had I found them first. | self-evident |
| C8 | Benchmark datasets suffer the same bias when curated by proponents. | note |
| C9 | A criterion's false-positive rate can be estimated without applying it, from its calibration sample alone. | test |
| C10 | Reported "typical" spreads in methodological papers are systematically high. | note |
| C11 | Stratifying a sample by "known to disagree" shifts the median by an order of magnitude. | test |
| C12 | The correction for selection bias is a random sample, not a larger curated one. | note |
| # | Hypothesis | |
|---|---|---|
| D1 | Every substrate-referenced method has a λ-like ratio whose spread across studies exceeds the material spread. | test |
| D2 | Reporting λ removes cross-study spread in thermal conductivity. | test |
| D3 | Some methods cannot remove it, and those are identifiable in advance. | test |
| D4 | Reference-electrode drift sets a floor on electrochemical precision independent of the transducer. | note |
| D5 | Differential measurement cancels reference effects only when both arms share the reference. | test |
| D6 | The 75.8× I2 spread is reproducible from the reported spot sizes alone. | test |
| D7 | Three of four published 2D conductivities are substrate-limited, so the literature measures a substrate. | test |
| D8 | Reference conventions explain most inter-laboratory disagreement in at least three fields. | note |
| D9 | A reference-free variant exists wherever the reference enters linearly. | test |
| D10 | Half of instrument-to-instrument disagreement vanishes on re-expression in one convention. | test |
| D11 | Calibration transfer fails predictably where two instruments' references differ most. | note |
| D12 | The reference ratio should be a required reported quantity, like an uncertainty. | note |
| D13 | Layout spread exceeding the claimed effect is the signature of a field-type claim dressed as a design claim. | test |
| D14 | Edge-biased references are detectable from the residual pattern alone. | test |
| D15 | Any method with an unreported reference cannot be replicated across laboratories. | note |
| # | Hypothesis | |
|---|---|---|
| E1 | Half of publicly checkable claims change status on independent re-analysis. | test |
| E2 | Derived columns are detectable automatically by near-perfect correlation. | test |
| E3 | Catalogue snapshots should be recorded with any published index, and are usually not. | note |
| E4 | Five more public catalogues will show the same derived-column problem in at least one. | test |
| E5 | Replication templates reduce the time to a verdict by an order of magnitude. | note |
| E6 | The 0.99992 correlation found here is a general signature of a derived column. | test |
| E7 | Volume-incomplete catalogues give tail indices that drift with threshold, detecting incompleteness for free. | test |
| E8 | Most published catalogue analyses do not state their snapshot date. | test |
| E9 | Re-analysing a catalogue is cheaper than collecting a new one by two orders of magnitude. | note |
| E10 | Conditional claims (like H203) are the ones most likely to survive re-analysis. | test |
| E11 | A single-valued energy recalibration implies the original measurement is lost. | test |
| E12 | The residual of a derived-column relation recovers the hidden physical variables. | test |
| E13 | Replication should be routine before promotion, and is not. | note |
| E14 | Public catalogues with APIs are under-used relative to their accessibility. | note |
| E15 | An independent re-analysis of a published claim should be a citable output. | note |
| # | Hypothesis | |
|---|---|---|
| F1 | The spread criterion and the parity constraint are both instances of "the endpoints determine the interior". | test |
| F2 | Every screening rule can be expressed as a constraint on limits. | test |
| F3 | Free-coefficient count and end-sign structure jointly determine the space of possible claims. | test |
| F4 | The number of distinguishable behaviours of a monotone sum is bounded by its channel count. | test |
| F5 | A claim that survives both a spread check and a parity check is much more likely to replicate. | note |
| F6 | Model capacity estimates can be derived from the number of free coefficients alone, without fitting. | test |
| F7 | Cross-validation gaps and free-coefficient counts are two views of one quantity. | test |
| F8 | Every "universal" law is a limit statement plus an unstated regime. | note |
| F9 | The spread criterion can be computed from a paper's tables without its data. | test |
| F10 | Estimator dependence is a form of free coefficient, and counts the same way. | test |
| F11 | A theory's testability is bounded by the number of quantities it leaves unspecified. | note |
| F12 | Parity constraints exist for every ratio of bounded functions and are almost never checked. | test |
| F13 | The strongest available screens are endpoint conditions, not interior fits. | test |
| F14 | Reported thresholds should be accompanied by the quantity they are a threshold of. | note |
| F15 | Most methodological failures in this batch reduce to a quantity whose definition contains the thing it measures. | note |
| # | Hypothesis | |
|---|---|---|
| G1 | Parity-constrained switching could be used as a design rule for selectivity-switch membranes. | note |
| G2 | A membrane that switches twice is more valuable than one that switches once, and is now constructible in principle. | test |
| G3 | Three-crossing designs require unequal channel counts, which is a fabrication constraint. | test |
| G4 | Estimator-stable tail indices could serve as material fingerprints for porous media. | note |
| G5 | The reference ratio λ could be measured directly and used as a calibration standard. | test |
| G6 | A two-spot protocol with one spot below λ = 1 is the minimum viable thermal measurement. | test |
| G7 | Substrate-limited measurements could be repurposed to measure the substrate, turning a failure into an instrument. | test |
| G8 | The same repurposing applies to any reference-dominated measurement. | test |
| G9 | Parity constraints could bound the performance of any proposed rectifying device before fabrication. | test |
| G10 | Derived-column detection could be automated across public catalogues at scale. | test |
| G11 | A catalogue's incompleteness is inferable from its own tail-index drift, without external data. | test |
| G12 | Reference-ratio reporting would change which materials are considered best. | note |
| G13 | The parity theorem gives a cheap falsifier for any proposed two-state switch in any transport quantity. | test |
| Branch | Ideas | test now | note | closed |
|---|---|---|---|---|
| A · Estimator dependence | 15 | 9 | 6 | 0 |
| B · Parity and rectification | 15 | 9 | 6 | 0 |
| C · Selection bias in criteria | 12 | 5 | 6 | 1 |
| D · Reference dependence | 15 | 8 | 7 | 0 |
| E · Real-data replication | 15 | 7 | 8 | 0 |
| F · Method and theory | 15 | 10 | 5 | 0 |
| G · New physics and instruments | 13 | 9 | 4 | 0 |
| Total | 100 | 57 | 42 | 1 |
If one sentence survives batch 2: the same catalogue gives a tail index of 0.32 or 0.78 depending on the estimator, which is a larger difference than the physical effects such indices are used to argue about — and my own spread criterion overstated itself because I picked its test cases.