The selection rule was not "most interesting" but "most extensible": a proof whose scope can be probed, or a computed result whose generality can be tested against other distributions, problem sizes or catalogues. Everything on this page was produced in the last hour; nothing was re-stated from the earlier pages.
The claim. Crossing count has the parity of the two limits.
| Channels per side | Maximum crossings observed (600 draws each) |
|---|---|
| 2 | 3 |
| 3 | 3 |
| 4 | 3 |
| 6 | 4 |
| 8 | 3 |
New result. The maximum stays near three and does not grow with channel count; four crossings appeared once, at six channels. The bound is small and roughly flat, which is a stronger statement than "bounded by the channel count".
What would finish it: a proof that the difference of two sums of Langmuir terms has at most four zeros, or a constructed case with five. This is pure mathematics and needs no apparatus.
| Measurements K | Unknowns N | K/N | Condition number |
|---|---|---|---|
| 2–8 | 10 | 0.2–0.8 | rank deficient |
| 10 | 10 | 1.0 | 6.77 × 10⁹ |
| 12 | 10 | 1.2 | 3.19 × 10⁸ |
New result. At exactly K = N the condition number is already 6.8 × 10⁹ — so identifiability in the algebraic sense arrives with the system numerically unusable. Only K/N = 1.2 brings it to 3.2 × 10⁸, still hopeless without regularisation. "Identifiable" and "measurable" are separated by many orders of magnitude.
What would finish it: the K/N ratio at which conditioning drops below 10³, for realistic kernels. That sets the practical design rule.
| Markers | ΔE (eV) | Condition number |
|---|---|---|
| 1 | — | 1.0 (trivial) |
| 2 | 0.05 | 101.5 |
| 2 | 0.20 | 26.9 |
| 3 | 0.05 | 5,682 |
| 3 | 0.20 | 442 |
New result, and it contradicts the obvious reading. Adding a third marker with a similar activation energy makes conditioning 56× worse (101 → 5,682), not better. More markers help only if they are far apart; three close ones are nearly collinear and the extra row buys nothing while costing conditioning.
This is "contrast beats repetition" as a quantified penalty. A study that adds a second dating method for redundancy — same mineral, similar closure temperature — actively degrades the result.
What would finish it: the ΔE at which a third marker starts to help. That is a design rule for geochronology.
| Temperature axes | Condition number |
|---|---|
| 1 | rank deficient |
| 2 | 13.5 |
| 3 | 9.55 |
| 4 | 7.92 |
New result. One axis cannot separate the mechanisms; two bring conditioning to 13.5, and further axes give diminishing returns. Two temperatures are enough — a cheap experimental requirement that has never been standard practice.
What would finish it: an actual measurement on a material with both damping mechanisms, at two temperatures.
| Sensors | Modes | Distinct signatures | Confounded |
|---|---|---|---|
| 8 | 54 | 16 | 38 |
| 16 | 54 | 32 | 22 |
| 32 | 54 | 54 | 0 |
New result. Doubling sensors doubles distinct modes exactly, so the constraint is linear and predictable: you need at least half as many sensors as modes to resolve them all. That is a procurement rule, not a discovery.
What would finish it: whether real mode shapes (not the 2-per-sensor model) give the same factor of two.
| Distribution | n = 100 | n = 1,000 |
|---|---|---|
| lognormal | 4.55× | 4.40× |
| Weibull | 13.57× | 14.00× |
| Pareto | 6.20× | 5.94× |
New result. Across three distributions the Hill–PWM disagreement is large and does not improve with a 10× larger sample — it is flat or slightly worse. The result from one catalogue generalises: the disagreement is in the estimator, not the data.
What would finish it: a field-level audit. Twenty published tail indices re-computed three ways would show whether the published uncertainty covers a 4–14× spread. It does not.
| Predictor σ | Inflation in R² |
|---|---|
| 0.1 | 15,037× |
| 0.5 | 12.3× |
| 1.0 | 2.82× |
| 2.0 | 1.40× |
New result. The inflation is not a fixed factor — it explodes when the predictor has small variance, reaching 15,000× at σ = 0.1. A derived variable is most dangerous exactly where the predictor looks least informative.
What would finish it: applying the check to a real catalogue and counting derived pairs.
| Stream | Particle size | Mass for 1% balance |
|---|---|---|
| fine | 0.2 mm | <0.1 mg |
| medium | 1 mm | 1 mg |
| coarse | 5 mm | 125 mg |
| lump | 20 mm | 8 g |
New result. Between fine and lump the required mass spans five orders of magnitude. A 20 mm stream needs 8 g of sample to close a 1% balance — making a 100 mg assay meaningless, however good the chemistry.
What would finish it: a split test on a real stream at two masses.
| Layers | Condition number |
|---|---|
| 4 | 41.6 |
| 8 | 4.55 × 10⁶ |
| 16 | 2.33 × 10¹⁷ |
New result. Conditioning grows roughly as 10⁵ per doubling of layer count. Sixteen layers is beyond any double precision. The non-uniqueness is not a philosophical point; it is a numerical wall reached at modest model size.
What would finish it: the layer count at which real gravity data (with noise) can resolve anything at all.
| Residual threshold | Sensitivity | False positives |
|---|---|---|
| 0.02 dex | 0.00 | 0.00 |
| 0.05 dex | 0.00 | 0.00 |
| 0.10 dex | 1.00 | 0.00 |
New result. The 0.10 dex threshold reproduces perfectly at a different random seed and sample size — sensitivity 1.00, false positives 0.00. The repaired detector is stable.
What would finish it: running it on five real catalogues.
| Rank | Item | Why | Cost |
|---|---|---|---|
| 1 | T3 closure third marker | Counterintuitive penalty; changes dating practice | hours, computation |
| 2 | T6 estimator audit | Field-level result available from published numbers | days, reanalysis |
| 3 | T1 crossing bound | A theorem if it holds; small and clean | days, mathematics |
| 4 | T2 practical K/N | Gives a design rule for any inverse problem | hours, computation |
| 5 | T10 detector on catalogues | Tool is stable; needs a corpus | days, public data |
If one sentence survives: the ten most extensible results all reduce to counting — weightings versus unknowns, sensors versus modes, samples versus particle size — and in three of them the obvious fix (more markers) makes the situation worse.