Aperiodic Multiplying Media defines a multiplying medium as one in which a quantity is generated as well as transported, with a Perron–Frobenius eigenvalue λPF governing both geometry and growth. Keratin is not one. The shaft is dead: generation within the medium is exactly zero, and material is advected from a boundary source at the matrix. There is no inflation and no eigenvalue to take.
A medium that generates within itself overwrites anything travelling through it — a gain medium amplifies power and destroys the record. Fidelity requires λPF = 1. What makes keratin fail the multiplying-media test is exactly what makes it a faithful archive. Elongation is not inflation, and the absence of inflation is the specification, not a deficiency.
Keratinisation does not complete at a point. It completes over a zone of width w, so the shaft carries the true signal convolved with a kernel of that width; in time the kernel is T = w/v wide. Taking the conservative case of a box kernel, the transfer function is a sinc, and the half-power point solves sinc²(πfT) = ½. Bisection gives fT = 0.4429 — computed in block [2] rather than quoted.
| w (mm) | T = w/v (days) | shortest period resolved (days) |
|---|---|---|
| 1.0 | 2.9 | 6.5 |
| 2.0 | 5.7 | 12.9 |
| 3.0 | 8.6 | 19.4 |
| 5.0 | 14.3 | 32.3 |
At scalp hair’s 0.35 mm/day and a nominal 2 mm zone, anything faster than a period of about thirteen days is attenuated below half power. It is not in the record at any reading resolution. The table is printed as a function of w because w is the least well pinned input in the paper.
Segmental analysis cuts the hair at fixed intervals, conventionally 1 cm, with the first cut placed at the scalp and therefore at an arbitrary phase of whatever is being recorded. That is systematic sampling with a random start of a one-dimensional signal — Cavalieri, on the time axis.
And the signal does not vanish at either cut. WP-112 shows that the asymptotic error exponent is fixed by the order at which the measured function vanishes at the ends of its support: linear vanishing gives γ = 2, ends that do not vanish give γ = 1. A hair segment series is squarely the second case. Block [3] re-measures it on the same construction:
So the error falls as 1/n, not 1/n². Quadrupling the number of segments halves the error rather than quartering it. Anyone carrying precision intuitions across from whole-organ Cavalieri — where the object has poles and the exponent is 2 — is a full power optimistic about a segmental series.
If the true rate differs from the assumed rate by a factor 1+ε, an event read at distance x is dated x/vassumed instead of x/vtrue. The error is εt — linear in the age of the event, so the far end of the record is always the worst dated.
| ε | 1 month | 3 months | 6 months | 12 months |
|---|---|---|---|---|
| 5% | 1.5 d | 4.6 d | 9.1 d | 18.3 d |
| 10% | 3.0 d | 9.1 d | 18.3 d | 36.5 d |
| 20% | 6.1 d | 18.3 d | 36.5 d | 73.1 d |
That ε is not hypothetical. Nail rate varies with site, season, age and health, and the two sites of one body differ from each other by a factor of 2.5 on the nominal figures — fingernail 3 mm/month against toenail 1.2. A subject carries at least two clocks.
| regime | limit set by | scale |
|---|---|---|
| events shorter than about two weeks | the write head | 13 days, irrecoverable |
| two weeks to about a month | the segment length | 29 days per cut |
| dating an event months back | the warp | εt, linear |
A 1 cm segment at 0.35 mm/day is 29 days. The write head at a nominal 2 mm is 13. Those are within a factor of about two of each other, which says something practical: the conventional segment length already sits close to the physical write limit, and cutting finer buys resolution that is not in the record. Effort spent on finer segmentation is better spent on the warp term, which is unbounded in the age of the event and is the only one of the three that grows.
documented Two nail-matrix impulses in one subject in 2019, one fingernail and one toenail, with arrival of each mark at the free edge observed at 6 and 12 months respectively. Against commonly quoted complete-replacement times of 4–6 months and 12–18 months, both sit inside the range and at its fast end, and the observed transit ratio is 2.00 against a nominal rate ratio of 2.50.
It is n = 1, retrospective, with arrival times observed rather than distances measured, so it constrains transit and calibrates no rate. A matrix impulse can also perturb the rate it marks, a confound the fluorochrome version of this measurement — two labelled doses a known interval apart, distance measured between them — does not have. What it does show is the shape a real calibration would take: dated impulse, observed transit, two sites, one subject, one year. That is the measurement segmental chronology assumes and almost never makes.
No published chronology is reanalysed here. The claim that the three terms are separable in practice is a claim about the model and not a demonstration on data; what would test it is a series with an independently dated event in it, where the write-head width and the warp can be fitted rather than assumed. The width of the keratinisation zone is the weakest input and every consequence in section 2 is printed as a function of it for that reason. The box kernel is the conservative choice: a smoother write head cuts off sooner, so the resolution limit here is optimistic.