THE RADIOACTIVE DECAY

A single nucleus dies at a random instant that no one can predict. In bulk, a trillion coin-flips per second, that randomness becomes the most reliable clock in nature: an exponential with a half-life that never changes. Death per nucleus, clockwork per gram. Rendered, not quoted.

source Rutherford & Soddy, "The Cause and Nature of Radioactivity," Phil. Mag. Ser. 6, 4 (1902) 370 & 569 (Parts I & II) — free full text: web.lemoyne.edu/giunta/ruthsod.html amber·paywalled orig.

Blue Team · builds & defends
3

THE MODEL

Each nucleus has a constant probability per unit time λ of decaying, independent of its age and of every other nucleus. That single assumption forces the whole law.

Expected number surviving obeys dN/dt = −λN, whose only solution is

N(t) = N₀·e^(−λt)

Half-life t½ = ln2/λ. Mean lifetime τ = 1/λ = t½/ln2. Activity (counts/s) A = λN, itself decaying with the same exponential.

5

THE LINEAGE

A clockwork from randomness. Rutherford & Soddy (1902) saw that the Curie activity curves were pure exponentials and read off the law N = N₀e^(−λt) with a constant half-life ln2/λ.

That one exponential is the trunk from which grow the-decay-chain (Bateman, daughters in secular equilibrium), the-alpha-decay (Gamow tunnelling sets λ), and the-beta-decay (λ from the weak coupling). Same law, different λ.

7

THE WITNESS

A live re-check, re-run on every render and after any tamper. It re-derives N(t½) from the current engine and confirms it equals N₀/2. If the Red Team swaps the law, this flips red.

witness idle
The Machine · the exponential clock
4

DATA IN in ↓

A starting count N₀, a half-life t½, and an elapsed time t. Drag to feed the engine.

↓ ↓ ↓
0

THE PANEL lit

The engine computes live from N₀e^(−λt), no baked numbers. The curve draws remaining fraction over several half-lives; the dashed rungs mark 1/2, 1/4, 1/8.

decay constant λ = ln2/t½— N(t) surviving— fraction remaining N/N₀— half-lives elapsed t/t½— activity A = λN (/s)— mean lifetime τ = 1/λ—
booting…
↓ ↓ ↓
8

DATA OUT out ↓

Proven at boot: after one half-life exactly half remains; after n, a factor 2^(−n); the half-life ln2/λ does not depend on N₀; decay is memoryless. The number below is recomputed, not stored.

surviving after this t—
Red Team · attacks & breaks
1

THE ADVERSARY wall

"Then tell me which atom decays next, and when." The law cannot — and that is the point. It predicts only the ensemble. The wall here is single-event unpredictability: λ gives a probability, never a schedule. A Geiger counter's clicks are genuinely irregular; only their rate is lawful.

Second attack: "your curve assumes N is huge and continuous." True — for a handful of atoms the exponential is only the mean; real counts are Poisson with fluctuations √N. The law is a statement about expectation.

2

THE GRAVEYARD

  • "Radioactivity fades because the atoms wear out with age."
    → Memoryless. A nucleus that has survived a billion years has the same chance of decaying next second as a fresh one. No ageing.
  • "A constant number of atoms decays each second."
    → That is the linear fallacy (the planted tamper, window 6). The rate A = λN falls as N falls; decay is per-capita, not per-second-fixed.
  • "Half-life depends on how much material you start with."
    → t½ = ln2/λ is intrinsic to the isotope; identical for a microgram or a tonne.
  • "Heat, pressure, or chemistry can speed decay."
    → For the classical modes, essentially no — λ is set by the nucleus. amber tiny exceptions exist (electron-capture rates shift by ~1% under extreme ionisation).
6

THE TAMPER

Replace the true per-capita law with a linear one, N = N₀(1−λt) — a constant number lost per second. Now the half-life is no longer constant and N crosses below zero. The Witness (7) catches it live.