Two books, and only one of them balances

Run the two books separately.

1. For the equation: count nucleons ($A$) and charge ($Z$), and require each to match across the arrow. That is what identifies the products.
2. For the energy: take reactant mass minus product mass and multiply by $c^2$. A positive defect means energy released, as kinetic energy of the products or as photons.

$$E = \Delta m c^2, \qquad 1\ \text{u} \approx 931\ \text{MeV}.$$

Totalling the masses on each side and expecting them to agree reports zero energy released, or takes the difference with the sign reversed and claims energy was absorbed. Balanced nucleons and balanced mass are different claims, and only the first one is true.

Total binding energy simply keeps growing as nuclei get bigger, so ranking by it makes uranium the most stable nucleus in existence. What actually tracks stability is binding energy per nucleon, which peaks near iron at roughly $8.8$ MeV.

Divide by the nucleon number before comparing anything, and both nuclear energy sources fall out of one curve:

1. Fission. Uranium sits near $7.6$ MeV per nucleon; its fragments sit near $8.5$. Splitting moves toward the peak, so it releases the difference.
2. Fusion. Light nuclei sit far below the peak, so joining them moves toward it too, releasing energy.

Moving toward the iron peak from either side releases energy. That single sentence explains why fission works for heavy nuclei, fusion for light ones, and neither one for iron itself.

A half-life removes half of what is left, not a fixed amount. So a sample runs to one half, then one quarter, then one eighth, and never reaches zero.

Declaring the sample gone after two half-lives is the error, and so is dividing the starting amount by the number of half-lives. Two half-lives leave $25\%$.

For a non-whole number of half-lives, either form works and they agree everywhere:

$$N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}} = N_0 e^{-\lambda t}, \qquad \lambda = \frac{0.693}{t_{1/2}}.$$

Constant proportion per interval, never a constant amount. That is why the decay curve flattens as it goes instead of striking the axis, and why a graph of counts against time is never a straight line.

The amount present, the temperature, and the chemical form do not change it. A sample that has already lost $90\%$ of its material halves the remainder in exactly the same time as when it was fresh.

So doubling the sample does not double the time to halve it. It doubles the number decaying per second and leaves the half-life untouched.

The underlying reason is worth stating, because it also settles a related question. No nucleus tracks its own age. Each one has the same chance of decaying in the next interval, whether it was formed a second ago or a million years ago, and the smooth exponential is simply what an enormous number of independent chances looks like from outside.

Half-lives span an extraordinary range, from fractions of a second to billions of years, and every one of them is fixed by the isotope alone.