The simplest way would be to prevent mutation and modify sex (you do not need sex in the world, but if you must...) in such a way that, for example, the male contributes nothing except the initial stimulus (which makes this essentially parthenogenesis, and that's why sex, or better, gender is not mandatory - you always can have homosexual intercourse as a matter of course, as it happens with Eric Flint's gukuys in Mother of Demons).
All individuals are XXY triploids, and inactivation of the Y chromosome happens "at random" (this allows any ratio of male to female, depending on the probability of inactivation). XXY genotypes have male phenotype, XXy genotypes are naturally female. This kind of genetic setup is perfectly possible and naturally occurs (except for the parthenogenetic reproduction twist) in humans, where it is an anomaly. But there's no reason for it to be.
All creatures would then be identical clones genetically, but could (and probably would) sport differences, even large ones, due to epigenetic factors.
Also, depending on what the cue actually is, it might pose an interesting problem for birth control. Since all the needed machinery is self-contained in the female, sex for nonreproductive purposes could in effect be made impossible (meaning that any intercourse results in conception), or so awkward as to negate any recreational value, which might appeal to both deities and advanced aliens.
This removes most mutations due to recombination between different DNA sets.
As to how to prevent other kinds of mutation, the DNA helix gets routinely unwound and split in order to allow cell duplication. There already are enzymes that correct some common DNA replication errors. "All" that would be needed would be a super-enzyme, which we could call DNA-reed-solomonase :-), capable of detecting errors and either repairing them if possibile, or otherwise triggering the cellular self-destruct mechanism.
Such an enzyme could never have evolved (naturally), but a sufficiently advanced alien Seeder species might not find it difficult to build it from scratch.
A possible side effect would be longevity, perhaps even immortality, and complete immunity to most forms of cancer, not unlike the aliens in Asimov's Hostess.
Another effect would be a strongly reduced capability for the species to cope with environmental changes through natural selection and random mutation. This might be no great problem for a technological civilization, but for 99.999% of Earth history, it would have been a sure ticket for racial extinction.
Update: couldn't the repair mechanism itself fail?
Yes, and on a cell-by-cell basis, it will often do just that.
But the "mechanism" is not a simple reasonably-soft-fail enzyme system such as evolved creatures possess. It is rather a designed computer program implemented with amino acids.
First of all it would have to fail in a zygote, otherwise the host would only get common cancer, and anyway the mutation would not get inherited.
Then, by definition the change would need to break the main cellular repair mechanism, and the organism has no others. God or our aliens never saw the need for them, and actually had a good reason not to provide them: we want defective cells to die.
So this pro-evolutionary change would actually be counterevolutionary, since it would expose the host to all kind of cellular damage against which it would have no resistance. Given the rate at which random mutations would occur and accumulate during its early development, it would be extremely unlikely that a mutated foetus could even come to term.
We (the aliens or god) can further improve our game in two ways. One: since the child is a genetic clone, we need no placental barrier. The foetus is inundated by the mother's enzyme in addition to its own. This has no effect on perfect replicas, but mutated children die stillborn.
Two: the planet itself could be abundant in any one of (or several) mutagenic compounds or phenomena (UV radiation, natural radioactivity...). Protected individuals get no cancer, while any unprotected individual will quickly develop several.
To have a mutation in the enzyme, a point mutation or even a series of point mutations would not be enough. We'd need for the enzyme to change in such a way that some further mutations will be permitted, while oncogenic ones will still get eliminated.
This is on the same scale of an English spell checker that somehow gets corrupted during the copy, but its SHA256 hash remains the same, and the resulting program turns out to not only still work but to have become a working German spell checker (I've heard this kind of hypothetical occurrence be referred to as a Minerva mutation, from the Roman goddess that was believed to have sprung, already adult and clad in armour, from Jupiter's head).
Chances of a Minerva event are in theory not zero, but I feel they're vanishingly small. This species' designers would have worried much more about the possibility of, say, a Chicxulub impact.
A closer look
This is a bit of a hen and egg problem, so let's see it in practice.
Our alien engineered DNA can be represented like this (actual order is not important):
[H][CHECK][ BODY ]
where BODY is the DNA required by the cell, CHECK is the DNA that codes the "compare to plan, then repair or kill" mechanism, and H is the "plan" hash.
The CHECK part translates into a very large molecule (megaDalton range) with helicase capability, a sort of specialized polymerase. The molecule attaches to a DNA strand and "walks" through it generating a hashing/correcting bubble. At the end of the process it has calculated the DNA "hash" and compares it to its expected value; if the check fails, cellular death is triggered. At any one moment every DNA molecule in the body could be examined by up to a dozen such correctases.
A random mutation can then occur in one of three places:
- B mutation. This is the most likely, since the C (CHECK) part will probably be no more than 5% of the total DNA. Helicases and polymerases occupy around 1% of human DNA, and this engineered DNA is very likely to be much more compact. In percent, I think a 1:20 relation is a good ballpark figure. Anyway, a B mutation will be caught by the intact C molecule and either repaired or, if not possible, killed.
- H mutation. Enormously unlikely due to its small size, it will nonetheless happen than the H sequence mutates. When it happens, lots of corrupted C molecules are generated that will routinely misinterpret cellular DNA as corrupted itself, thereby behaving like a fast-acting cellular poison. The mutated cell will be the first to die, and will likely bring down several hundred of the nearby cells before the mistaken correctase is finally degraded. Something remotely similar happens, on a much larger scale, with some kinds of poison (e.g. that of some snakes or that of the brown recluse spider; if you google that, be aware that the images may be quite disturbing). However, the net result is that this kind of mutation can't be inherited.
- C (CHECK) mutation. This is more promising. There are several kinds of mutations which can affect the correctase molecule:
- mutations that make it believe any DNA is always corrupt. Same as the H mutation case.
- mutations that stop it working altogether (e.g. it can no longer bind to the DNA strand).
- mutations that stop it from being lethal (either by stopping it from detecting changes, stopping it from initiating cellular death) but keep it working as a correcting enzyme.
But the main problem here, which would make it impossible as a natural occurrence, is that since every cell is routinely drenched in the correctase produced by itself and its immediate neighbours, any lessening in any one cell's correctase lethality would avail nothing. The cell would still be killed by its neighbours.
To survive, all cells must be mutated (or synthesized) in the same way, all at once or after being kept separate. Or you need to have a single cell with no neighbours. Even the zygote cell is not "alone", it is connected to the mother organism.
"Life will find a way"
This is @MikeNichols' conclusion, and from the above scenario I would conclude that he's wrong... except he's not. He would be (of that I'm quite certain) if the organism existed alone, in vacuo. But no organism ever exists in an ecological vacuum (the closest approximation I'm aware of are Leo Frankowski's Mitchegai, an eptalogy euthanised in 2004).
And the correctase mechanism is expensive - it needs specialized machinery that has an operating as well as a replication cost. A sizeable organism would have no trouble in keeping up the whole show, but a micro-organism would be hard pressed to do the same. So our alien engineers may have stopped evolution in higher organisms, but they can't reasonably stop evolution in microbes. And as far as we know, without a (healthy) microscopic biota, life is not possible.
So we have a life pyramid where the top 10-15% is immune from evolution, and the lower organisms are free to evolve. While the middle layers may still be controllable by the 15% nobility, I suspect that a good 50% of the total planetary biomass would be logistically unreachable.
And let's not forget that this setup is the exact opposite of biodiversity. Sooner or later some pathogen will evolve that finds a suitable pabulum in those perfectly engineered, static, possibly unageing higher organisms, and will kill them all. Won't they develop genetic immunity? Well, any other imperfect organism very likely would. But the Creators made sure this couldn't happen...