Math Fell Fast. Life Falls Last

On the evening Anthropic released a complete computer-checked formalization of Fermat's Last Theorem, I was at dinner with one of my mentors, Hiroaki Kitano, and two colleagues, one of them a senior machine learning leader. The conversation went where most conversations in this field now go, to how quickly AI will take over whole disciplines, and the mood at the table, as I heard it, was that mathematics, of all things, was close to done, and that biology, my field, would not be far behind.

I left with a question rather than a conclusion. The order in which fields yield to AI is not a ranking by difficulty. It follows something more specific.

What decides the order

The sciences have long been divided into those that discover laws and those that reconstruct histories. Jacob put the biological version in 1977: evolution tinkers rather than engineers, so organisms are historical structures. Gould called it contingency. Adler has recently formalized the split as two epistemic regimes, one in which fixed rules govern each episode and one in which each step rewrites the conditions for the next, and argued that in the second, predicting the future cannot in general cost less than simulating the history that produced it. Wolfram's computational irreducibility is the same idea stated for computation generally. None of that is new, and this essay does not claim it.

The proposal here is narrower: treat a problem's position on that continuum, from answers fixed by reusable constraint to answers fixed by a particular trajectory, as a predictor of how fast AI will master it. This is not a claim that historical outcomes are incompressible; trajectories often have regularities, attractors, and sufficient statistics. It is a claim about a gradient. The more the information needed to determine an outcome depends on its own particular path, the less room there is for a compact, general shortcut, and the slower AI's progress.

Two further factors complete the hypothesis: AI tractability rises with the strength of reusable constraints, with the availability of exact, cheap verification, and with how much of the relevant history can be recovered, whether from a redundant record or because the system's present state carries that history within it. Mathematics is the limiting case. Its truths follow from axioms and depend on nothing that occurred in the physical world, so constraint is maximal; a formal proof can be checked mechanically, so verification is exact; and no historical trajectory needs to be recovered at all.

The verification objection

A skeptic in machine learning would stop me here, and rightly. Perhaps mathematics is falling not because it lacks history but because it has an oracle. Proof assistants give a near-perfect reward signal: propose a step, the checker accepts or rejects, iterate. That day's result is a case in point. The Fermat formalization, thirteen million lines of Lean produced largely autonomously in eleven days, formalized Wiles's existing proof, building on years of upstream work by Kevin Buzzard's group and the Mathlib community; it did not settle an open problem. Its engine was the checker.

The objection is correct, and complementary rather than competing. Verification explains why search in mathematics is so efficient. Mathematics has the unusual combination of strong reusable constraint and an exact internal verifier: a candidate proof can be checked against the same formal system that defines the problem. Historical biological claims rarely offer anything comparable. Their verification is usually partial, experimental, expensive, and itself dependent on incomplete observations of the trajectory. Constraint and verification are therefore distinct factors, even though mathematics happens to sit at the favorable extreme of both.

The ladder inside biology

Biology runs the gradient in the other direction, and it helps to read its rungs as constraint, then context, then history.

At the base is the prediction of native structure from sequence for well-folded proteins. Anfinsen's thermodynamic hypothesis holds for this class: the native state is largely the minimum-free-energy conformation determined by the sequence. It was the first major biological rung to fall, to AlphaFold in 2020, and it was favorable on all three factors: strong physical constraint, a sharply defined target that experiment could verify, and an unusually rich recoverable history: millions of natural sequence variants filtered by evolution under related structural constraints. AlphaFold did not solve folding kinetics, pathways, alternative states, or dynamics. It solved the rung where the three factors align.

One step up, the structures of complexes and their binding. Mostly constraint, more context, falling now.

Then context. Intrinsically disordered proteins do not adopt a single native structure; they exist as ensembles that shift with cellular environment and binding partners. Static structure predictors handle them poorly. But this is context dependence, not history dependence: the ensemble is in principle fixed by present sequence and present environment, with no need to know how the protein got there.

History enters one rung higher, with cell state. Two cells with identical genomes can occupy different stable states because of the regulatory trajectories they have taken. Their genome does not determine their present state; part of their history has been written into the state itself. That is a subtlety the third factor has to carry. History need not be observed from the beginning if the present system holds a sufficient record of it, which is why cell state can often be inferred from present molecular measurements even though it cannot be derived from the genome. Above that, development, the mapping from genome to organism, where each step conditions the next. Then whole-organism pharmacology, where a compound's behavior depends on metabolism, toxicity, and individual variation, and where target structure is seldom the bottleneck. Then disease as it unfolds in a single person: weak reusable constraint at the level that matters, expensive and noisy verification, and a poorly observed individual history, unfavorable on all three factors at once.

At the top sit ecology and evolution. They are not without laws; population genetics is mathematics, and ecology has regularities. What distinguishes them is that the realized outcome depends heavily on contingent initial conditions and accumulated history. The share of the prediction that is contingent, not the presence of laws, sets the rung.

The AlphaFold objection

The strongest counterexample appears to cut the other way. AlphaFold succeeds partly by reading evolutionary history directly, through the co-variation of related sequences across species. Does this not show AI can master a history-dependent problem? It shows where history is recoverable. Folding sits on the rung where the record is redundant enough to learn from. From inside a model, that redundancy is easy to mistake for a universal rule, when it is in fact a richly recorded history. Higher on the ladder the record thins. Development, disease, and individual physiology are sparse, singular, and entangled with context. The method that works where history is recoverable fails where it is not; recoverability is what separates the two.

The wormhole view

The machine learning leader at that dinner, a person I admire for his wisdom as much as his technical judgment, read an early version of this essay and told me, kindly, that I had rendered his view too crudely. He had not been saying that mathematics was finished and biology was next. His point was sharper. Algorithms find wormholes, shortcuts through challenges that looked impassable from a distance. Symbolic manipulation was thought impossible for autoregressive models, and the capability arrived anyway, verifier or not; the checker made search at the frontier efficient, but it did not create the ability. The primitives are more universal than most of us grasp. What looks intractable in biology may fall to the same primitives, because nothing in the architecture rules it out.

I think he is right about the learner, and that this essay is a claim about the problem, which is why the two views fit together rather than collide. Universality says what a system could compute given the right inputs. The third factor asks whether the inputs survive. A universal approximator still needs samples of the function it is learning; folding supplied millions, an individual's disease supplies one. So far the wormholes have opened where constraint, verification, and recoverable history were favorable and experts underestimated the tractability anyway, which is the surprise this essay began with. The deepest of those cases points the way. Go did not wait for a record to exist. Self-play manufactured one. The analogue in biology is not a smarter model but AI as experimenter: closed-loop laboratories, controlled perturbations at scale, cell atlases, digital twins, generating the redundant history that nature never wrote down. Recoverability is not fixed. AI can raise it. That is where I expect his wormholes to open on the upper rungs, and it is the part of the ladder I most want to watch.

What the ordering predicts

If the ordering holds, it is testable. The next parts of biology to yield should disproportionately be tasks with strong physical constraints, experimentally verifiable outputs, and abundant repeated observations: molecular interactions, binding, local molecular properties, and other chemistry-like components of the cell. The parts that stand longest should be those least favorable: development, whole-organism response, the course of disease in an individual, the structure of ecosystems. And if the wormhole view is right, the fastest movement on the upper rungs will come wherever AI can manufacture its own record through experiment.

It also changes what "solved" means along the ladder. At the base, to solve a problem is to find its rule. At the top there may be no compact rule to find, and the closest available thing to understanding is to reconstruct the history in enough detail to replay it: simulation rather than derivation. That is a gradient, not a wall. Progress will continue, but the slope steepens, and near the top the task changes in kind.

Systems biology, where I work, studies that upper end: networks, robustness, behavior that emerges from the whole and is present in no part. It is the portion of biology most fully built from history, and the portion I would least expect to yield to the methods that predicted a protein's structure.

Back to the table

The wormhole view is right about the machine, and I have come to think the useful question is where the wormholes open. Mathematics yielded fast not because it was the hardest field and the machines had at last become strong enough, but because it is the field in which nothing ever happened and everything can be checked, and a system that lives by finding rules and testing them finds them most easily where there is nothing else. Life is the opposite case, the accumulated residue of every accident that survived. AI will climb toward it, and the wormholes there will most likely be dug by AI making the record that nature did not leave. Wherever that record cannot be made, the climb will be slow. The summit was never where the surprise placed it. Mathematics was the base camp.

Key takeaways

• AI advances by tractability, not difficulty. Three factors set the pace: reusable constraint, cheap exact verification, and how much of the relevant history can be recovered.
• Mathematics is the limiting case. Constraint is maximal, verification is exact, and no history needs recovering. Protein structure prediction fell first in biology because it is favorable on all three. An individual's disease trajectory is unfavorable on all three and stands.
• The rule-versus-history distinction is old; the ordering is the claim. A problem's position on the continuum predicts when AI will master it, which makes the conjecture testable. Recoverability is the one factor AI can move, by generating history through experiment.

@misc{palaniappan2026mathfellfast,
  author       = {Palaniappan, Sucheendra Kumar},
  title        = {Math Fell Fast. Life Falls Last: How Constraint, Verification, and History Set the Pace of AI},
  year         = {2026},
  howpublished = {\url{https://suchee.org/musings/math-fell-fast-life-falls-last/}},
  note         = {Essay, suchee.org}
}