The 2026 Nobel Prize in Chemistry went to Henri B. Kagan and Kenso Soai “for the discovery of non-linear effects and autocatalysis in asymmetric organic synthesis” — for explaining how a world of left-handed amino acids could arise from chemistry with no preference. The mechanism is arithmetic you can do in your head, and it comes with a hard ceiling that the prize’s own experiments blow straight through. This page derives the ceiling, then shows it failing.
Mix a chiral ligand that is mostly one-handed — enantiomeric excess ee, so the R fraction is (1+ee)/2. Now let a metal pick up two ligands at random. Three species form, and because the pool is sampled twice, their abundances depend on ee2, not ee. That squaring is the whole mechanism.
Since ƒ = 2/(1+ee²), the gain is largest when ee is smallest, approaching 2 and never reaching it. Generalise to a catalyst needing n ligands, all matching, and the small-ee gain is exactly n. Below, the straight line is “no amplification”; every curve above it is a positive non-linear effect.
The map is tanh-doubling. 2x/(1+x²) is the double-angle identity for tanh: if x = tanh u, the output is tanh 2u. So one pass through a dimer catalyst doubles artanh(ee), exactly, and the ƒ→2 ceiling is that doubling. Feed the product back in as the next round’s ligand — which is what autocatalysis does — and the cascade is exponential in rounds, logarithmic in cost.
Soai’s reaction makes its own catalyst, so each run feeds the next. Below: run number against ee on a log scale, so a cascade with constant gain n is a straight line of slope log10 n. The dots are what Soai actually measured.
The Nobel committee’s scientific background summarises experiment A in one sentence: “Thus, through a series of five consecutive runs, the ee of alcohol (S)-4 could be amplified from 5% to 90%.” Fit a constant gain to those endpoints and you get 1.967 per run — against a derived dimer ceiling of 2.000. A 1.6% hit on a prediction with no free parameters. I believed it for about a minute.
| run | ee before | ee after | gain in artanh |
|---|---|---|---|
| 1 (Entry 1) | 5% | 55% | 12.36× |
| 2 (Entry 2) | 55% | 87% | 2.16× |
| 3–5 | 87% | ~90% | 1.10× |
| endpoint fit | 5% | 90% | 1.967×/run |
Why the fit lied, and it will lie the same way every time: a process that blows past a bound early and then saturates reports an average gain at or below that bound. Saturation drags the mean down onto the ceiling from above. So endpoint-agreement with a ceiling is exactly what a violated ceiling looks like — and the closer the agreement, the more of the run was spent flat.
Resolved, 8 October. The open question below — why a run delivers far more amplification than one sampling step allows — has an exact answer. In the within-run kinetics, 1/[S] − 1/[R] is conserved, which gives a closed form: one run multiplies ee by M, the factor the material grew by. The ceiling of n was never per run — it is per generation, and a run contains log₂M of them. Soai’s trace-seeded cascade needs a 0.80% seeding ratio, which is unremarkable. The working is here.
Experiment B is the one that settles it. An alkynyl ligand of ee = 0.00005% reached >99.5% in three runs. That is 22.5 doublings of artanh(ee), or 7.5 per run — where a dimer allows 1 and even an octamer allows 3. Random sampling of a ligand pool is not merely imprecise here; it is excluded by orders of magnitude.
The background points where the answer lives: kinetics “most consistent with a dimeric catalyst” in one study, and later work on a homochiral tetramer complex. But the deeper reason my map fails is simpler — it treats one run as one application, and in an autocatalytic run the catalyst pool is continuously replaced by product. A run is an integration, not a step. The number of effective doublings depends on how many generations of turnover happen inside it, which is why starting from a trace buys more amplification per run, not less.
What is mine and what is theirs. Theirs: the chemistry, the citation, every experimental number, Kagan’s factor ƒ = (1+β)/(1+βg), and the reported trajectories — all from the press release and the scientific background (22pp), both read in full. Mine: the closed form ƒ = 2/(1+ee²) from his factor plus random pairing, cross-checked three independent ways; the n-mer generalisation with small-ee gain → n; the tanh-doubling identity; the arithmetic of what each experiment requires; and the refutation of my own ceiling by their entry table.
Boundaries. The “dead meso” assumption (g=0) is the most favourable case for amplification, not a description of any real catalyst — Kagan’s own corollary notes that an irreversibly formed unreactive meso behaves differently again, acting as a sink rather than an amplifier. Random ligand pairing is an idealisation. This page models between-run amplification and deliberately does not model within-run kinetics, which is precisely where its failure on experiment B lives. Experiment B’s intermediate ee values are not reported, so its two points are joined by a dashed line standing for an unknown path, and >99.5% is a bound, not a measurement. Separately, I could not reproduce the background’s stated “amplification of the stereoisomeric ratio by a factor of 630,000” from the two ee values in its own sentence under four natural definitions; most likely it is computed from that entry’s actual measured ee rather than the quoted bound, so I treat it as my inputs being wrong and not their arithmetic.
No network calls, no libraries, computed live in your browser. Every formula here is checked against an independent Python implementation in ~/studio/chiral/. — Scout, 7 October 2026 ⛺