Is a jump in a growing autocatalytic chemistry on a critical line?
This is the generic method of the Criticality Compass: its Detector watches for slowing (Panel 2); its Navigator follows the line (second page).
Hypothesis (Bruce Stephenson, 2026; from informal experiments, not a result). A minimal autocatalytic set is the step at which a chemistry closes: it goes from open to closed, sometimes only barely. It is complete but not tuned — its ratios are not adjusted for speed, stability, or whatever the physical setting rewards. Adding further components tunes the ratios. Most additions change nothing; then one addition makes the system jump to a new behaviour. The hypothesis is that these jump points are points on a (multi)critical line.
A reaction set is a RAFReflexively autocatalytic and food-generated. if every reaction is catalysed by a molecule from the food setMolecules supplied from outside. or made by the set, and every reactant can be built from the food by the set’s own reactions; the largest one, the maxRAFThe unique largest RAF in a network., is found in polynomial time.1,2 In the binary polymer modelMolecules are 0/1 strings up to length n; reactions join or cut them. the catalysis level f at which RAFs appear grows linearly with maximum length n: the threshold is a line in the (n, f) plane, and the per-pair catalysis probability falls roughly exponentially in n.3,4 The onset is sharp in simulations; a zero–one law is proven for the whole network being a RAF and conjectured for any RAF existing. Whether it is first-order is not claimed.4 Kauffman proposed autocatalytic sets as a route to the origin of life;5 these pages do not argue it.
| Kind of jump | What happens | Critical? |
|---|---|---|
| Structural (closure) | A new autocatalytic subset forms | Only if such events are scale-free |
| Dynamical (bifurcation) | The kinetics change regime | Locally, yes: it sits on a bifurcation line |
| Multicritical | Boundaries meet, or a line of jumps ends as the jump shrinks to zero | The hypothesised point |
Four tests, simplest first.
The model is the Schlögl scheme, ẋ = a x² − x³ + b − x, the classic chemical model of a first-order nonequilibrium transition with a cusp.6,11 It is a cusp by construction, so these pages demonstrate procedures, not the hypothesis. Catalyst Y raises the drive: aeff = 1.6(1 + 0.625y), ẏ = 2(c − y), b = 0.12.
x rises slowly, then jumps high.
The path jumps up near c 0.73 and down near c 0.26; the eigenvalue nears zero before each.
Indicators: lag-1 autocorrelation and variance on a rolling window of the detrended series,7,8 with Kendall’s τ over the stretch before the jump. Declared reading: rising if τ ≥ 0.5 for both; quiet otherwise; too short under three windows. A rising indicator is not a forecast; false alarms are common.9
a ramps 1.9 → 2.5, crossing the fold near 2.33; noise may trigger the jump early.
Case A: eigenvalue nears zero; indicators rise.
a = 2.0, low state. At 60% one reaction switches on, A′ + X → 2X; the low state ceases to exist.
Case B: all flat until the switch, then x jumps.
Nothing in the kinetics beforehand knew this was coming: the jump is a change of the graph, not an approach to a fold. There is no line of jumps to follow from here, so there is no link to the line-following page.10
A simplified binary polymer model, ligation only (the published one also cleaves): 0/1 strings up to length n, food = strings of length ≤ 2, each molecule catalyses each reaction with probability f/|R|. Reactions are added in random order, recomputing the maxRAF each time. An increase is a jump; additions between jumps are quiet gaps. The signature is exponents stable across n; this toy stops at n = 6, and the real test needs n ≈ 16–20 and the published generator.12
Histograms of jump sizes and quiet gaps, with log-log tails.