Autocatalytic Set
A reaction that catalyzes its own formation, the barest test of a loop that reproduces itself, and of what it takes to keep the loop closed.
Layer 1 of 2 · the accepted science
The classical picture
A reaction is autocatalytic when one of its products speeds up the reaction that makes it. The simplest case is A + X → 2X: one molecule of X meets a molecule of raw material A and leaves as two molecules of X. Chemists write the rate of such a reaction with the law of mass action: the rate is proportional to the product of the concentrations that have to meet, here k [A] [X].
Alfred Lotka chained two of these steps together in a 1920 paper on chemical oscillations. The rate equations it produces have the same form as the ones later known as the Lotka-Volterra predator-prey equations:
The raw material A is held constant by a steady supply. The panel below runs the same kind of chain one step longer, with raw material F (empty, fed space) and a product that returns to the pool when it breaks down:
Why these terms
- Products of concentrations. A reaction that needs two molecules to meet runs at a rate proportional to how many of each there are. That is the whole content of mass action, and it is why every step here is written as a product.
- Each step is proportional to its own product. Because B is made from A only when a B is there to help, the rate of making B rises with B. With the raw material held roughly constant, an isolated autocatalytic step of this kind starts out growing exponentially. In the full network, on a lattice, it does not stay that way: A is used up, B is converted onward, and sites fill. The same fact has a consequence that matters for the whole panel: a species at zero cannot make itself. Every term that produces it is multiplied by zero.
- A constant supply. Lotka held A fixed, and the panel feeds empty space at a set rate. This is not a convenience. At chemical equilibrium there is no sustained net flow around a reaction cycle, so a loop that keeps turning needs a steady flow of matter or energy through it, which is the subject of the thermodynamics of open systems (Ilya Prigogine's dissipative structures).
- A decay step. C → F returns material to the supply, which is what turns a one-way chain into a cycle that can run indefinitely.
Where the idea goes from here
Lotka's chain is the smallest case of a much larger theory. Manfred Eigen and Peter Schuster's hypercycle, developed through the 1970s, links self-replicators so that each catalyzes the next, and Stuart Kauffman's autocatalytic sets (1986) ask when a large random network of reactions contains a subset that catalyzes its own formation from a food supply. The weak C-helps-make-A link in this panel is the kind of cross-catalysis those theories are about.
Where the classical equations stop
The rate equations assume a well-mixed pot and smooth concentrations. The panel is a lattice: a molecule can only react with its neighbours, counts are whole numbers, and a species that reaches zero is gone for good, as the mass-action terms already say it must be. On a lattice a reaction can also stall for a purely local reason: an A with no B anywhere near it never converts, however much B there is elsewhere.
What you are looking at
A lattice seen from above. Each square is one site that holds one molecule or nothing: green is A, gold is B, red is C, and black is empty space. The feed slider sets how much of the empty space carries raw material at each step, loop turnover sets how readily A becomes B and B becomes C, and Cut the feed sets the feed to zero until you press Reset.
The Reaction Loop
Duncan(√5). No sovereignty verdict is
claimed, that is gated.
Layer 2 of 2 · the framework's reading
The wrapped reading
Everything below is written for readers working through Principia Attractum. If you came for the chemistry, you have already had it.
This is the deepest question in the gallery, made runnable. Species on the lattice react so that a product helps make more of itself. Material moves around a short loop: fed empty space becomes A, an A becomes B, a B becomes C, and a C breaks down back into empty space. Each step is helped by more of the species it makes, and C also weakly helps make A. When the loop closes, a self-sustaining reaction network lights up across the grid. The framework's word for this is autocatalysis, and its central caution rides on top of it: a loop that runs is not the same as a loop that pays its own way.
Real molecules run on a feed: a supplied flow of raw substrate and energy. A bare lattice has none, so this panel authors one, visibly, the feed slider on the panel above. That authored gate is shown on the panel itself, never buried, so an imported gate is never mistaken for an emergent one.
Which is exactly what makes the honest test possible: cut the feed and watch.
A loop that kept closing without feed would have passed the one test of independence from
supply that this panel can run, though passing it would still not show that the loop produces
everything else it depends on, such as the clock and the rules. This one does not pass. It breaks: within about eighty
steps of the cut the last C is gone, and because only a C can help make a C, the loop can never
close again. Some A and B cells stay on the board as a remnant, and the status line reports the
loop as ruptured. That break is T0-Φ (throughput-conditionality) made visible,
not a sovereignty verdict, which stays gated.
The Shape of the Basin
At the default settings (feed 0.90, loop turnover 0.14) every run that lights settles into one observed population regime: about 2,880 A cells, 825 B cells and 3,480 C cells out of 8,100, with runs from very different starts landing within about one percent of each other. That is a statement about the counts. Many different arrangements of cells on the board give the same counts, so it does not show that the cell-by-cell dynamics have only one attractor.
This basin is a matter of odds, not a clean line. The rules are probabilistic, so the same starting state can light the loop in one run and fail in the next. The counts below ("7 of 12") are measured rates of establishing or recovering, not a list of which states are inside the basin and which are outside.
A seed has to be big enough to hold all of the loop. The panel starts from a patch of mixed species in the centre. Patches 7 cells across or larger, up to 89 across (nearly the whole board), lit 12 runs out of 12. A patch 5 across lit 7 of 12, and a patch 3 across lit 1 of 12. The small patches that failed did so within the first six steps, because they held no C, or lost their last C at once.
Which species are missing decides the rest. A seed with no C never lit, 0 of 12, and the board filled with B. A seed with no B never lit either, 0 of 12, and the board filled with A. A seed with no A lit 12 of 12, because the weak C-helps-make-A link remade A from the feed. That asymmetry is written into the rules: B and C can each only be made next to one that already exists, so either one at zero is permanent, while A can be rebuilt. What the runs add is which parts of the loop the rest of the network can regenerate and which it cannot.
Against the push, the wall is measured in time. The Cut the feed button stops the feed until you press Reset, so it is not a bounded push on its own. A bounded version cuts the feed for a set number of steps and then restores it to 0.90, applied here to twelve settled worlds at each length:
| feed cut for | loop recovered |
|---|---|
| 10, 20 or 30 steps | 36 of 36, to the same levels |
| 40 steps | 10 of 12 |
| 50 steps | 4 of 12 |
| 60 steps | 2 of 12 |
| 80 steps | 1 of 12 |
Two different edges show up here, and they should not be confused. The first is authored: C at zero is permanent because no rule makes C without C, so a run that loses its last C cannot recover. The second is what the runs measure: how long the feed can be withheld before a run is likely to reach that edge. A settled loop survived 30 steps without feed every time, and the chance of recovery then fell quickly, to 10 in 12 at 40 steps and 4 in 12 at 50. Every failure lost its last C between 46 and 73 steps after the cut, and once the feed came back the loop could not restart: the board filled with a solid sheet of B, 8,003 to 8,100 cells. Left cut for good, all twelve worlds lost C within 80 steps and kept a remnant of 445 to 636 A cells and 4 to 133 B cells that can never close the loop again. The status line reports every one of these cases as ruptured.
The Morph of the Basin
The feed is the dial that moves the regime. Changing it changes the rules the loop runs under, and so the attractor and its basin with them. Twelve worlds were run for 1,500 steps at each setting:
| feed | loop held | settled levels (A / B / C) |
|---|---|---|
| 1.00 | 12 of 12 | 2,893 / 829 / 3,543 |
| 0.90 | 12 of 12 | 2,880 / 826 / 3,479 |
| 0.70 | 12 of 12 | 2,838 / 812 / 3,312 |
| 0.50 | 12 of 12 | 2,771 / 792 / 3,035 |
| 0.30 | 12 of 12 | 2,578 / 757 / 2,524 |
| 0.20 | 12 of 12 | 2,347 / 726 / 2,047 |
| 0.10 | 11 of 12 | 1,695 / 697 / 1,140 |
| 0.05 | 0 of 12 | C lost between step 24 and 435 |
The home sinks first, and mostly in C. Going from full feed to 0.10 cuts C to about a third and A to under two thirds, while B moves least. C is the species that returns material to the feed, and losing it is what broke the loop in the failures below.
Then the regime breaks. At 0.10 one world in twelve lost its C, and at 0.05 all twelve did, mostly within the first 130 steps while the loop was still lighting. Eleven of those twelve left a scattering of 102 to 929 B cells and nothing else; the twelfth lost B as well and filled the board with A.
Loop turnover reshapes the home without breaking it. Across the whole slider, 0.04 to 0.40, all 96 worlds held. Faster turnover moves material along the loop and into C: at 0.04 the home is about 2,800 A, 1,970 B and 2,600 C, and at 0.40 it is about 1,650 A, 720 B and 4,460 C. Within the range the slider offers, this dial changes what the basin holds and not whether it holds.
What the morph can and cannot say here. The feed dial is the one that can break this basin, and the feed is authored: empty space is resupplied at a rate written into the panel. No cost of keeping the loop going is represented in the model at all. The basin's shape tracks that supply directly, which is the throughput-conditionality the disclosure card below records under condition 4. A loop that could keep its shape as the supply thinned would be a different kind of structure, and this panel does not model one.
Tier-0-A §4 · T0-M). An authored design claim, not a measurement.
SOV §7.2), what this
panel supplies, not how it scored- exercised 1 · Recursion lock. The loop A→B→C→A catalyzes its own formation; it either stays closed or it does not, and cutting the feed lets you watch it break: species C runs out and cannot be remade, and the cells left on the board can never close the loop again.
- exercised 2 · Internal recurcline persistence. The population of the loop either sustains itself against decay or collapses; you can watch whether it holds. Caveat: The update runs synchronously at u=1, so part of the temporal order is supplied by the clock.
- out of scope 3 · Boundary retention. No boundary is modeled. Cell states are chemical species, not a membrane, and the lattice wraps.
- supplied 4 · Maintenance-bearing continuation. The loop only forms where there is feed, supplied raw substrate. Cutting the feed breaks it within about eighty steps (this is T0-Φ throughput-conditionality made visible), so persistence is bought with an authored feed, not paid by the structure.
No sovereignty verdict is claimed here or anywhere in this gallery. The Sovereign column is empty, and that emptiness is the honest reading: no panel here has been shown to produce its own order.