Two-Species Loop (A ⇄ B)
The shortest possible mutual catalysis, two species, each helping make the other. The barest closed loop there is, and the cleanest place to watch a loop live or die on its feed.
Layer 1 of 2 · the accepted science
The classical picture
Two molecules are cross-catalytic when each speeds up the reaction that makes the other. Neither can make copies of itself alone, but together they form a closed loop in which more of either leads to more of both. It is the smallest case of Manfred Eigen and Peter Schuster's hypercycle, a ring of replicators each helping make the next, which they developed through the 1970s as a model of how cooperating molecules might have held together before cells existed.
Written as reactions, with F for fed empty space that raw material can occupy:
Why these terms
- Formation is a product, k f E B. By the law of mass action, the rate at which A forms is proportional to how much empty space there is to form in (E), how much of that space is fed (f), and how much catalyst B is present. The catalyst appears in the rate but is not used up, which is what the B on both sides of the reaction means.
- Each species appears only in the other's formation term. That is the cross-catalysis. It also means a species at zero stops the other from forming, so the loss of either one ends both.
- Decay, δ A and δ B. Molecules break down at a constant rate and return their material to the pool. Without decay the board would simply fill up and freeze; with it, the loop has to keep making molecules to stay present.
- A supply, f. A closed chemical system relaxes to equilibrium, where no cycle can keep turning. A loop that persists needs a flow of raw material through it, the subject of the thermodynamics of open systems.
What the equations predict
The empty state A = B = 0 is always a solution. The equations have exactly one other steady state, with equal amounts of both species:
This is a threshold. When the supply-weighted formation rate k f is larger than the decay rate, a small seed of the pair grows and settles at the level above. When it is smaller, the empty state is stable and any seed dies away, however large. Turning the supply down moves the steady state smoothly toward zero until it meets the empty state at k f = δ and disappears. Mathematicians call this meeting a transcritical bifurcation.
Where the classical equations stop
The equations assume a well-mixed pot. On a lattice a molecule can only be made next to its partner, so each species grows in patches rather than everywhere at once, and a thin population can lose all its partners locally even when the average says it should survive. Lattice models of this kind (the best-known is the contact process from probability theory) share the threshold, but typically need a stronger formation rate than the well-mixed equations to cross it. The panel below is such a lattice.
What you are looking at
A lattice seen from above. Each square is one site holding one molecule or nothing: green is A, blue is B, 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 each species helps make the other, 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.
Strip the three-species loop down to its irreducible core. Here there are only two reacting species: A catalyzes the formation of B, and B catalyzes the formation of A. Each needs the other to be made, so the pair catalyzes its own formation around a loop of length two, the smallest autocatalytic set that can exist. The same 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.
With only two species there is nowhere to hide. A mutual loop that kept closing without feed
would have passed the one test of independence from supply that this panel can run, though that
alone would not show it produces everything else it depends on; one that goes dark once the feed
stops was living on what was handed to it. This one goes dark: within about seventy to ninety steps of the cut one species
runs out, the other can no longer be made, and the board empties. That collapse 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 0.12) every run that lights settles into one observed population regime: about 3,600 A cells and 3,600 B cells out of 8,100, held equal by the symmetry of the rule, with runs from every start that lit landing within a few cells of each other. That is a statement about the counts, not a proof 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 are measured rates of lighting or recovering, not a list of which states are inside the basin and which are outside.
It is nearly as wide as a basin can be in starting states. A single A cell alone lit the loop in 24 runs out of 24, because one A is enough to catalyze B into the empty space around it (a single B cell, which happened in four runs, lit three). Patches of A only or B only lit 12 of 12, mixed patches from 3 to 89 cells across lit 12 of 12 at every size, and so did two molecules in a thousand scattered across the whole board. The only start that fell out was a single B cell, once in four runs.
The floor is written into the rules. A species can only be made next to its partner, so if either one reaches zero the other can no longer be made, decays, and the board goes dark. Nothing brings the loop back from there. That edge was authored, not discovered; what the runs measure is how hard the loop can be pushed toward it and still come back.
Against the push, the wall is measured in time. The Cut the feed button stops the feed until you press Reset. 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 |
|---|---|
| 5 to 65 steps (11 lengths) | 132 of 132, to the same levels |
| 70 steps | 8 of 12 |
| 75 steps | 4 of 12 |
| 80 steps | 3 of 12 |
| 90 steps | 0 of 12 |
The wall is the moment the first species runs out. Without feed both populations decay by about a tenth each step, and in every failure one of them reached zero between 66 and 86 steps into the cut. A world that still held even a few of each when the feed returned rebuilt the full home. Left cut for good, all twelve worlds lost a species 66 to 86 steps after the cut, and the status line read dark 81 to 94 steps after it.
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. The right-hand column is what the well-mixed equations in the classical layer predict for the same rule, taking k as 0.12 per neighbour times 20 neighbours and δ as 0.10:
| feed | loop held | settled level of each species | well-mixed prediction |
|---|---|---|---|
| 1.00 | 12 of 12 | 3,645 | 3,881 |
| 0.90 | 12 of 12 | 3,603 | 3,862 |
| 0.50 | 12 of 12 | 3,294 | 3,712 |
| 0.30 | 12 of 12 | 2,891 | 3,488 |
| 0.20 | 12 of 12 | 2,464 | 3,206 |
| 0.10 | 12 of 12 | 1,481 | 2,362 |
| 0.08 | 12 of 12 | 1,092 | 1,941 |
| 0.06 | 12 of 12 | 306 | 1,237 |
| 0.055 | 8 of 12 (still alive at step 1,500) | about 25 | 982 |
| 0.05 | 0 of 12 | lost by step 744 | 675 |
| 0.045 | 0 of 12 | lost by step 327 | 300 |
| 0.04 | 0 of 12 | lost by step 139 | none |
The regime shrinks smoothly to nothing. It thins continuously as the feed falls: from 3,600 of each species to 1,481 at 0.10, to 306 at 0.06, to about 25 at 0.055, and then to zero. That is the shape of the threshold in the classical layer, where the steady state slides down to meet the empty state and vanishes.
The lattice moves the threshold. The well-mixed equations put the edge at a feed of about 0.042. On the lattice the loop failed at 0.045 and 0.05, where the equations still predict hundreds of each species, and the real edge sits between 0.055 and 0.06. At every feed the lattice holds fewer molecules than the equations predict, and the gap widens toward the edge. Because each molecule can only be made next to its partner, local shortages that the well-mixed equations average away are real on the lattice.
Loop turnover barely moves it. Across the whole slider, 0.04 to 0.40, all 108 worlds held at feed 0.90, and the home ranged only from about 3,290 to 3,645 of each species.
What the morph can and cannot say here. The one dial that breaks this basin is the feed, and the feed is authored: empty space is resupplied at a rate written into the panel, and no cost of keeping the loop going is represented in the model. The basin's size tracks that supply all the way down to the threshold, which is the throughput-conditionality the disclosure card below records under condition 4.
SOV §7.2), what this
panel supplies, not how it scored- exercised 1 · Recursion lock. A catalyzes B and B catalyzes A, a mutual loop of length two that makes more of itself; cut the feed and neither can be made, so you can watch the loop fail to close and go dark.
- exercised 2 · Internal recurcline persistence. The pair's population either sustains itself against decay or collapses. 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. A and B form only where there is feed, supplied raw substrate. Cutting the feed starves the loop (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.