Things you can run
The arguments elsewhere on this site are easier to feel than to read. This is where you can move the rule and watch the behaviour change.
Relational coherence
An action is already underway, and it consumes clearance at a rate set by a relationship you cannot see directly. You have one correction available. It protects that clearance by spending a second margin — and where the safe correction lies depends on the relationship you cannot see.
What the research asks is what understanding that relationship is actually worth. So you can change what the observer is allowed to know, and watch the set of safe starting states move.
Every figure below is computed by backward induction over the game tree, not illustrated. It is an independent implementation of the problem class in 10.5281/zenodo.23372032, not a reproduction of that paper’s saved experiment — and where it disagrees with the published table it says so.
Which corrections are safe
Two constraints. Correcting the first spends the second. Where the safe corrections lie depends on the hidden relationship coefficient — so each value of it has its own window, and they do not sit on top of each other.
At these margins there is a correction that is safe whatever the relationship turns out to be. Information is worth nothing in this state, and a system with room to spare does not need to understand itself.
How small a margin you can start from
The smallest starting margin from which the intended action can be carried through the whole horizon without either constraint being violated — computed by backward induction over the game tree, once for each thing the observer is allowed to know.
| The observer knows | Relationship fixed | Relationship may change |
|---|---|---|
| NothingThe coefficient is never measured. The policy runs open-loop. | 12 | 12 |
| NoisyEach reading is within one level of the truth, adversarially. | 12 | 12 |
| ExactThe coefficient is read exactly, at the end of every transition. | 10 | 10 |
Exact knowledge of the relationship expands the viable set: 12 down to 10. This is the direction the paper's second proposition requires, and it is the one result here I would defend.
A noisy reading is worth exactly nothing here — 12, the same as measuring nothing at all. An error of one level is enough to leave the windows unseparated, which is the paper's own finding and the least intuitive thing on this page.
This model disagrees with the paper here. The published table reports that an adversarially changing relationship removes the benefit of exact observation entirely; in this implementation it survives — 10 against 12. Knowing θ now still rules out the levels it cannot reach by the next step, so the information is degraded rather than destroyed.
304 INFORMATION STATES EVALUATEDThe cost is shown because it is part of the finding. Exact belief-state viability grows combinatorially, and that growth is the sixth counterexample — a guarantee you cannot compute in time is not a guarantee.
The model, the two constraints, and the reason the second one has to be there are documented in the functional and where knowing stops helping.
Amplification and restraint
One quantity, growing in proportion to itself. One stabiliser that engages once a threshold is crossed. Some noise for everything the model leaves out.
That is the smallest system with both forces in it, and it is enough to produce runaway growth, collapse, oscillation and stability — depending only on the relationship between the two numbers.
Growth and restraint are in proportion. The system settles at a level it can hold — which is the interesting case, because nothing dramatic happens and it is easy to mistake for luck.
Try this: set stabilisation to zero and watch what any amount of amplification does. Then raise the threshold — the system survives longer and fails harder. Then set stabilisation high and amplification low, and watch a system get destroyed by its own safety mechanism — the restraint costs more than the growth was worth.
This is a toy, and it is not a claim about any real system. It is a model small enough to be understood completely, which is the only kind worth reasoning from. The related thinking is in Thresholds and feedback.